Sample records for collisional boltzmann equation

  1. Boltzmann rovibrational collisional coarse-grained model for internal energy excitation and dissociation in hypersonic flows. (United States)

    Munafò, A; Panesi, M; Magin, T E


    A Boltzmann rovibrational collisional coarse-grained model is proposed to reduce a detailed kinetic mechanism database developed at NASA Ames Research Center for internal energy transfer and dissociation in N(2)-N interactions. The coarse-grained model is constructed by lumping the rovibrational energy levels of the N(2) molecule into energy bins. The population of the levels within each bin is assumed to follow a Boltzmann distribution at the local translational temperature. Excitation and dissociation rate coefficients for the energy bins are obtained by averaging the elementary rate coefficients. The energy bins are treated as separate species, thus allowing for non-Boltzmann distributions of their populations. The proposed coarse-grained model is applied to the study of nonequilibrium flows behind normal shock waves and within converging-diverging nozzles. In both cases, the flow is assumed inviscid and steady. Computational results are compared with those obtained by direct solution of the master equation for the rovibrational collisional model and a more conventional multitemperature model. It is found that the proposed coarse-grained model is able to accurately resolve the nonequilibrium dynamics of internal energy excitation and dissociation-recombination processes with only 20 energy bins. Furthermore, the proposed coarse-grained model provides a superior description of the nonequilibrium phenomena occurring in shock heated and nozzle flows when compared with the conventional multitemperature models.

  2. The Boltzmann equation from quantum field theory

    Energy Technology Data Exchange (ETDEWEB)

    Drewes, Marco, E-mail: [Institut fuer Theoretische Teilchenphysik und Kosmologie, RWTH Aachen, 52056 Aachen (Germany); Physik Department T70, Technische Universitaet Muenchen, James Franck Strasse 1, D-85748 Garching (Germany); Mendizabal, Sebastian [Institut fuer Theoretische Physik, Goethe-Universitaet, 60438 Frankfurt am Main (Germany); Weniger, Christoph [Max-Planck-Institut fuer Physik, Foehringer Ring 6, 80805 Muenchen (Germany)


    We show from first principles the emergence of classical Boltzmann equations from relativistic nonequilibrium quantum field theory as described by the Kadanoff-Baym equations. Our method applies to a generic quantum field, coupled to a collection of background fields and sources, in a homogeneous and isotropic spacetime. The analysis is based on analytical solutions to the full Kadanoff-Baym equations, using the WKB approximation. This is in contrast to previous derivations of kinetic equations that rely on similar physical assumptions, but obtain approximate equations of motion from a gradient expansion in momentum space. We show that the system follows a generalized Boltzmann equation whenever the WKB approximation holds. The generalized Boltzmann equation, which includes off-shell transport, is valid far from equilibrium and in a time dependent background, such as the expanding universe.

  3. An introduction to the theory of the Boltzmann equation

    CERN Document Server

    Harris, Stewart


    Boltzmann's equation (or Boltzmann-like equations) appears extensively in such disparate fields as laser scattering, solid-state physics, nuclear transport, and beyond the conventional boundaries of physics and engineering, in the fields of cellular proliferation and automobile traffic flow. This introductory graduate-level course for students of physics and engineering offers detailed presentations of the basic modern theory of Boltzmann's equation, including representative applications using both Boltzmann's equation and the model Boltzmann equations developed within the text. It emphasizes

  4. The Boltzmann Equation in Fluorescent Lamp Theory (United States)

    Lister, Graeme

    Ken Hines was a wonderful mentor in my formative years as a physicist, and a great friend and companion throughout the rest of his life. He introduced me to the Fokker-Planck equation, which formed the basis of my Masters thesis, and showed me how to use it to derive the Boltzmann equation. Many years later, when my research took me into the field of gas discharge lighting, I was to re-discover the Boltzmann equation and apply it to fluorescent lamp modelling. Herein I discuss the important role the electron energy distribution function plays in understanding the physics of fluorescent lamps, and I describe some of the important insights gained from interpreting the Boltzmann equation.

  5. Fractional Diffusion Limit for Collisional Kinetic Equations

    KAUST Repository

    Mellet, Antoine


    This paper is devoted to diffusion limits of linear Boltzmann equations. When the equilibrium distribution function is a Maxwellian distribution, it is well known that for an appropriate time scale, the small mean free path limit gives rise to a diffusion equation. In this paper, we consider situations in which the equilibrium distribution function is a heavy-tailed distribution with infinite variance. We then show that for an appropriate time scale, the small mean free path limit gives rise to a fractional diffusion equation. © 2010 Springer-Verlag.

  6. The Quantum Boltzmann Equation in Semiconductor Physics


    Snoke, D. W.


    The quantum Boltzmann equation, or Fokker-Planck equation, has been used to successfully explain a number of experiments in semiconductor optics in the past two decades. This paper reviews some of the developments of this work, including models of excitons in bulk materials, electron-hole plasmas, and polariton gases.

  7. The Boltzmann equation in the difference formulation

    Energy Technology Data Exchange (ETDEWEB)

    Szoke, Abraham [Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States); Brooks III, Eugene D. [Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)


    First we recall the assumptions that are needed for the validity of the Boltzmann equation and for the validity of the compressible Euler equations. We then present the difference formulation of these equations and make a connection with the time-honored Chapman - Enskog expansion. We discuss the hydrodynamic limit and calculate the thermal conductivity of a monatomic gas, using a simplified approximation for the collision term. Our formulation is more consistent and simpler than the traditional derivation.

  8. Full Boltzmann equations for leptogenesis including scattering

    CERN Document Server

    Hahn-Woernle, F; Wong, Y Y Y


    We study the evolution of a cosmological baryon asymmetry produced via leptogenesis by means of the full classical Boltzmann equations, without the assumption of kinetic equilibrium and including all quantum statistical factors. Beginning with the full mode equations we derive the usual equations of motion for the right-handed neutrino number density and integrated lepton asymmetry, and show explicitly the impact of each assumption on these quantities. For the first time, we investigate also the effects of scattering of the right-handed neutrino with the top quark to leading order in the Yukawa couplings by means of the full Boltzmann equations. We find that in our full Boltzmann treatment the final lepton asymmetry can be suppressed by as much as a factor of 1.5 in the weak wash-out regime (K1), the full Boltzmann treatment and the integrated approach give nearly identical final lepton asymmetries (within 10 % of each other at K>3). Finally, we show that the opposing effects of quantum statistics on decays/i...

  9. The quantum Boltzmann equation in semiconductor physics

    Energy Technology Data Exchange (ETDEWEB)

    Snoke, D.W. [Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, PA, 15260 (United States)


    The quantum Boltzmann equation, or Fokker-Planck equation, has been used to successfully explain a number of experiments in semiconductor optics in the past two decades. This paper reviews some of the developments of this work, including models of excitons in bulk materials, electron-hole plasmas, and polariton gases. (Copyright copyright 2011 WILEY-VCH Verlag GmbH and Co. KGaA, Weinheim)

  10. Generalized Boltzmann equation for neutron stars

    Energy Technology Data Exchange (ETDEWEB)

    Kaniadakis, G. [Politecnico di Torino (Italy). Dipt. di Fisica]|[Istituto Nazionale di Fisica Nucleare, Sezione di Torino (Italy)]|[INFM, Torino (Italy); Lavagno, A. [Politecnico di Torino (Italy). Dipt. di Fisica]|[Istituto Nazionale di Fisica Nucleare, Sezione di Torino (Italy)]|[INFM, Torino (Italy); Quarati, P. [Politecnico di Torino (Italy). Dipt. di Fisica]|[Istituto Nazionale di Fisica Nucleare, Sezione di Cagliari (Italy)]|[INFM, Torino (Italy)


    Baryon pairing and neutron superfluidity are believed to play an important role in the evolution of neutron stars. The pairing interaction provides a partial antisymmetrization of the nucleons in the stars with the evidence that fractional statistics must hold. By using a kinetic approach recently proposed [G. Kaniadakis, A. Lavagno and P. Quarati, Nucl. Phys. B 466 (1996) 527], we derive a non-linear Boltzmann equation which takes into account collective effects introduced by an exclusion-inclusion principle. This equation describes the dynamics of particles ruled by a fractional statistics. In addition, we extend this Boltzmann equation to the relativistic case and discuss the relevance of the quark matter in the star core. (orig.).

  11. Generalized Boltzmann Equation for Neutron Stars (United States)

    Kaniadakis, G.; Lavagno, A.; Quarati, P.


    Baryon pairing and neutron superfluidity are believed to play an important role in the evolution of neutron stars. The pairing interaction provides a partial antisymmetrization of the nucleons in the stars with the evidence that fractional statistics must hold. By using a kinetic approach recently proposed [G. Kaniadakis, A. Lavagno and P. Quarati, Nucl. Phys. B 466 (1996) 527], we derive a non-linear boltzmann equation which takes into account collective effects introduced by an exclusion-inclusion principle. This equation describes the dynamics of particles ruled by a fractional statistics. In addition, we extend this Boltzmann equation to the relativistic case and discuss the relevance of the quark matter in the star core.

  12. On the full Boltzmann equations for leptogenesis

    Energy Technology Data Exchange (ETDEWEB)

    Garayoa, J.; Pastor, S.; Pinto, T.; Rius, N.; Vives, O., E-mail:, E-mail:, E-mail:, E-mail:, E-mail: [Depto. de Física Teórica and IFIC, Universidad de Valencia-CSIC, Edificio de Institutos de Paterna, Apt. 22085, 46071 Valencia (Spain)


    We consider the full Boltzmann equations for standard and soft leptogenesis, instead of the usual integrated Boltzmann equations which assume kinetic equilibrium for all species. Decays and inverse decays may be inefficient for thermalising the heavy-(s)neutrino distribution function, leading to significant deviations from kinetic equilibrium. We analyse the impact of using the full kinetic equations in the case of a previously generated lepton asymmetry, and find that the washout of this initial asymmetry due to the interactions of the right-handed neutrino is larger than when calculated via the integrated equations. We also solve the full Boltzmann equations for soft leptogenesis, where the lepton asymmetry induced by the soft SUSY-breaking terms in sneutrino decays is a purely thermal effect, since at T = 0 the asymmetry in leptons cancels the one in sleptons. In this case, we obtain that in the weak washout regime (K ∼< 1) the final lepton asymmetry can change up to a factor four with respect to previous estimates.

  13. The Einstein-Boltzmann equations revisited (United States)

    Nadkarni-Ghosh, Sharvari; Refregier, Alexandre


    The linear Einstein-Boltzmann (E-B) equations describe the evolution of perturbations in the universe and its numerical solutions play a central role in cosmology. We revisit this system of differential equations and present a detailed investigation of its mathematical properties. For this purpose, we focus on a simplified set of equations aimed at describing the broad features of the matter power spectrum. We first perform an eigenvalue analysis and study the onset of oscillations in the system signalled by the transition from real to complex eigenvalues. We then provide a stability criterion of different numerical schemes for this linear system and estimate the associated step size. We elucidate the stiffness property of the E-B system and show how it can be characterized in terms of the eigenvalues. While the parameters of the system are time dependent making it non-autonomous, we define an adiabatic regime where the parameters vary slowly enough for the system to be quasi-autonomous. We summarize the different regimes of the system for these different criteria as function of wavenumber k and scalefactor a. We also provide a compendium of analytic solutions for all perturbation variables in six limits on the k-a plane and express them explicitly in terms of initial conditions. These results are aimed to help the further development and testing of numerical cosmological Boltzmann solvers.

  14. A Boltzmann Equation for the QCD Plasma

    CERN Document Server

    Blaizot, Jean-Paul; Blaizot, Jean-Paul; Iancu, Edmond


    We present a derivation of a Boltzmann equation for the QCD plasma, starting from the quantum field equations. The derivation is based on a gauge covariant gradient expansion which takes consistently into account all possible dependences on the gauge coupling assumed to be small. We point out a limitation of the gradient expansion arising when the range of the interactions becomes comparable with that of the space-time inhomogeneities of the system. The method is first applied to the case of scalar electrodynamics, and then to the description of long wavelength colour fluctuations in the QCD plasma, where our equation coincides with that recently proposed by Arnold, Son and Yaffe. We discuss interesting cancellations among various collision terms, which occur in the calculation of most transport coefficients, but not in that of the quasiparticle lifetime, or in that of the relaxation time of colour excitations.

  15. Soluble Boltzmann equations for internal state and Maxwell models

    NARCIS (Netherlands)

    Futcher, E.; Hoare, M.R.; Hendriks, E.M.; Ernst, M.H.

    We consider a class of scalar nonlinear Boltzmann equations describing the evolution of a microcanonical ensemble in which sub-systems exchange internal energy ‘randomly’ in binary interactions. In the continuous variable version these models can equally be interpreted as Boltzmann equations for

  16. Boltzmann equation with double-well potentials (United States)

    Chiacchiera, Silvia; Macrı, Tommaso; Trombettoni, Andrea


    We study the dynamics of an interacting classical gas trapped in a double-well potential at finite temperature. Two model potentials are considered: a cubic box with a square barrier in the middle, and a harmonic trap with a Gaussian barrier along one direction. The study is performed using the Boltzmann equation, solved numerically via the test-particle method. We introduce and discuss a simple analytical model that allows one to provide estimates of the relaxation time, which are compared with numerical results. Finally, we use our findings to make numerical and analytical predictions for the case of a fermionic mixture in the normal-fluid phase in a realistic double-well potential relevant for experiments with cold atoms.

  17. Hot electrons in superlattices: quantum transport versus Boltzmann equation

    DEFF Research Database (Denmark)

    Wacker, Andreas; Jauho, Antti-Pekka; Rott, S.


    A self-consistent solution of the transport equation is presented for semiconductor superlattices within different approaches: (i) a full quantum transport model based on nonequilibrium Green functions, (ii) the semiclassical Boltzmann equation for electrons in a miniband, and (iii) Boltzmann...... equation for electrons in Wannier-Stark states. We find good quantitative agreement of the approximations (ii) and (iii) with (i) in their respective ranges of validity. (C) 1999 Elsevier Science B.V. All rights reserved....

  18. Analysis of spectral methods for the homogeneous Boltzmann equation

    KAUST Repository

    Filbet, Francis


    The development of accurate and fast algorithms for the Boltzmann collision integral and their analysis represent a challenging problem in scientific computing and numerical analysis. Recently, several works were devoted to the derivation of spectrally accurate schemes for the Boltzmann equation, but very few of them were concerned with the stability analysis of the method. In particular there was no result of stability except when the method was modified in order to enforce the positivity preservation, which destroys the spectral accuracy. In this paper we propose a new method to study the stability of homogeneous Boltzmann equations perturbed by smoothed balanced operators which do not preserve positivity of the distribution. This method takes advantage of the "spreading" property of the collision, together with estimates on regularity and entropy production. As an application we prove stability and convergence of spectral methods for the Boltzmann equation, when the discretization parameter is large enough (with explicit bound). © 2010 American Mathematical Society.

  19. Metamaterial characterization using Boltzmann's kinetic equation for electrons

    DEFF Research Database (Denmark)

    Novitsky, Andrey; Zhukovsky, Sergei; Novitsky, D.


    Statistical properties of electrons in metals are taken into consideration to describe the microscopic motion of electrons. Assuming degenerate electron gas in metal, we introduce the Boltzmann kinetic equation to supplement Maxwell's equations. The solution of these equations clearly shows the r...

  20. Celebrating Cercignani's conjecture for the Boltzmann equation

    KAUST Repository

    Villani, Cédric


    Cercignani\\'s conjecture assumes a linear inequality between the entropy and entropy production functionals for Boltzmann\\'s nonlinear integral operator in rarefied gas dynamics. Related to the field of logarithmic Sobolev inequalities and spectral gap inequalities, this issue has been at the core of the renewal of the mathematical theory of convergence to thermodynamical equilibrium for rarefied gases over the past decade. In this review paper, we survey the various positive and negative results which were obtained since the conjecture was proposed in the 1980s. © American Institute of Mathematical Sciences.

  1. Two generalizations of the Boltzmann equation


    Biro, T. S.; Kaniadakis, G.


    We connect two different generalizations of Boltzmann's kinetic theory by requiring the same stationary solution. Non-extensive statistics can be produced by either using corresponding collision rates nonlinear in the one-particle densities or equivalently by using nontrivial energy composition rules in the energy conservation constraint. Direct transformation formulas between key functions of the two approaches are given.

  2. Two generalizations of the Boltzmann equation

    Energy Technology Data Exchange (ETDEWEB)

    Biro, T.S. [KFKI Research Institute for Particle and Nuclear Physics, Budapest (Hungary); Kaniadakis, G. [Instituto Nazionale di Fisica della Materia, Politecnico di Torino, Dipt. di Fisica, Torino (Italy)


    We connect two different extensions of Boltzmann's kinetic theory by requiring the same stationary solution. Non-extensive statistics can be produced by either using corresponding collision rates nonlinear in the one-particle densities or equivalently by using nontrivial energy composition rules in the energy conservation constraint part. Direct transformation formulas between key functions of the two approaches are given. (authors)

  3. Using NEKTON to solve systems of discrete Boltzmann equations

    Energy Technology Data Exchange (ETDEWEB)

    Torczynski, J.R.


    A discrete-velocity-gas (DVG) model of the Boltzmann equation that employs four velocity states has been implemented numerically by using the computational fluid dynamics code NEKTON to solve the DVG species-transport (discrete Boltzmann) equations. The model is applicable to rarefied two-dimensional isothermal flow and is used to simulate flow through a channel. As expected, the velocity profile is found to be uniform for large Knudsen numbers (free molecular flow) and parabolic for small Knudsen numbers (near continuum flow). Since there are no conceptual differences between the four-state model and models employing more velocity states to better represent the Boltzmann equation, implementation of models with more velocity states appear to be straightforward.

  4. Using NEKTON to solve systems of discrete Boltzmann equations

    Energy Technology Data Exchange (ETDEWEB)

    Torczynski, J.R.


    A discrete-velocity-gas (DVG) model of the Boltzmann equation that employs four velocity states has been implemented numerically by using the computational fluid dynamics code NEKTON to solve the DVG species-transport (discrete Boltzmann) equations. The model is applicable to rarefied two-dimensional isothermal flow and is used to simulate flow through a channel. As expected, the velocity profile is found to be uniform for large Knudsen numbers (free molecular flow) and parabolic for small Knudsen numbers (near continuum flow). Since there are no conceptual differences between the four-state model and models employing more velocity states to better represent the Boltzmann equation, implementation of models with more velocity states appear to be straightforward.

  5. The relativistic Boltzmann equation on a spherically symmetric gravitational field (United States)

    Takou, Etienne; Ciake Ciake, Fidèle L.


    In this paper, we consider the Cauchy problem for the relativistic Boltzmann equation with near vacuum initial data where the distribution function depends on the time, the position and the impulsion. We consider this equation on a spherically symmetric gravitational field spacetime. The collision kernel considered here is for the hard potentials case. We prove the existence of a unique global (in time) mild solution in a suitable weighted space.

  6. Efficient Asymptotic Preserving Deterministic methods for the Boltzmann Equation (United States)


    a density function f(x, v, t) describing the time evolution of a monoatomic rarefied gas of particles which move with velocity v ∈ IR3 in the position...density, and total energy density, satisfy the compressible Euler equations for a monoatomic gas, therefore the model describes the correct fluid dynamic...RTO-EN-AVT-194 8 - 13 1 THE BOLTZMANN EQUATION 1.5 Other collision operators model for a monoatomic gas leads to a Prandtl number very close to

  7. An efficient numerical method for solving the Boltzmann equation in multidimensions (United States)

    Dimarco, Giacomo; Loubère, Raphaël; Narski, Jacek; Rey, Thomas


    In this paper we deal with the extension of the Fast Kinetic Scheme (FKS) (Dimarco and Loubère, 2013 [26]) originally constructed for solving the BGK equation, to the more challenging case of the Boltzmann equation. The scheme combines a robust and fast method for treating the transport part based on an innovative Lagrangian technique supplemented with conservative fast spectral schemes to treat the collisional operator by means of an operator splitting approach. This approach along with several implementation features related to the parallelization of the algorithm permits to construct an efficient simulation tool which is numerically tested against exact and reference solutions on classical problems arising in rarefied gas dynamic. We present results up to the 3 D × 3 D case for unsteady flows for the Variable Hard Sphere model which may serve as benchmark for future comparisons between different numerical methods for solving the multidimensional Boltzmann equation. For this reason, we also provide for each problem studied details on the computational cost and memory consumption as well as comparisons with the BGK model or the limit model of compressible Euler equations.

  8. Simple and robust solver for the Poisson-Boltzmann equation (United States)

    Baptista, M.; Schmitz, R.; Dünweg, B.


    A variational approach is used to develop a robust numerical procedure for solving the nonlinear Poisson-Boltzmann equation. Following Maggs and Rossetto [Phys. Rev. Lett. 88, 196402 (2002)], we construct an appropriate constrained free energy functional such that its Euler-Lagrange equations are equivalent to the Poisson-Boltzmann equation. This is a formulation that searches for a true minimum in function space, in contrast to previous variational approaches that rather searched for a saddle point. We then develop, implement, and test an algorithm for its numerical minimization, which is quite simple and unconditionally stable. The analytic solution for planar geometry is used for validation. Some results are presented for a charged colloidal sphere surrounded by counterions and optimizations based upon fast Fourier transforms and hierarchical preconditioning are briefly discussed.

  9. Distributional monte carlo methods for the boltzmann equation (United States)

    Schrock, Christopher R.

    Stochastic particle methods (SPMs) for the Boltzmann equation, such as the Direct Simulation Monte Carlo (DSMC) technique, have gained popularity for the prediction of flows in which the assumptions behind the continuum equations of fluid mechanics break down; however, there are still a number of issues that make SPMs computationally challenging for practical use. In traditional SPMs, simulated particles may possess only a single velocity vector, even though they may represent an extremely large collection of actual particles. This limits the method to converge only in law to the Boltzmann solution. This document details the development of new SPMs that allow the velocity of each simulated particle to be distributed. This approach has been termed Distributional Monte Carlo (DMC). A technique is described which applies kernel density estimation to Nanbu's DSMC algorithm. It is then proven that the method converges not just in law, but also in solution for Linfinity(R 3) solutions of the space homogeneous Boltzmann equation. This provides for direct evaluation of the velocity density function. The derivation of a general Distributional Monte Carlo method is given which treats collision interactions between simulated particles as a relaxation problem. The framework is proven to converge in law to the solution of the space homogeneous Boltzmann equation, as well as in solution for Linfinity(R3) solutions. An approach based on the BGK simplification is presented which computes collision outcomes deterministically. Each technique is applied to the well-studied Bobylev-Krook-Wu solution as a numerical test case. Accuracy and variance of the solutions are examined as functions of various simulation parameters. Significantly improved accuracy and reduced variance are observed in the normalized moments for the Distributional Monte Carlo technique employing discrete BGK collision modeling.

  10. On the conditions of validity of the Boltzmann equation and Boltzmann H-theorem (United States)

    Tessarotto, Massimo; Cremaschini, Claudio; Tessarotto, Marco


    In this paper the problem is posed of the formulation of the so-called ab initio approach to the statistical description of the Boltzmann-Sinai N -body classical dynamical system (CDS) formed by identical smooth hard spheres. This amounts to introducing a suitably generalized version of the axioms of Classical Statistical Mechanics. The latter involve a proper definition of the functional setting for the N -body probability density function (PDF), so that it includes also the case of the deterministic N -body PDF. In connection with this issue, a further development concerns the introduction of modified collision boundary conditions which differ from the usual ones adopted in previous literature. Both features are proved to be consistent with the validity of exact H-theorems for the N -body and 1 -body PDFs, respectively. Consequences of the axiomatic approach which concern the conditions of validity of the Boltzmann kinetic equation and the Boltzmann H-theorem are investigated. In particular, the role of the modified boundary conditions is discussed. It is shown that both theorems fail in the case in which the N -body PDF is identified with the deterministic PDF. Finally, the issue of applicability of the Zermelo and Loschmidt paradoxes to the ab initio approach presented here is discussed.

  11. Lattice Boltzmann method for the fractional advection-diffusion equation. (United States)

    Zhou, J G; Haygarth, P M; Withers, P J A; Macleod, C J A; Falloon, P D; Beven, K J; Ockenden, M C; Forber, K J; Hollaway, M J; Evans, R; Collins, A L; Hiscock, K M; Wearing, C; Kahana, R; Villamizar Velez, M L


    Mass transport, such as movement of phosphorus in soils and solutes in rivers, is a natural phenomenon and its study plays an important role in science and engineering. It is found that there are numerous practical diffusion phenomena that do not obey the classical advection-diffusion equation (ADE). Such diffusion is called abnormal or superdiffusion, and it is well described using a fractional advection-diffusion equation (FADE). The FADE finds a wide range of applications in various areas with great potential for studying complex mass transport in real hydrological systems. However, solution to the FADE is difficult, and the existing numerical methods are complicated and inefficient. In this study, a fresh lattice Boltzmann method is developed for solving the fractional advection-diffusion equation (LabFADE). The FADE is transformed into an equation similar to an advection-diffusion equation and solved using the lattice Boltzmann method. The LabFADE has all the advantages of the conventional lattice Boltzmann method and avoids a complex solution procedure, unlike other existing numerical methods. The method has been validated through simulations of several benchmark tests: a point-source diffusion, a boundary-value problem of steady diffusion, and an initial-boundary-value problem of unsteady diffusion with the coexistence of source and sink terms. In addition, by including the effects of the skewness β, the fractional order α, and the single relaxation time τ, the accuracy and convergence of the method have been assessed. The numerical predictions are compared with the analytical solutions, and they indicate that the method is second-order accurate. The method presented will allow the FADE to be more widely applied to complex mass transport problems in science and engineering.

  12. Distributional Monte Carlo Methods for the Boltzmann Equation (United States)


    real valued function over R3 such that ∫ R3 φ (v) Q ( f , f ) dv exists. Cercignani shows [29] ∫ R3 φ (v) Q ( f , f ) dv = 1 4 ∫ R3 ∫ R3 ∫ S + [ f ( v...1957. [26] Cercignani , C. “Existence and Uniqueness in the Large for Boundary Value Problems in Kinetic Theory”. Journal of Mathematical Physics, 8(8...1653–1656, 1967. [27] Cercignani , C. The Boltzmann Equation and Its Applications. Springer-Verlag, 1988. [28] Cercignani , C. Mathematical Methods in

  13. Generalizing the Boltzmann equation in complex phase space (United States)

    Zadehgol, Abed


    In this work, a generalized form of the BGK-Boltzmann equation is proposed, where the velocity, position, and time can be represented by real or complex variables. The real representation leads to the conventional BGK-Boltzmann equation, which can recover the continuity and Navier-Stokes equations. We show that the complex representation yields a different set of equations, and it can also recover the conservation and Navier-Stokes equations, at low Mach numbers, provided that the imaginary component of the macroscopic mass can be neglected. We briefly review the Constant Speed Kinetic Model (CSKM), which was introduced in Zadehgol and Ashrafizaadeh [J. Comp. Phys. 274, 803 (2014), 10.1016/] and Zadehgol [Phys. Rev. E 91, 063311 (2015), 10.1103/PhysRevE.91.063311]. The CSKM is then used as a basis to show that the complex-valued equilibrium distribution function of the present model can be identified with a simple singularity in the complex phase space. The virtual particles, in the present work, are concentrated on virtual "branes" which surround the computational nodes. Employing the Cauchy integral formula, it is shown that certain variations of the "branes," in the complex phase space, do not affect the local kinetic states. This property of the new model, which is referred to as the "apparent jumps" in the present work, is used to construct new models. The theoretical findings have been tested by simulating three benchmark flows. The results of the present simulations are in excellent agreement with the previous results reported by others.

  14. Exact solutions to the Boltzmann equation by mapping the scattering integral into a differential operator

    Energy Technology Data Exchange (ETDEWEB)

    Zabadal, Jorge; Borges, Volnei; Van der Laan, Flavio T., E-mail:, E-mail:, E-mail: [Universidade Federal do Rio Grande do Sul (UFRGS), Porto Alegre, RS (Brazil). Departamento de Engenharia Mecanica. Grupo de Pesquisas Radiologicas; Ribeiro, Vinicius G., E-mail: [Centro Universitario Ritter dos Reis (UNIRITTER), Porto Alegre, RS (Brazil); Santos, Marcio G., E-mail: [Universidade Federal do Rio Grande do Sul (UFRGS), Tramandai, RS (Brazil). Departamento Interdisciplinar do Campus Litoral Norte


    This work presents a new analytical method for solving the Boltzmann equation. In this formulation, a linear differential operator is applied over the Boltzmann model, in order to produce a partial differential equation in which the scattering term is absent. This auxiliary equation is solved via reduction of order. The exact solution obtained is employed to define a precursor for the buildup factor. (author)

  15. Lattice Boltzmann equation method for multiple immiscible continuum fluids (United States)

    Spencer, T. J.; Halliday, I.; Care, C. M.


    This paper generalizes the two-component algorithm of Sec. , extending it, in Sec. , to describe N>2 mutually immiscible fluids in the isothermal continuum regime. Each fluid has an independent interfacial tension. While retaining all its computational advantages, we remove entirely the empiricism associated with contact behavior in our previous multiple immiscible fluid models [M. M. Dupin , Phys. Rev. E 73, 055701(R) (2006)10.1103/PhysRevE.73.055701; Med. Eng. Phys. 28, 13 (2006)10.1016/j.medengphy.2005.04.015] while solidifying the physical foundations. Moreover, the model relies upon a fluid-fluid segregation which is simpler, computationally faster, more free of artifacts (i.e., the interfacial microcurrent), and upon an interface-inducing force distribution which is analytic. The method is completely symmetric between any numbers of immiscible fluids and stable over a wide range of directly input interfacial tension. We present data on the steady-state properties of multiple interface model, which are in good agreement with theory [R. E. Johnson and S. S. Sadhal, Annu. Rev. Fluid Mech. 17, 289 (1985)10.1146/annurev.fl.17.010185.001445], specifically on the shapes of multidrop systems. Section is an analysis of the kinetic and continuum-scale descriptions of the underlying two-component lattice Boltzmann model for immiscible fluids, extendable to more than two immiscible fluids. This extension requires (i) the use of a more local kinetic equation perturbation which is (ii) free from a reliance on measured interfacial curvature. It should be noted that viewed simply as a two-component method, the continuum algorithm is inferior to our previous methods, reported by Lishchuk [Phys. Rev. E 67, 036701 (2003)]10.1103/PhysRevE.76.036701 and Halliday [Phys. Rev. E 76, 026708 (2007)]10.1103/PhysRevE.76.026708. Greater stability and parameter range is achieved in multiple drop simulations by using the forced multi-relaxation-time lattice Boltzmann method developed

  16. On kinetic Boltzmann equations and related hydrodynamic flows with dry viscosity

    Directory of Open Access Journals (Sweden)

    Nikolai N. Bogoliubov (Jr.


    Full Text Available A two-component particle model of Boltzmann-Vlasov type kinetic equations in the form of special nonlinear integro-differential hydrodynamic systems on an infinite-dimensional functional manifold is discussed. We show that such systems are naturally connected with the nonlinear kinetic Boltzmann-Vlasov equations for some one-dimensional particle flows with pointwise interaction potential between particles. A new type of hydrodynamic two-component Benney equations is constructed and their Hamiltonian structure is analyzed.

  17. Diffusion and transport phenomena in a collisional magnetoplasma ...

    Indian Academy of Sciences (India)

    Abstract. Boltzmann-transport equation is analytically solved for two-component mag- netoplasma using Chapman–Enskog analysis to include collisional diffusion transport hav- ing anisotropies in both streaming velocity and temperature components. The modified collisional integrals are analytically solved with flux ...

  18. Diffusion and transport phenomena in a collisional magnetoplasma ...

    Indian Academy of Sciences (India)

    Boltzmann-transport equation is analytically solved for two-component magnetoplasma using Chapman-Enskog analysis to include collisional diffusion transport having anisotropies in both streaming velocity and temperature components. The modified collisional integrals are analytically solved with flux integrals and ...

  19. High-order regularization in lattice-Boltzmann equations (United States)

    Mattila, Keijo K.; Philippi, Paulo C.; Hegele, Luiz A.


    A lattice-Boltzmann equation (LBE) is the discrete counterpart of a continuous kinetic model. It can be derived using a Hermite polynomial expansion for the velocity distribution function. Since LBEs are characterized by discrete, finite representations of the microscopic velocity space, the expansion must be truncated and the appropriate order of truncation depends on the hydrodynamic problem under investigation. Here we consider a particular truncation where the non-equilibrium distribution is expanded on a par with the equilibrium distribution, except that the diffusive parts of high-order non-equilibrium moments are filtered, i.e., only the corresponding advective parts are retained after a given rank. The decomposition of moments into diffusive and advective parts is based directly on analytical relations between Hermite polynomial tensors. The resulting, refined regularization procedure leads to recurrence relations where high-order non-equilibrium moments are expressed in terms of low-order ones. The procedure is appealing in the sense that stability can be enhanced without local variation of transport parameters, like viscosity, or without tuning the simulation parameters based on embedded optimization steps. The improved stability properties are here demonstrated using the perturbed double periodic shear layer flow and the Sod shock tube problem as benchmark cases.

  20. An introduction to the Boltzmann equation and transport processes in gases

    CERN Document Server

    Kremer, Gilberto M; Colton, David


    This book covers classical kinetic theory of gases, presenting basic principles in a self-contained framework and from a more rigorous approach based on the Boltzmann equation. Uses methods in kinetic theory for determining the transport coefficients of gases.

  1. Computational Aeroacoustics Using the Generalized Lattice Boltzmann Equation Project (United States)

    National Aeronautics and Space Administration — The overall objective of the proposed project is to develop a generalized lattice Boltzmann (GLB) approach as a potential computational aeroacoustics (CAA) tool for...

  2. Fractional Boltzmann equation for multiple scattering of resonance radiation in low-temperature plasma

    Energy Technology Data Exchange (ETDEWEB)

    Uchaikin, V V; Sibatov, R T, E-mail:, E-mail: [Ulyanovsk State University, 432000, 42 Leo Tolstoy str., Ulyanovsk (Russian Federation)


    The fractional Boltzmann equation for resonance radiation transport in plasma is proposed. We start with the standard Boltzmann equation; averaging over photon frequencies leads to the appearance of a fractional derivative. This fact is in accordance with the conception of latent variables leading to hereditary and non-local dynamics (in particular, fractional dynamics). The presence of a fractional material derivative in the equation is concordant with heavy tailed distribution of photon path lengths and with spatiotemporal coupling peculiar to the process. We discuss some methods of solving the obtained equation and demonstrate numerical results in some simple cases.

  3. The Initial Boundary Value Problem for the Boltzmann Equation with Soft Potential (United States)

    Liu, Shuangqian; Yang, Xiongfeng


    Boundary effects are central to the dynamics of the dilute particles governed by the Boltzmann equation. In this paper, we study both the diffuse reflection and the specular reflection boundary value problems for the Boltzmann equation with a soft potential, in which the collision kernel is ruled by the inverse power law. For the diffuse reflection boundary condition, based on an L 2 argument and its interplay with intricate {L^∞} analysis for the linearized Boltzmann equation, we first establish the global existence and then obtain the exponential decay in {L^∞} space for the nonlinear Boltzmann equation in general classes of bounded domain. It turns out that the zero lower bound of the collision frequency and the singularity of the collision kernel lead to some new difficulties for achieving the a priori {L^∞} estimates and time decay rates of the solution. In the course of the proof, we capture some new properties of the probability integrals along the stochastic cycles and improve the {L^2-L^∞} theory to give a more direct approach to overcome those difficulties. As to the specular reflection condition, our key contribution is to develop a new time-velocity weighted {L^∞} theory so that we could deal with the greater difficulties stemming from the complicated velocity relations among the specular cycles and the zero lower bound of the collision frequency. From this new point, we are also able to prove that the solutions of the linearized Boltzmann equation tend to equilibrium exponentially in {L^∞} space with the aid of the L 2 theory and a bootstrap argument. These methods, in the latter case, can be applied to the Boltzmann equation with soft potential for all other types of boundary condition.

  4. An overview of the Boltzmann transport equation solution for neutrons, photons and electrons in cartesian geometry

    Energy Technology Data Exchange (ETDEWEB)

    Rodriguez, Barbara D. do Amaral, E-mail: [Universidade Federal do Rio Grande do Sul (UFRGS), Porto Alegre, RS (Brazil). Programa de Pos-Graduacao em Engenharia Mecanica; Vilhena, Marco Tullio, E-mail: vilhena@mat.ufrgs.b [Universidade Federal do Rio Grande do Sul (UFRGS), Porto Alegre, RS (Brazil). Dept. de Matematica Pura e Aplicada


    Questions regarding accuracy and efficiency of deterministic transport methods are still on our mind today, even with modern supercomputers. The most versatile and widely used deterministic methods are the P{sub N} approximation, the S{sub N} method (discrete ordinates method) and their variants. In the discrete ordinates (S{sub N}) formulations of the transport equation, it is assumed that the linearized Boltzmann equation only holds for a set of distinct numerical values of the direction-of-motion variables. In this work, looking forward to confirm the capabilities of deterministic methods in obtaining accurate results, we present a general overview of deterministic methods to solve the Boltzmann transport equation for neutral and charged particles. First, we describe a review in the Laplace transform technique applied to S{sub N} two dimensional transport equation in a rectangular domain considering Compton scattering. Next, we solved the Fokker-Planck (FP) equation, an alternative approach for the Boltzmann transport equation, assuming a monoenergetic electron beam in a rectangular domain. The main idea relies on applying the P{sub N} approximation, a recent advance in the class of deterministic methods, in the angular variable, to the two dimensional Fokker-Planck equation and then applying the Laplace Transform in the spatial x-variable. Numerical results are given to illustrate the accuracy of deterministic methods presented. (author)

  5. An Eulerian description of the streaming process in the lattice Boltzmann equation

    CERN Document Server

    Lee Tae Hun


    This paper presents a novel strategy for solving discrete Boltzmann equation (DBE) for simulation of fluid flows. This strategy splits the solution procedure into streaming and collision steps as in the lattice Boltzmann equation (LBE) method. The streaming step can then be carried out by solving pure linear advection equations in an Eulerian framework. This offers two significant advantages over previous methods. First, the relationship between the relaxation parameter and the discretization of the collision term developed from the LBE method is directly applicable to the DBE method. The resulting DBE collision step remains local and poses no constraint on time step. Second, decoupling of the advection step from the collision step facilitates implicit discretization of the advection equation on arbitrary meshes. An implicit unstructured DBE method is constructed based on this strategy and is evaluated using several test cases of flow over a backward-facing step, lid-driven cavity flow, and flow past a circul...

  6. Stable lattice Boltzmann model for Maxwell equations in media (United States)

    Hauser, A.; Verhey, J. L.


    The present work shows a method for stable simulations via the lattice Boltzmann (LB) model for electromagnetic waves (EM) transiting homogeneous media. LB models for such media were already presented in the literature, but they suffer from numerical instability when the media transitions are sharp. We use one of these models in the limit of pure vacuum derived from Liu and Yan [Appl. Math. Model. 38, 1710 (2014), 10.1016/j.apm.2013.09.009] and apply an extension that treats the effects of polarization and magnetization separately. We show simulations of simple examples in which EM waves travel into media to quantify error scaling, stability, accuracy, and time scaling. For conductive media, we use the Strang splitting and check the simulations accuracy at the example of the skin effect. Like pure EM propagation, the error for the static limits, which are constructed with a current density added in a first-order scheme, can be less than 1 % . The presented method is an easily implemented alternative for the stabilization of simulation for EM waves propagating in spatially complex structured media properties and arbitrary transitions.

  7. Fixed points and flow analysis on off-equilibrium dynamics in the boson Boltzmann equation (United States)

    Fukushima, Kenji; Murase, Koichi; Pu, Shi


    We consider fixed points of steady solutions and flow directions using the boson Boltzmann equation that is a one-dimensionally reduced kinetic equation after the angular integration. With an elastic collision integral of the two-to-two scattering process, in the dense (dilute) regime where the distribution function is large (small), the boson Boltzmann equation has approximate fixed points with a power-law spectrum in addition to the thermal distribution function. We argue that the power-law fixed point can be exact in special cases. We elaborate a graphical presentation to display evolving flow directions similarly to the renormalization group flow, which explicitly exhibits how fixed points are connected and parameter space is separated by critical lines. We discuss that such a flow diagram contains useful information on thermalization processes out of equilibrium.

  8. Finite element method for neutron transport. 7. Completely boundary-free maximum principle for the first-order Boltzmann equation

    Energy Technology Data Exchange (ETDEWEB)

    Ackroyd, R.T. (UKAEA Risley Nuclear Power Development Establishment. Process Technology and Safety Directorate)


    A completely boundary-free maximum principle for the first-order Boltzmann equation is derived from the completely boundary-free maximum principle for the mixed-parity Boltzmann equation. When continuity is imposed on the trial function for directions crossing interfaces the completely boundary-free principle for the first-order Boltzmann equation reduces to a maximum principle previously established directly from first principles and indirectly by the Euler-Lagrange method. Present finite element methods for the first-order Boltzmann equation are based on a weighted-residual method which permits the use of discontinuous trial functions. The new principle for the first-order equation can be used as a basis for finite-element methods with the same freedom from boundary conditions as those based on the weighted-residual method. The extremum principle as the parent of the variationally-derived weighted-residual equations ensures their good behaviour.

  9. Quantum Boltzmann equations for electroweak baryogenesis including gauge fields

    CERN Document Server

    Kainulainen, K; Schmidt, M G; Weinstock, S; Kainulainen, Kimmo; Prokopec, Tomislav; Schmidt, Michael G.; Weinstock, Steffen


    We review and extend to include the gauge fields our derivation of the semiclassical limit of the collisionless quantum transport equations for the fermions in presence of a CP-violating bubble wall at a first order electroweak phase transition. We show how the (gradient correction modified) Lorenz-force appears both in the Schwinger-Keldysh approach and in the semiclassical WKB-treatment. In the latter approach the inclusion of gauge fields removes the apparent phase reparametrization dependence of the intermediate calculations. We also discuss setting up the fluid equations for practical calculations in electroweak baryogenesis including the self-consistent (hyper)electric field and the anomaly.

  10. Numerical Treatment of the Boltzmann Equation for Self-Propelled Particle Systems (United States)

    Thüroff, Florian; Weber, Christoph A.; Frey, Erwin


    Kinetic theories constitute one of the most promising tools to decipher the characteristic spatiotemporal dynamics in systems of actively propelled particles. In this context, the Boltzmann equation plays a pivotal role, since it provides a natural translation between a particle-level description of the system's dynamics and the corresponding hydrodynamic fields. Yet, the intricate mathematical structure of the Boltzmann equation substantially limits the progress toward a full understanding of this equation by solely analytical means. Here, we propose a general framework to numerically solve the Boltzmann equation for self-propelled particle systems in two spatial dimensions and with arbitrary boundary conditions. We discuss potential applications of this numerical framework to active matter systems and use the algorithm to give a detailed analysis to a model system of self-propelled particles with polar interactions. In accordance with previous studies, we find that spatially homogeneous isotropic and broken-symmetry states populate two distinct regions in parameter space, which are separated by a narrow region of spatially inhomogeneous, density-segregated moving patterns. We find clear evidence that these three regions in parameter space are connected by first-order phase transitions and that the transition between the spatially homogeneous isotropic and polar ordered phases bears striking similarities to liquid-gas phase transitions in equilibrium systems. Within the density-segregated parameter regime, we find a novel stable limit-cycle solution of the Boltzmann equation, which consists of parallel lanes of polar clusters moving in opposite directions, so as to render the overall symmetry of the system's ordered state nematic, despite purely polar interactions on the level of single particles.

  11. Numerical Treatment of the Boltzmann Equation for Self-Propelled Particle Systems

    Directory of Open Access Journals (Sweden)

    Florian Thüroff


    Full Text Available Kinetic theories constitute one of the most promising tools to decipher the characteristic spatiotemporal dynamics in systems of actively propelled particles. In this context, the Boltzmann equation plays a pivotal role, since it provides a natural translation between a particle-level description of the system’s dynamics and the corresponding hydrodynamic fields. Yet, the intricate mathematical structure of the Boltzmann equation substantially limits the progress toward a full understanding of this equation by solely analytical means. Here, we propose a general framework to numerically solve the Boltzmann equation for self-propelled particle systems in two spatial dimensions and with arbitrary boundary conditions. We discuss potential applications of this numerical framework to active matter systems and use the algorithm to give a detailed analysis to a model system of self-propelled particles with polar interactions. In accordance with previous studies, we find that spatially homogeneous isotropic and broken-symmetry states populate two distinct regions in parameter space, which are separated by a narrow region of spatially inhomogeneous, density-segregated moving patterns. We find clear evidence that these three regions in parameter space are connected by first-order phase transitions and that the transition between the spatially homogeneous isotropic and polar ordered phases bears striking similarities to liquid-gas phase transitions in equilibrium systems. Within the density-segregated parameter regime, we find a novel stable limit-cycle solution of the Boltzmann equation, which consists of parallel lanes of polar clusters moving in opposite directions, so as to render the overall symmetry of the system’s ordered state nematic, despite purely polar interactions on the level of single particles.

  12. From Conformal Invariance towards Dynamical Symmetries of the Collisionless Boltzmann Equation

    Directory of Open Access Journals (Sweden)

    Stoimen Stoimenov


    Full Text Available Dynamical symmetries of the collisionless Boltzmann transport equation, or Vlasov equation, but under the influence of an external driving force, are derived from non-standard representations of the 2D conformal algebra. In the case without external forces, the symmetry of the conformally-invariant transport equation is first generalized by considering the particle momentum as an independent variable. This new conformal representation can be further extended to include an external force. The construction and possible physical applications are outlined.

  13. Application of Littlewood-Paley decomposition to the regularity of Boltzmann type kinetic equations; Application de la decomposition de Littlewood-Paley a la regularite pour des equations cinetiques de type Boltzmann

    Energy Technology Data Exchange (ETDEWEB)

    EL Safadi, M


    We study the regularity of kinetic equations of Boltzmann type.We use essentially Littlewood-Paley method from harmonic analysis, consisting mainly in working with dyadics annulus. We shall mainly concern with the homogeneous case, where the solution f(t,x,v) depends only on the time t and on the velocities v, while working with realistic and singular cross-sections (non cutoff). In the first part, we study the particular case of Maxwellian molecules. Under this hypothesis, the structure of the Boltzmann operator and his Fourier transform write in a simple form. We show a global C{sup {infinity}} regularity. Then, we deal with the case of general cross-sections with 'hard potential'. We are interested in the Landau equation which is limit equation to the Boltzmann equation, taking in account grazing collisions. We prove that any weak solution belongs to Schwartz space S. We demonstrate also a similar regularity for the case of Boltzmann equation. Let us note that our method applies directly for all dimensions, and proofs are often simpler compared to other previous ones. Finally, we finish with Boltzmann-Dirac equation. In particular, we adapt the result of regularity obtained in Alexandre, Desvillettes, Wennberg and Villani work, using the dissipation rate connected with Boltzmann-Dirac equation. (author)

  14. A new lattice Boltzmann equation to simulate density-driven convection of carbon dioxide

    KAUST Repository

    Allen, Rebecca


    The storage of CO2 in fluid-filled geological formations has been carried out for more than a decade in locations around the world. After CO2 has been injected into the aquifer and has moved laterally under the aquifer\\'s cap-rock, density-driven convection becomes an important transport process to model. However, the challenge lies in simulating this transport process accurately with high spatial resolution and low CPU cost. This issue can be addressed by using the lattice Boltzmann equation (LBE) to formulate a model for a similar scenario when a solute diffuses into a fluid and density differences lead to convective mixing. The LBE is a promising alternative to the traditional methods of computational fluid dynamics. Rather than discretizing the system of partial differential equations of classical continuum mechanics directly, the LBE is derived from a velocity-space truncation of the Boltzmann equation of classical kinetic theory. We propose an extension to the LBE, which can accurately predict the transport of dissolved CO2 in water, as a step towards fluid-filled porous media simulations. This is achieved by coupling two LBEs, one for the fluid flow and one for the convection and diffusion of CO2. Unlike existing lattice Boltzmann equations for porous media flow, our model is derived from a system of moment equations and a Crank-Nicolson discretization of the velocity-truncated Boltzmann equation. The forcing terms are updated locally without the need for additional central difference approximation. Therefore our model preserves all the computational advantages of the single-phase lattice Boltzmann equation and is formally second-order accurate in both space and time. Our new model also features a novel implementation of boundary conditions, which is simple to implement and does not suffer from the grid-dependent error that is present in the standard "bounce-back" condition. The significance of using the LBE in this work lies in the ability to efficiently

  15. Spherical-harmonic type expansion for the Boltzmann equation in semiconductor devices

    Directory of Open Access Journals (Sweden)

    Armando Majorana


    Full Text Available The Boltzmann equation for an electron gas in a semiconductor is considered. The electron energy is assumed to have a very general form, so that, for instance, parabolic or non parabolic band approximations can be treated. A technique, which recalls the classical moment method due to Grad, to deduce an approximate quasi-hydrodynamical model is shown and compared with the spherical harmonic expansion. Some characteristics of the model, as entropy inequality, are explicitly presented.

  16. Parallel FE Approximation of the Even/Odd Parity Form of the Linear Boltzmann Equation

    Energy Technology Data Exchange (ETDEWEB)

    Drumm, Clifton R.; Lorenz, Jens


    A novel solution method has been developed to solve the linear Boltzmann equation on an unstructured triangular mesh. Instead of tackling the first-order form of the equation, this approach is based on the even/odd-parity form in conjunction with the conventional mdtigroup discrete-ordinates approximation. The finite element method is used to treat the spatial dependence. The solution method is unique in that the space-direction dependence is solved simultaneously, eliminating the need for the conventional inner iterations, and the method is well suited for massively parallel computers.

  17. Numerical homogenization of electrokinetic equations in porous media using lattice-Boltzmann simulations (United States)

    Obliger, Amaël; Duvail, Magali; Jardat, Marie; Coelho, Daniel; Békri, Samir; Rotenberg, Benjamin


    We report the calculation of all the transfer coefficients which couple the solvent and ionic fluxes through a charged pore under the effect of pressure, electrostatic potential, and concentration gradients. We use a combination of analytical calculations at the Poisson-Nernst-Planck and Navier-Stokes levels of description and mesoscopic lattice simulations based on kinetic theory. In the absence of added salt, i.e., when the only ions present in the fluid are the counterions compensating the charge of the surface, exact analytical expressions for the fluxes in cylindrical pores allow us to validate a new lattice-Boltzmann electrokinetics (LBE) scheme which accounts for the osmotic contribution to the transport of all species. The influence of simulation parameters on the numerical accuracy is thoroughly investigated. In the presence of an added salt, we assess the range of validity of approximate expressions of the fluxes computed from the linearized Poisson-Boltzmann equation by a systematic comparison with LBE simulations.

  18. Conservative phase-field lattice Boltzmann model for interface tracking equation. (United States)

    Geier, Martin; Fakhari, Abbas; Lee, Taehun


    Based on the phase-field theory, we propose a conservative lattice Boltzmann method to track the interface between two different fluids. The presented model recovers the conservative phase-field equation and conserves mass locally and globally. Two entirely different approaches are used to calculate the gradient of the phase field, which is needed in computation of the normal to the interface. One approach uses finite-difference stencils similar to many existing lattice Boltzmann models for tracking the two-phase interface, while the other one invokes central moments to calculate the gradient of the phase field without any finite differences involved. The former approach suffers from the nonlocality of the collision operator while the latter is entirely local making it highly suitable for massive parallel implementation. Several benchmark problems are carried out to assess the accuracy and stability of the proposed model.

  19. From Newton's Law to the Linear Boltzmann Equation Without Cut-Off (United States)

    Ayi, Nathalie


    We provide a rigorous derivation of the linear Boltzmann equation without cut-off starting from a system of particles interacting via a potential with infinite range as the number of particles N goes to infinity under the Boltzmann-Grad scaling. More particularly, we will describe the motion of a tagged particle in a gas close to global equilibrium. The main difficulty in our context is that, due to the infinite range of the potential, a non-integrable singularity appears in the angular collision kernel, making no longer valid the single-use of Lanford's strategy. Our proof relies then on a combination of Lanford's strategy, of tools developed recently by Bodineau, Gallagher and Saint-Raymond to study the collision process, and of new duality arguments to study the additional terms associated with the long-range interaction, leading to some explicit weak estimates.

  20. Hybrid simulation combining two space-time discretization of the discrete-velocity Boltzmann equation (United States)

    Horstmann, Jan Tobias; Le Garrec, Thomas; Mincu, Daniel-Ciprian; Lévêque, Emmanuel


    Despite the efficiency and low dissipation of the stream-collide scheme of the discrete-velocity Boltzmann equation, which is nowadays implemented in many lattice Boltzmann solvers, a major drawback exists over alternative discretization schemes, i.e. finite-volume or finite-difference, that is the limitation to Cartesian uniform grids. In this paper, an algorithm is presented that combines the positive features of each scheme in a hybrid lattice Boltzmann method. In particular, the node-based streaming of the distribution functions is coupled with a second-order finite-volume discretization of the advection term of the Boltzmann equation under the Bhatnagar-Gross-Krook approximation. The algorithm is established on a multi-domain configuration, with the individual schemes being solved on separate sub-domains and connected by an overlapping interface of at least 2 grid cells. A critical parameter in the coupling is the CFL number equal to unity, which is imposed by the stream-collide algorithm. Nevertheless, a semi-implicit treatment of the collision term in the finite-volume formulation allows us to obtain a stable solution for this condition. The algorithm is validated in the scope of three different test cases on a 2D periodic mesh. It is shown that the accuracy of the combined discretization schemes agrees with the order of each separate scheme involved. The overall numerical error of the hybrid algorithm in the macroscopic quantities is contained between the error of the two individual algorithms. Finally, we demonstrate how such a coupling can be used to adapt to anisotropic flows with some gradual mesh refinement in the FV domain.

  1. The Master kinetic equation for the statistical treatment of the Boltzmann-Sinai classical dynamical system (United States)

    Tessarotto, Massimo; Cremaschini, Claudio


    In this investigation, exact particular realizations are sought for the microscopic statistical description which is associated with the classical dynamical system (CDS) formed by N identical smooth hard spheres subject to elastic collisions ( S N -CDS). The problem is posed in the framework of the ab initio statistical description of S N -CDS recently developed. It is shown that the Liouville equation associated with SN-CDS admits an exact particular solution for the N-body probability density function (PDF). This is factorized in terms of the i-th particle 1-body PDF (for all i = 1, N) via suitable weighting factors, which are denoted here as particle occupation coefficients. The latter are found to depend functionally only on the 1-body PDFs which are associated with each of the remaining particles belonging to S N -CDS. Furthermore, the 1-body PDF is proved to obey a well-defined statistical equation, referred to here as Master kinetic equation. This is an exact kinetic equation which takes into account the occurrence of configuration-space correlations due to the finite size of the extended particles, while depending functionally on the same 1-body PDF only. The asymptotic approximation of the Master equation, which holds in validity of the Boltzmann-Grad limit, is shown to recover in a suitable asymptotic sense the customary Boltzmann equation. Finally, a critical analysis is presented of the original and modified versions of the Enskog kinetic equation, as well as of some of the non-linear kinetic approaches formulated in the past for dense granular gases. Their conditions of validity and main differences with respect to the present theory are pointed out.

  2. Chequerboard effects on spurious currents in the lattice Boltzmann equation for two-phase flows. (United States)

    Guo, Zhaoli; Shi, Baochang; Zheng, Chuguang


    Spurious currents near an interface between different phases are a common undesirable feature of the lattice Boltzmann equation (LBE) method for two-phase systems. In this paper, we show that the spurious currents of a kinetic theory-based LBE have a significant dependence on the parity of the grid number of the underlying lattice, which can be attributed to the chequerboard effect. A technique that uses a Lax-Wendroff streaming is proposed to overcome this anomaly, and its performance is verified numerically.

  3. Solitary wave of the Korteweg-de Vries equation based on lattice Boltzmann model with three conservation laws (United States)

    Wang, Huimin


    In this paper, a new lattice Boltzmann model for the Korteweg-de Vries (KdV) equation is proposed. By using the Chapman-Enskog expansion and the multi-scale time expansion, a series of partial differential equations in different time scales and several higher- order moments of equilibrium distribution functions are obtained. In order to make the scheme obey the three conservation laws of the KdV equation, two equilibrium distribution functions are used and a correlation between the first conservation law and the second conservation law is constructed. In numerical examples, the numerical results of the KdV equation obtained by this scheme are compared with those results obtained by the previous lattice Boltzmann model. Numerical experiments demonstrate this scheme can be used to reduce the truncation error of the lattice Boltzmann scheme and preserve the three conservation laws.

  4. Relativistic entropy and related Boltzmann kinetics

    Energy Technology Data Exchange (ETDEWEB)

    Kaniadakis, G. [Politecnico di Torino (Italy). Dipartimento di Fisica


    It is well known that the particular form of the two-particle correlation function, in the collisional integral of the classical Boltzmann equation, fixes univocally the entropy of the system, which turns out to be the Boltzmann-Gibbs-Shannon entropy. In the ordinary relativistic Boltzmann equation, some standard generalizations, with respect to its classical version, imposed by the special relativity, are customarily performed. The only ingredient of the equation, which tacitly remains in its original classical form, is the two-particle correlation function, and this fact imposes that also the relativistic kinetics is governed by the Boltzmann-Gibbs-Shannon entropy. Indeed the ordinary relativistic Boltzmann equation admits as stationary stable distribution, the exponential Juttner distribution. Here, we show that the special relativity laws and the maximum entropy principle suggest a relativistic generalization also of the two-particle correlation function and then of the entropy. The so obtained, fully relativistic Boltzmann equation, obeys the H-theorem and predicts a stationary stable distribution, presenting power law tails in the high-energy region. The ensued relativistic kinetic theory preserves the main features of the classical kinetics, which recovers in the c{yields}{infinity} limit. (orig.)

  5. Probabilistic Interpretation for the Nonlinear Poisson-Boltzmann Equation in Molecular Dynamics

    Directory of Open Access Journals (Sweden)

    Perrin Nicolas


    Full Text Available The Poisson-Boltzmann (PB equation describes the electrostatic potential of a biomolecular system composed by a molecule in a solvent. The electrostatic potential is involved in biomolecular models which are used in molecular simulation. In consequence, finding an efficient method to simulate the numerical solution of PB equation is very useful. As a first step, we establish in this paper a probabilistic interpretation of the nonlinear PB equation with Backward Stochastic Differential Equations (BSDEs. This interpretation requires an adaptation of existing results on BSDEs. En dynamique moléculaire, l’équation de Poisson-Boltzmann (PB permet de décrire le potentiel électrostatique d’un système moléculaire composé d’une molécule dans un solvant. Ce potentiel électrostatique intervient dans les modèles de simulation numérique permettant de comprendre la structure, la dynamique et le fonctionnement des protéines. La résolution numérique de l’équation de PB est donc une étape importante de ces simulations. Aussi, nous proposons dans un premier temps, une interprétation probabiliste de l’équation de PB non-linéaire à l’aide des Equations Différentielles Stochastiques Rétrogrades (EDSR. Cette interprétation nécessite une adaptation des résultats d’existence et d’unicité des solutions d’EDSR.

  6. Sensitivities to parameterization in the size-modified Poisson-Boltzmann equation (United States)

    Harris, Robert C.; Boschitsch, Alexander H.; Fenley, Marcia O.


    Experimental results have demonstrated that the numbers of counterions surrounding nucleic acids differ from those predicted by the nonlinear Poisson-Boltzmann equation, NLPBE. Some studies have fit these data against the ion size in the size-modified Poisson-Boltzmann equation, SMPBE, but the present study demonstrates that other parameters, such as the Stern layer thickness and the molecular surface definition, can change the number of bound ions by amounts comparable to varying the ion size. These parameters will therefore have to be fit simultaneously against experimental data. In addition, the data presented here demonstrate that the derivative, SK, of the electrostatic binding free energy, ΔGel, with respect to the logarithm of the salt concentration is sensitive to these parameters, and experimental measurements of SK could be used to parameterize the model. However, although better values for the Stern layer thickness and ion size and better molecular surface definitions could improve the model's predictions of the numbers of ions around biomolecules and SK, ΔGel itself is more sensitive to parameters, such as the interior dielectric constant, which in turn do not significantly affect the distributions of ions around biomolecules. Therefore, improved estimates of the ion size and Stern layer thickness to use in the SMPBE will not necessarily improve the model's predictions of ΔGel.

  7. Sensitivities to parameterization in the size-modified Poisson-Boltzmann equation. (United States)

    Harris, Robert C; Boschitsch, Alexander H; Fenley, Marcia O


    Experimental results have demonstrated that the numbers of counterions surrounding nucleic acids differ from those predicted by the nonlinear Poisson-Boltzmann equation, NLPBE. Some studies have fit these data against the ion size in the size-modified Poisson-Boltzmann equation, SMPBE, but the present study demonstrates that other parameters, such as the Stern layer thickness and the molecular surface definition, can change the number of bound ions by amounts comparable to varying the ion size. These parameters will therefore have to be fit simultaneously against experimental data. In addition, the data presented here demonstrate that the derivative, SK, of the electrostatic binding free energy, ΔGel, with respect to the logarithm of the salt concentration is sensitive to these parameters, and experimental measurements of SK could be used to parameterize the model. However, although better values for the Stern layer thickness and ion size and better molecular surface definitions could improve the model's predictions of the numbers of ions around biomolecules and SK, ΔGel itself is more sensitive to parameters, such as the interior dielectric constant, which in turn do not significantly affect the distributions of ions around biomolecules. Therefore, improved estimates of the ion size and Stern layer thickness to use in the SMPBE will not necessarily improve the model's predictions of ΔGel.

  8. A New 2D-Transport, 1D-Diffusion Approximation of the Boltzmann Transport equation

    Energy Technology Data Exchange (ETDEWEB)

    Larsen, Edward


    The work performed in this project consisted of the derivation, implementation, and testing of a new, computationally advantageous approximation to the 3D Boltz- mann transport equation. The solution of the Boltzmann equation is the neutron flux in nuclear reactor cores and shields, but solving this equation is difficult and costly. The new “2D/1D” approximation takes advantage of a special geometric feature of typical 3D reactors to approximate the neutron transport physics in a specific (ax- ial) direction, but not in the other two (radial) directions. The resulting equation is much less expensive to solve computationally, and its solutions are expected to be sufficiently accurate for many practical problems. In this project we formulated the new equation, discretized it using standard methods, developed a stable itera- tion scheme for solving the equation, implemented the new numerical scheme in the MPACT code, and tested the method on several realistic problems. All the hoped- for features of this new approximation were seen. For large, difficult problems, the resulting 2D/1D solution is highly accurate, and is calculated about 100 times faster than a 3D discrete ordinates simulation.

  9. Lid-driven cavity flow using a discrete velocity method for solving the Boltzmann equation (United States)

    Sekaran, Aarthi; Varghese, Philip; Estes, Samuel; Goldstein, David


    We extend the discrete velocity method for solving the Boltzmann equation previously used for one-dimensional problems to two spatial dimensions. The collision integral is computed using collisions between velocity classes selected randomly using a Monte Carlo method. Arbitrary post-collision velocities are mapped back onto the grid using a projection scheme which conserves mass, momentum, and energy. In addition, a variance reduction scheme is implemented to decrease noise and further reduce computational effort. The convection part of the equation is computed using first order upwind finite differences. We apply this discrete velocity scheme to the 2D lid-driven square cavity flow problem with Ar as the fluid medium and explore the effect of the additional flexibility available in this quasi-particle based stochastic method on the accuracy and noise level in the solutions obtained.

  10. On Stable Wall Boundary Conditions for the Hermite Discretization of the Linearised Boltzmann Equation (United States)

    Sarna, Neeraj; Torrilhon, Manuel


    We define certain criteria, using the characteristic decomposition of the boundary conditions and energy estimates, which a set of stable boundary conditions for a linear initial boundary value problem, involving a symmetric hyperbolic system, must satisfy. We first use these stability criteria to show the instability of the Maxwell boundary conditions proposed by Grad (Commun Pure Appl Math 2(4):331-407, 1949). We then recognise a special block structure of the moment equations which arises due to the recursion relations and the orthogonality of the Hermite polynomials; the block structure will help us in formulating stable boundary conditions for an arbitrary order Hermite discretization of the Boltzmann equation. The formulation of stable boundary conditions relies upon an Onsager matrix which will be constructed such that the newly proposed boundary conditions stay close to the Maxwell boundary conditions at least in the lower order moments.

  11. Opinion dynamics: inhomogeneous Boltzmann-type equations modelling opinion leadership and political segregation

    CERN Document Server

    Düring, Bertram


    We propose and investigate different kinetic models for opinion formation, when the opinion formation process depends on an additional independent variable, e.g. a leadership or a spatial variable. More specifically, we consider:(i) opinion dynamics under the effect of opinion leadership, where each individual is characterised not only by its opinion, but also by another independent variable which quantifies leadership qualities; (ii) opinion dynamics modelling political segregation in the `The Big Sort', a phenomenon that US citizens increasingly prefer to live in neighbourhoods with politically like-minded individuals. Based on microscopic opinion consensus dynamics such models lead to inhomogeneous Boltzmann-type equations for the opinion distribution. We derive macroscopic Fokker-Planck-type equations in a quasi-invariant opinion limit and present results of numerical experiments.

  12. Lattice Boltzmann model for high-order nonlinear partial differential equations (United States)

    Chai, Zhenhua; He, Nanzhong; Guo, Zhaoli; Shi, Baochang


    In this paper, a general lattice Boltzmann (LB) model is proposed for the high-order nonlinear partial differential equation with the form ∂tϕ +∑k=1mαk∂xkΠk(ϕ ) =0 (1 ≤k ≤m ≤6 ), αk are constant coefficients, Πk(ϕ ) are some known differential functions of ϕ . As some special cases of the high-order nonlinear partial differential equation, the classical (m)KdV equation, KdV-Burgers equation, K (n ,n ) -Burgers equation, Kuramoto-Sivashinsky equation, and Kawahara equation can be solved by the present LB model. Compared to the available LB models, the most distinct characteristic of the present model is to introduce some suitable auxiliary moments such that the correct moments of equilibrium distribution function can be achieved. In addition, we also conducted a detailed Chapman-Enskog analysis, and found that the high-order nonlinear partial differential equation can be correctly recovered from the proposed LB model. Finally, a large number of simulations are performed, and it is found that the numerical results agree with the analytical solutions, and usually the present model is also more accurate than the existing LB models [H. Lai and C. Ma, Sci. China Ser. G 52, 1053 (2009), 10.1007/s11433-009-0149-3; H. Lai and C. Ma, Phys. A (Amsterdam) 388, 1405 (2009), 10.1016/j.physa.2009.01.005] for high-order nonlinear partial differential equations.

  13. A Combined MPI-CUDA Parallel Solution of Linear and Nonlinear Poisson-Boltzmann Equation

    Directory of Open Access Journals (Sweden)

    José Colmenares


    Full Text Available The Poisson-Boltzmann equation models the electrostatic potential generated by fixed charges on a polarizable solute immersed in an ionic solution. This approach is often used in computational structural biology to estimate the electrostatic energetic component of the assembly of molecular biological systems. In the last decades, the amount of data concerning proteins and other biological macromolecules has remarkably increased. To fruitfully exploit these data, a huge computational power is needed as well as software tools capable of exploiting it. It is therefore necessary to move towards high performance computing and to develop proper parallel implementations of already existing and of novel algorithms. Nowadays, workstations can provide an amazing computational power: up to 10 TFLOPS on a single machine equipped with multiple CPUs and accelerators such as Intel Xeon Phi or GPU devices. The actual obstacle to the full exploitation of modern heterogeneous resources is efficient parallel coding and porting of software on such architectures. In this paper, we propose the implementation of a full Poisson-Boltzmann solver based on a finite-difference scheme using different and combined parallel schemes and in particular a mixed MPI-CUDA implementation. Results show great speedups when using the two schemes, achieving an 18.9x speedup using three GPUs.

  14. A combined MPI-CUDA parallel solution of linear and nonlinear Poisson-Boltzmann equation. (United States)

    Colmenares, José; Galizia, Antonella; Ortiz, Jesús; Clematis, Andrea; Rocchia, Walter


    The Poisson-Boltzmann equation models the electrostatic potential generated by fixed charges on a polarizable solute immersed in an ionic solution. This approach is often used in computational structural biology to estimate the electrostatic energetic component of the assembly of molecular biological systems. In the last decades, the amount of data concerning proteins and other biological macromolecules has remarkably increased. To fruitfully exploit these data, a huge computational power is needed as well as software tools capable of exploiting it. It is therefore necessary to move towards high performance computing and to develop proper parallel implementations of already existing and of novel algorithms. Nowadays, workstations can provide an amazing computational power: up to 10 TFLOPS on a single machine equipped with multiple CPUs and accelerators such as Intel Xeon Phi or GPU devices. The actual obstacle to the full exploitation of modern heterogeneous resources is efficient parallel coding and porting of software on such architectures. In this paper, we propose the implementation of a full Poisson-Boltzmann solver based on a finite-difference scheme using different and combined parallel schemes and in particular a mixed MPI-CUDA implementation. Results show great speedups when using the two schemes, achieving an 18.9x speedup using three GPUs.

  15. Computational and analytical comparison of flux discretizations for the semiconductor device equations beyond Boltzmann statistics (United States)

    Farrell, Patricio; Koprucki, Thomas; Fuhrmann, Jürgen


    We compare three thermodynamically consistent numerical fluxes known in the literature, appearing in a Voronoï finite volume discretization of the van Roosbroeck system with general charge carrier statistics. Our discussion includes an extension of the Scharfetter-Gummel scheme to non-Boltzmann (e.g. Fermi-Dirac) statistics. It is based on the analytical solution of a two-point boundary value problem obtained by projecting the continuous differential equation onto the interval between neighboring collocation points. Hence, it serves as a reference flux. The exact solution of the boundary value problem can be approximated by computationally cheaper fluxes which modify certain physical quantities. One alternative scheme averages the nonlinear diffusion (caused by the non-Boltzmann nature of the problem), another one modifies the effective density of states. To study the differences between these three schemes, we analyze the Taylor expansions, derive an error estimate, visualize the flux error and show how the schemes perform for a carefully designed p-i-n benchmark simulation. We present strong evidence that the flux discretization based on averaging the nonlinear diffusion has an edge over the scheme based on modifying the effective density of states.

  16. Simulation of non-ideal gases and liquid-gas phase transitions by lattice Boltzmann equation

    CERN Document Server

    Shan, X; Xiaowen Shan; Hudong Chen


    We describe in detail a recently proposed lattice-Boltzmann model for simulating flows with multiple phases and components. In particular, the focus is on the modeling of one-component fluid systems which obey non-ideal gas equations of state and can undergo a liquid-gas type phase transition. The model is shown to be momentum-conserving. From the microscopic mechanical stability condition, the densities in bulk liquid and gas phases are obtained as functions of a temperature-like parameter. Comparisons with the thermodynamic theory of phase transition show that the LBE model can be made to correspond exactly to an isothermal process. The density profile in the liquid-gas interface is also obtained as function of the temperature-like parameter and is shown to be isotropic. The surface tension, which can be changed independently, is calculated. The analytical conclusions are verified by numerical simulations. (To appear in Phys. Rev. E)

  17. Maximum-entropy reconstruction method for moment-based solution of the Boltzmann equation (United States)

    Summy, Dustin; Pullin, Dale


    We describe a method for a moment-based solution of the Boltzmann equation. This starts with moment equations for a 10 + 9 N , N = 0 , 1 , 2 . . . -moment representation. The partial-differential equations (PDEs) for these moments are unclosed, containing both higher-order moments and molecular-collision terms. These are evaluated using a maximum-entropy construction of the velocity distribution function f (c , x , t) , using the known moments, within a finite-box domain of single-particle-velocity (c) space. Use of a finite-domain alleviates known problems (Junk and Unterreiter, Continuum Mech. Thermodyn., 2002) concerning existence and uniqueness of the reconstruction. Unclosed moments are evaluated with quadrature while collision terms are calculated using a Monte-Carlo method. This allows integration of the moment PDEs in time. Illustrative examples will include zero-space- dimensional relaxation of f (c , t) from a Mott-Smith-like initial condition toward equilibrium and one-space dimensional, finite Knudsen number, planar Couette flow. Comparison with results using the direct-simulation Monte-Carlo method will be presented.

  18. Fast Maximum Entropy Moment Closure Approach to Solving the Boltzmann Equation (United States)

    Summy, Dustin; Pullin, Dale


    We describe a method for a moment-based solution of the Boltzmann Equation (BE). This is applicable to an arbitrary set of velocity moments whose transport is governed by partial-differential equations (PDEs) derived from the BE. The equations are unclosed, containing both higher-order moments and molecular-collision terms. These are evaluated using a maximum-entropy reconstruction of the velocity distribution function f (c , x , t) , from the known moments, within a finite-box domain of single-particle velocity (c) space. Use of a finite-domain alleviates known problems (Junk and Unterreiter, Continuum Mech. Thermodyn., 2002) concerning existence and uniqueness of the reconstruction. Unclosed moments are evaluated with quadrature while collision terms are calculated using any desired method. This allows integration of the moment PDEs in time. The high computational cost of the general method is greatly reduced by careful choice of the velocity moments, allowing the necessary integrals to be reduced from three- to one-dimensional in the case of strictly 1D flows. A method to extend this enhancement to fully 3D flows is discussed. Comparison with relaxation and shock-wave problems using the DSMC method will be presented. Partially supported by NSF grant DMS-1418903.

  19. Consistent Algorithms Marching Along Characteristics for the Numerical Solution of the Boltzmann Equation

    Directory of Open Access Journals (Sweden)

    Nilson C. Roberty


    Full Text Available We introduce algorithms marching over a polygonal mesh with elements consistent with the propagation directions of the particle (radiation flux. The decision for adopting this kind of mesh to solve the one-speed Boltzmann transport equation is due to characteristics of the domain of the transport operator which controls derivatives only in the direction of propagation of the particles (radiation flux in the absorbing and scattering media. This a priori adaptivity has the advantages that it formulates a consistent scheme which makes appropriate the application of the Lax equivalence theorem framework to the problem. In this work, we present the main functional spaces involved in the formalism and a description of the algorithms for the mesh generation and the transport equation solution. Some numerical examples related to the solution of a transmission problem in a high-contrast model with absorption and scattering are presented. Also, a comparison with benchmarks problems for source and reactor criticality simulations shows the compatibility between calculations with the algorithms proposed here and theoretical results.

  20. Asymptotic analysis of the Poisson-Boltzmann equation describing electrokinetics in porous media (United States)

    Allaire, Grégoire; Dufrêche, Jean-François; Mikelić, Andro; Piatnitski, Andrey


    We consider the Poisson-Boltzmann equation in a periodic cell, representative of a porous medium. It is a model for the electrostatic distribution of N chemical species diluted in a liquid at rest, occupying the pore space with charged solid boundaries. We study the asymptotic behaviour of its solution depending on a parameter β, which is the square of the ratio between a characteristic pore length and the Debye length. For small β we identify the limit problem which is still a nonlinear Poisson equation involving only one species with maximal valence, opposite to the average of the given surface charge density. This result justifies the Donnan effect, observing that the ions for which the charge is that of the solid phase are expelled from the pores. For large β we prove that the solution behaves like a boundary layer near the pore walls and is constant far away in the bulk. Our analysis is valid for Neumann boundary conditions (namely for imposed surface charge densities) and establishes rigorously that solid interfaces are uncoupled from the bulk fluid so that simplified additive theories, such as the popular Derjaguin, Landau, Verwey and Overbeek approach, can be used. We show that the asymptotic behaviour is completely different in the case of Dirichlet boundary conditions (namely for imposed surface potential).




    We consider the design of an effective and reliable adaptive finite element method (AFEM) for the nonlinear Poisson-Boltzmann equation (PBE). We first examine the two-term regularization technique for the continuous problem recently proposed by Chen, Holst, and Xu based on the removal of the singular electrostatic potential inside biomolecules; this technique made possible the development of the first complete solution and approximation theory for the Poisson-Boltzmann equation, the first provably convergent discretization, and also allowed for the development of a provably convergent AFEM. However, in practical implementation, this two-term regularization exhibits numerical instability. Therefore, we examine a variation of this regularization technique which can be shown to be less susceptible to such instability. We establish a priori estimates and other basic results for the continuous regularized problem, as well as for Galerkin finite element approximations. We show that the new approach produces regularized continuous and discrete problems with the same mathematical advantages of the original regularization. We then design an AFEM scheme for the new regularized problem, and show that the resulting AFEM scheme is accurate and reliable, by proving a contraction result for the error. This result, which is one of the first results of this type for nonlinear elliptic problems, is based on using continuous and discrete a priori L∞ estimates to establish quasi-orthogonality. To provide a high-quality geometric model as input to the AFEM algorithm, we also describe a class of feature-preserving adaptive mesh generation algorithms designed specifically for constructing meshes of biomolecular structures, based on the intrinsic local structure tensor of the molecular surface. All of the algorithms described in the article are implemented in the Finite Element Toolkit (FETK), developed and maintained at UCSD. The stability advantages of the new regularization scheme

  2. Solution of the Boltzmann-Fokker-Planck transport equation using exponential nodal schemes; Solucion de la ecuacion de transporte de Boltzmann-Fokker-Planck usando esquemas nodales exponenciales

    Energy Technology Data Exchange (ETDEWEB)

    Ortega J, R.; Valle G, E. del [IPN-ESFM, 07738 Mexico D.F. (Mexico)]. e-mail:


    There are carried out charge and energy calculations deposited due to the interaction of electrons with a plate of a certain material, solving numerically the electron transport equation for the Boltzmann-Fokker-Planck approach of first order in plate geometry with a computer program denominated TEOD-NodExp (Transport of Electrons in Discreet Ordinates, Nodal Exponentials), using the proposed method by the Dr. J. E. Morel to carry out the discretization of the variable energy and several spatial discretization schemes, denominated exponentials nodal. It is used the Fokker-Planck equation since it represents an approach of the Boltzmann transport equation that is been worth whenever it is predominant the dispersion of small angles, that is to say, resulting dispersion in small dispersion angles and small losses of energy in the transport of charged particles. Such electrons could be those that they face with a braking plate in a device of thermonuclear fusion. In the present work its are considered electrons of 1 MeV that impact isotropically on an aluminum plate. They were considered three different thickness of plate that its were designated as problems 1, 2 and 3. In the calculations it was used the discrete ordinate method S{sub 4} with expansions of the dispersion cross sections until P{sub 3} order. They were considered 25 energy groups of uniform size between the minimum energy of 0.1 MeV and the maximum of 1.0 MeV; the one spatial intervals number it was considered variable and it was assigned the values of 10, 20 and 30. (Author)

  3. ShengBTE: A solver of the Boltzmann transport equation for phonons (United States)

    Li, Wu; Carrete, Jesús; A. Katcho, Nebil; Mingo, Natalio


    ShengBTE is a software package for computing the lattice thermal conductivity of crystalline bulk materials and nanowires with diffusive boundary conditions. It is based on a full iterative solution to the Boltzmann transport equation. Its main inputs are sets of second- and third-order interatomic force constants, which can be calculated using third-party ab-initio packages. Dirac delta distributions arising from conservation of energy are approximated by Gaussian functions. A locally adaptive algorithm is used to determine each process-specific broadening parameter, which renders the method fully parameter free. The code is free software, written in Fortran and parallelized using MPI. A complementary Python script to help compute third-order interatomic force constants from a minimum number of ab-initio calculations, using a real-space finite-difference approach, is also publicly available for download. Here we discuss the design and implementation of both pieces of software and present results for three example systems: Si, InAs and lonsdaleite.

  4. pK(A) in proteins solving the Poisson-Boltzmann equation with finite elements. (United States)

    Sakalli, Ilkay; Knapp, Ernst-Walter


    Knowledge on pK(A) values is an eminent factor to understand the function of proteins in living systems. We present a novel approach demonstrating that the finite element (FE) method of solving the linearized Poisson-Boltzmann equation (lPBE) can successfully be used to compute pK(A) values in proteins with high accuracy as a possible replacement to finite difference (FD) method. For this purpose, we implemented the software molecular Finite Element Solver (mFES) in the framework of the Karlsberg+ program to compute pK(A) values. This work focuses on a comparison between pK(A) computations obtained with the well-established FD method and with the new developed FE method mFES, solving the lPBE using protein crystal structures without conformational changes. Accurate and coarse model systems are set up with mFES using a similar number of unknowns compared with the FD method. Our FE method delivers results for computations of pK(A) values and interaction energies of titratable groups, which are comparable in accuracy. We introduce different thermodynamic cycles to evaluate pK(A) values and we show for the FE method how different parameters influence the accuracy of computed pK(A) values. © 2015 Wiley Periodicals, Inc.

  5. Evaluation of outflow boundary conditions for two-phase lattice Boltzmann equation. (United States)

    Lou, Qin; Guo, Zhaoli; Shi, Baochang


    Outflow boundary condition (OBC) is a critical issue in computational fluid dynamics. As a type of numerical method for fluid flows, the lattice Boltzmann equation (LBE) method has gained much success in a variety of complex flows, and certain OBCs have been suggested for the LBE in simulating simple single-phase flows. However, very few discussions on the OBCs have been made for the two-phase LBE method. In this work, three types of OBCs that are widely used in the LBE for single-phase flows, i.e., the Neumann boundary condition, the convective boundary condition, and the extrapolation boundary condition, are extended to a two-phase LBE method and their performances are investigated. The comprehensive results of several two-phase flows show that these boundary conditions behave quite differently in the simulations of two-phase flows. Specifically, it is found that the Neumann boundary condition and the extrapolation boundary condition give rather poor predictions, while the type of convective boundary conditions work well, although the choice of the convection velocity has some slight influences on the results. We also apply these OBC schemes to some other two-phase models, and similar observations are found.

  6. Interfacial micro-currents in continuum-scale multi-component lattice Boltzmann equation hydrodynamics (United States)

    Halliday, I.; Lishchuk, S. V.; Spencer, T. J.; Burgin, K.; Schenkel, T.


    We describe, analyse and reduce micro-current effects in one class of lattice Boltzmann equation simulation method describing immiscible fluids within the continuum approximation, due to Lishchuk et al. (2003). This model's micro-current flow field and associated density adjustment, when considered in the linear, low-Reynolds number regime, may be decomposed into independent, superposable contributions arising from various error terms in its immersed boundary force. Error force contributions which are rotational (solenoidal) are mainly responsible for the micro-current (corresponding density adjustment). Rotationally anisotropic error terms arise from numerical derivatives and from the sampling of the interface-supporting force. They may be removed, either by eliminating the causal error force or by negating it. It is found to be straightforward to design more effective stencils with significantly improved performance. Practically, the micro-current activity arising in Lishchuk's method is reduced by approximately three quarters by using an appropriate stencil and approximately by an order of magnitude when the effects of sampling are removed.

  7. High Order Finite Volume Nonlinear Schemes for the Boltzmann Transport Equation

    Energy Technology Data Exchange (ETDEWEB)

    Bihari, B L; Brown, P N


    The authors apply the nonlinear WENO (Weighted Essentially Nonoscillatory) scheme to the spatial discretization of the Boltzmann Transport Equation modeling linear particle transport. The method is a finite volume scheme which ensures not only conservation, but also provides for a more natural handling of boundary conditions, material properties and source terms, as well as an easier parallel implementation and post processing. It is nonlinear in the sense that the stencil depends on the solution at each time step or iteration level. By biasing the gradient calculation towards the stencil with smaller derivatives, the scheme eliminates the Gibb's phenomenon with oscillations of size O(1) and reduces them to O(h{sup r}), where h is the mesh size and r is the order of accuracy. The current implementation is three-dimensional, generalized for unequally spaced meshes, fully parallelized, and up to fifth order accurate (WENO5) in space. For unsteady problems, the resulting nonlinear spatial discretization yields a set of ODE's in time, which in turn is solved via high order implicit time-stepping with error control. For the steady-state case, they need to solve the non-linear system, typically by Newton-Krylov iterations. There are several numerical examples presented to demonstrate the accuracy, non-oscillatory nature and efficiency of these high order methods, in comparison with other fixed-stencil schemes.

  8. Heat dissipation in the quasiballistic regime studied using the Boltzmann equation in the spatial frequency domain (United States)

    Hua, Chengyun; Minnich, Austin J.


    Quasiballistic heat conduction, in which some phonons propagate ballistically over a thermal gradient, has recently become of intense interest. Most works report that the thermal resistance associated with nanoscale heat sources is far larger than predicted by Fourier's law; however, recent experiments show that in certain cases the difference is negligible despite the heaters being far smaller than phonon mean-free paths. In this work, we examine how thermal resistance depends on the heater geometry using analytical solutions of the Boltzmann equation. We show that the spatial frequencies of the heater pattern play the key role in setting the thermal resistance rather than any single geometric parameter, and that for many geometries the thermal resistance in the quasiballistic regime is no different than the Fourier prediction. We also demonstrate that the spectral distribution of the heat source also plays a major role in the resulting transport, unlike in the diffusion regime. Our work provides an intuitive link between the heater geometry, spectral heating distribution, and the effective thermal resistance in the quasiballistic regime, a finding that could impact strategies for thermal management in electronics and other applications.

  9. Influence of Grid Spacing in Poisson-Boltzmann Equation Binding Energy Estimation. (United States)

    Harris, Robert C; Boschitsch, Alexander H; Fenley, Marcia O


    Grid-based solvers of the Poisson-Boltzmann, PB, equation are routinely used to estimate electrostatic binding, ΔΔGel, and solvation, ΔGel, free energies. The accuracies of such estimates are subject to grid discretization errors from the finite difference approximation to the PB equation. Here, we show that the grid discretization errors in ΔΔGel are more significant than those in ΔGel, and can be divided into two parts: (i) errors associated with the relative positioning of the grid and (ii) systematic errors associated with grid spacing. The systematic error in particular is significant for methods, such as the molecular mechanics PB surface area, MM-PBSA, approach that predict electrostatic binding free energies by averaging over an ensemble of molecular conformations. Although averaging over multiple conformations can control for the error associated with grid placement, it will not eliminate the systematic error, which can only be controlled by reducing grid spacing. The present study indicates that the widely-used grid spacing of 0.5 Å produces unacceptable errors in ΔΔGel, even though its predictions of ΔGel are adequate for the cases considered here. Although both grid discretization errors generally increase with grid spacing, the relative sizes of these errors differ according to the solute-solvent dielectric boundary definition. The grid discretization errors are generally smaller on the Gaussian surface used in the present study than on either the solvent-excluded or van der Waals surfaces, which both contain more surface discontinuities (e.g., sharp edges and cusps). Additionally, all three molecular surfaces converge to very different estimates of ΔΔGel.

  10. Heat transport in two-dimensional materials by directly solving the phonon Boltzmann equation under Callaway's dual relaxation model (United States)

    Guo, Yangyu; Wang, Moran


    The single mode relaxation time approximation has been demonstrated to greatly underestimate the lattice thermal conductivity of two-dimensional materials due to the collective effect of phonon normal scattering. Callaway's dual relaxation model represents a good approximation to the otherwise ab initio solution of the phonon Boltzmann equation. In this work we develop a discrete-ordinate-method (DOM) scheme for the numerical solution of the phonon Boltzmann equation under Callaway's model. Heat transport in a graphene ribbon with different geometries is modeled by our scheme, which produces results quite consistent with the available molecular dynamics, Monte Carlo simulations, and experimental measurements. Callaway's lattice thermal conductivity model with empirical boundary scattering rates is examined and shown to overestimate or underestimate the direct DOM solution. The length convergence of the lattice thermal conductivity of a rectangular graphene ribbon is explored and found to depend appreciably on the ribbon width, with a semiquantitative correlation provided between the convergence length and the width. Finally, we predict the existence of a phonon Knudsen minimum in a graphene ribbon only at a low system temperature and isotope concentration so that the average normal scattering rate is two orders of magnitude stronger than the intrinsic resistive one. The present work will promote not only the methodology for the solution of the phonon Boltzmann equation but also the theoretical modeling and experimental detection of hydrodynamic phonon transport in two-dimensional materials.

  11. On the transparent conducting oxide Al doped ZnO: First Principles and Boltzmann equations study

    Energy Technology Data Exchange (ETDEWEB)

    Slassi, A. [Institute of Nanomaterials and Nanotechnology, MAScIR, Rabat (Morocco); LMPHE (URAC 12), Faculté des Sciences, Université Mohammed V-Agdal, Rabat (Morocco); Naji, S. [LMPHE (URAC 12), Faculté des Sciences, Université Mohammed V-Agdal, Rabat (Morocco); Department of Physics, Faculty of Science, Ibb University, Ibb (Yemen); Benyoussef, A. [Institute of Nanomaterials and Nanotechnology, MAScIR, Rabat (Morocco); LMPHE (URAC 12), Faculté des Sciences, Université Mohammed V-Agdal, Rabat (Morocco); Hamedoun, M., E-mail: [Institute of Nanomaterials and Nanotechnology, MAScIR, Rabat (Morocco); El Kenz, A. [LMPHE (URAC 12), Faculté des Sciences, Université Mohammed V-Agdal, Rabat (Morocco)


    Highlights: • The incorporation of Al in ZnO increases the optical band edge absorption. • Incorporated Al creates shallow donor states of Al-3s around Fermi level. • Transmittance decreases in the visible and IR regions, while it increases in the UV region. • Electrical conductivity increases and reaches almost the saturation for high concentration of Al. - Abstract: We report, in this work, a theoretical study on the electronic, optical and electrical properties of pure and Al doped ZnO with different concentrations. In fact, we investigate these properties using both First Principles calculations within TB-mBJ approximation and Boltzmann equations under the constant relaxation time approximation for charge carriers. It is found out that, the calculated lattice parameters and the optical band gap of pure ZnO are close to the experimental values and in a good agreement with the other theoretical studies. It is also observed that, the incorporations of Al in ZnO increase the optical band edge absorption which leads to a blue shift and no deep impurities levels are induced in the band gap as well. More precisely, these incorporations create shallow donor states around Fermi level in the conduction band minimum from mainly Al-3s orbital. Beside this, it is found that, the transmittance is decreased in the visible and IR regions, while it is significantly improved in UV region. Finally, our calculations show that the electrical conductivity is enhanced as a result of Al doping and it reaches almost the saturation for high concentration of Al. These features make Al doped ZnO a transparent conducting electrode for optoelectronic device applications.

  12. SMPBS: Web server for computing biomolecular electrostatics using finite element solvers of size modified Poisson-Boltzmann equation. (United States)

    Xie, Yang; Ying, Jinyong; Xie, Dexuan


    SMPBS (Size Modified Poisson-Boltzmann Solvers) is a web server for computing biomolecular electrostatics using finite element solvers of the size modified Poisson-Boltzmann equation (SMPBE). SMPBE not only reflects ionic size effects but also includes the classic Poisson-Boltzmann equation (PBE) as a special case. Thus, its web server is expected to have a broader range of applications than a PBE web server. SMPBS is designed with a dynamic, mobile-friendly user interface, and features easily accessible help text, asynchronous data submission, and an interactive, hardware-accelerated molecular visualization viewer based on the 3Dmol.js library. In particular, the viewer allows computed electrostatics to be directly mapped onto an irregular triangular mesh of a molecular surface. Due to this functionality and the fast SMPBE finite element solvers, the web server is very efficient in the calculation and visualization of electrostatics. In addition, SMPBE is reconstructed using a new objective electrostatic free energy, clearly showing that the electrostatics and ionic concentrations predicted by SMPBE are optimal in the sense of minimizing the objective electrostatic free energy. SMPBS is available at the URL: © 2017 Wiley Periodicals, Inc. © 2017 Wiley Periodicals, Inc.

  13. Global weak solutions and uniform L-stability of the Boltzmann-Enskog equation (United States)

    Ha, Seung-Yeal; Noh, Se Eun

    We present the regularity theory of renormalized solutions and uniform L-stability estimates of the Boltzmann-Enskog model. We use a multi-dimensional Bony-type functional to control the time-phase space integral of a truncated collision operator. We employ Cercignani's arguments and a Bony-type functional to show that the renormalized solutions to the Boltzmann-Enskog model are weak solutions in usual distributional sense. For the uniform L-stability estimate, we use Lu's trick together with the Hölder inequality.

  14. Conjugate heat and mass transfer in the lattice Boltzmann equation method. (United States)

    Li, Like; Chen, Chen; Mei, Renwei; Klausner, James F


    An interface treatment for conjugate heat and mass transfer in the lattice Boltzmann equation method is proposed based on our previously proposed second-order accurate Dirichlet and Neumann boundary schemes. The continuity of temperature (concentration) and its flux at the interface for heat (mass) transfer is intrinsically satisfied without iterative computations, and the interfacial temperature (concentration) and their fluxes are conveniently obtained from the microscopic distribution functions without finite-difference calculations. The present treatment takes into account the local geometry of the interface so that it can be directly applied to curved interface problems such as conjugate heat and mass transfer in porous media. For straight interfaces or curved interfaces with no tangential gradient, the coupling between the interfacial fluxes along the discrete lattice velocity directions is eliminated and thus the proposed interface schemes can be greatly simplified. Several numerical tests are conducted to verify the applicability and accuracy of the proposed conjugate interface treatment, including (i) steady convection-diffusion in a channel containing two different fluids, (ii) unsteady convection-diffusion in the channel, (iii) steady heat conduction inside a circular domain with two different solid materials, and (iv) unsteady mass transfer from a spherical droplet in an extensional creeping flow. The accuracy and order of convergence of the simulated interior temperature (concentration) field, the interfacial temperature (concentration), and heat (mass) flux are examined in detail and compared with those obtained from the "half-lattice division" treatment in the literature. The present analysis and numerical results show that the half-lattice division scheme is second-order accurate only when the interface is fixed at the center of the lattice links, while the present treatment preserves second-order accuracy for arbitrary link fractions. For curved

  15. Trend to equilibrium of weak solutions of the Boltzmann equation in a slab with diffusive boundary conditions

    Energy Technology Data Exchange (ETDEWEB)

    Cercignani, C. [Politecnico di Milano, Milan (Italy)


    Recently R. Illner and the author proved that, under a physically realistic truncation on the collision kernel, the Boltzmann equation in the one-dimensional slab [0,1] with general diffusive boundary conditions at 0 and 1 has a global weak solution in the traditional sense. Here it is proved that when the Maxwellians associated with the boundary conditions at x=0 and x = 1 are the same Maxwellian M{sub w}, then the solution is uniformly bounded and tends to M{sub w} for t {r_arrow}{infinity}.

  16. Unified solution of the Boltzmann equation for electron and ion velocity distribution functions and transport coefficients in weakly ionized plasmas (United States)

    Konovalov, Dmitry A.; Cocks, Daniel G.; White, Ronald D.


    The velocity distribution function and transport coefficients for charged particles in weakly ionized plasmas are calculated via a multi-term solution of Boltzmann's equation and benchmarked using a Monte-Carlo simulation. A unified framework for the solution of the original full Boltzmann's equation is presented which is valid for ions and electrons, avoiding any recourse to approximate forms of the collision operator in various limiting mass ratio cases. This direct method using Lebedev quadratures over the velocity and scattering angles avoids the need to represent the ion mass dependence in the collision operator through an expansion in terms of the charged particle to neutral mass ratio. For the two-temperature Burnett function method considered in this study, this amounts to avoiding the need for the complex Talmi-transformation methods and associated mass-ratio expansions. More generally, we highlight the deficiencies in the two-temperature Burnett function method for heavy ions at high electric fields to calculate the ion velocity distribution function, even though the transport coefficients have converged. Contribution to the Topical Issue "Physics of Ionized Gases (SPIG 2016)", edited by Goran Poparic, Bratislav Obradovic, Dragana Maric and Aleksandar Milosavljevic.

  17. almaBTE : A solver of the space-time dependent Boltzmann transport equation for phonons in structured materials (United States)

    Carrete, Jesús; Vermeersch, Bjorn; Katre, Ankita; van Roekeghem, Ambroise; Wang, Tao; Madsen, Georg K. H.; Mingo, Natalio


    almaBTE is a software package that solves the space- and time-dependent Boltzmann transport equation for phonons, using only ab-initio calculated quantities as inputs. The program can predictively tackle phonon transport in bulk crystals and alloys, thin films, superlattices, and multiscale structures with size features in the nm- μm range. Among many other quantities, the program can output thermal conductances and effective thermal conductivities, space-resolved average temperature profiles, and heat-current distributions resolved in frequency and space. Its first-principles character makes almaBTE especially well suited to investigate novel materials and structures. This article gives an overview of the program structure and presents illustrative examples for some of its uses. PROGRAM SUMMARY Program Title:almaBTE Program Files doi: Licensing provisions: Apache License, version 2.0 Programming language: C++ External routines/libraries: BOOST, MPI, Eigen, HDF5, spglib Nature of problem: Calculation of temperature profiles, thermal flux distributions and effective thermal conductivities in structured systems where heat is carried by phonons Solution method: Solution of linearized phonon Boltzmann transport equation, Variance-reduced Monte Carlo

  18. Asymptotic Analysis of the Boltzmann Equation for Dark Matter Relics in the presence of a Running Dilaton and Space-Time Defects

    CERN Document Server

    Bender, Carl M; Sarkar, Sarben


    The interplay of dilatonic effects in dilaton cosmology and stochastic quantum space-time defects within the framework of string/brane cosmologies is examined. The Boltzmann equation describes the physics of thermal dark-matter-relic abundances in the presence of rolling dilatons. These dilatons affect the coupling of stringy matter to D-particle defects, which are generic in string theory. This coupling leads to an additional source term in the Boltzmann equation. The techniques of asymptotic matching and boundary-layer theory, which were recently applied by two of the authors (CMB and SS) to a Boltzmann equation, are used here to find the detailed asymptotic relic abundances for all ranges of the expectation value of the dilaton field. The phenomenological implications for the search of supersymmetric dark matter in current colliders, such as the LHC, are discussed.

  19. Effects of density and force discretizations on spurious velocities in lattice Boltzmann equation for two-phase flows

    KAUST Repository

    Xiong, Yuan


    Spurious current emerging in the vicinity of phase interfaces is a well-known disadvantage of the lattice Boltzmann equation (LBE) for two-phase flows. Previous analysis shows that this unphysical phenomenon comes from the force imbalance at discrete level inherited in LBE (Guo et al 2011 Phys. Rev. E 83 036707). Based on the analysis of the LBE free of checkerboard effects, in this work we further show that the force imbalance is caused by the different discretization stencils: the implicit one from the streaming process and the explicit one from the discretization of the force term. Particularly, the total contribution includes two parts, one from the difference between the intrinsically discretized density (or ideal gas pressure) gradient and the explicit ones in the force term, and the other from the explicit discretized chemical potential gradients in the intrinsically discretized force term. The former contribution is a special feature of LBE which was not realized previously.

  20. First-principles particle simulation and Boltzmann equation analysis of Negative Differential Conductivity and Transient Negative Mobility effects in xenon

    CERN Document Server

    Donko, Zoltan


    The Negative Differential Conductivity and Transient Negative Mobility effects in xenon gas are analyzed by a first-principles particle simulation technique and via an approximate solution of the Boltzmann transport equation (BE). The particle simulation method is devoid of the approximations that are traditionally adopted in the BE solutions in which (i) the distribution function is searched for in a two-term form, (ii) the Coulomb part of the collision integral for the anisotropic part of the distribution function is neglected, (iii) Coulomb collisions are treated as binary events, and (iv) the range of the electron-electron interaction is limited to a cutoff distance. The results obtained from the two methods are, for both effects, in good qualitative agreement, small differences are attributed to the approximations listed above.

  1. Bistable solutions for the electron energy distribution function in electron swarms in xenon via Boltzmann equation analysis and particle simulations

    CERN Document Server

    Dyatko, Nikolay


    At low reduced electric fields the electron energy distribution function in heavy noble gases can take two distinct shapes. This bistability effect - in which electron-electron (Coulomb) collisions play an essential role - is analyzed here with a Boltzmann equation approach and with a first principles particle simulation method. The latter is based on a combination of a molecular dynamics technique that accounts for the many-body interaction within the electron gas and a Monte Carlo treatment of the collisions between electrons and the background gas atoms. The good agreement found between the results of the two techniques confirms the existence of the two different stable solutions for the EEDF under swarm conditions at low electric fields.

  2. Boundary condition at a two-phase interface in the lattice Boltzmann method for the convection-diffusion equation. (United States)

    Yoshida, Hiroaki; Kobayashi, Takayuki; Hayashi, Hidemitsu; Kinjo, Tomoyuki; Washizu, Hitoshi; Fukuzawa, Kenji


    A boundary scheme in the lattice Boltzmann method (LBM) for the convection-diffusion equation, which correctly realizes the internal boundary condition at the interface between two phases with different transport properties, is presented. The difficulty in satisfying the continuity of flux at the interface in a transient analysis, which is inherent in the conventional LBM, is overcome by modifying the collision operator and the streaming process of the LBM. An asymptotic analysis of the scheme is carried out in order to clarify the role played by the adjustable parameters involved in the scheme. As a result, the internal boundary condition is shown to be satisfied with second-order accuracy with respect to the lattice interval, if we assign appropriate values to the adjustable parameters. In addition, two specific problems are numerically analyzed, and comparison with the analytical solutions of the problems numerically validates the proposed scheme.

  3. A variational treatment for the time dependent Boltzmann equation as a basis for numerical solutions conserving neutrons

    Energy Technology Data Exchange (ETDEWEB)

    Ackroyd, R.T.; Oliveira, C.R.E. de [Imperial Coll. of Science, Technology and Medicine, London (United Kingdom). Dept. of Mechanical Engineering


    A maximum principle for the time-dependent first-order Boltzmann equation is established in two independent ways:- by a generalized least squares method and by a method based on the properties of an appropriate bi-linear form. The second derivation suggests a metric for a Hilbert space which provides a geometrical interpretation of the variational principle. This interpretation leads to a Petrov-Galerkin method of Martin for time dependent transport. The maximum principle is used to define a figure of merit for the global error of any numerical solution for time dependent transport. The principle is used also to demonstrate the neutron conservation property of optimized numerical solutions, and the convergence of finite element methods based on the variational principle. (Author).

  4. Viscous-slip, thermal-slip, and temperature-jump coefficients as defined by the linearized Boltzmann equation and the Cercignani-Lampis boundary condition (United States)

    Siewert, C. E.


    A polynomial expansion procedure and an analytical discrete-ordinates method are used to evaluate the viscous-slip coefficient, the thermal-slip coefficient, and the temperature-jump coefficient as defined by a rigorous version of the linearized Boltzmann equation for rigid-sphere interactions and the Cercignani-Lampis boundary condition.

  5. Nonequilibrium scheme for computing the flux of the convection-diffusion equation in the framework of the lattice Boltzmann method. (United States)

    Chai, Zhenhua; Zhao, T S


    In this paper, we propose a local nonequilibrium scheme for computing the flux of the convection-diffusion equation with a source term in the framework of the multiple-relaxation-time (MRT) lattice Boltzmann method (LBM). Both the Chapman-Enskog analysis and the numerical results show that, at the diffusive scaling, the present nonequilibrium scheme has a second-order convergence rate in space. A comparison between the nonequilibrium scheme and the conventional second-order central-difference scheme indicates that, although both schemes have a second-order convergence rate in space, the present nonequilibrium scheme is more accurate than the central-difference scheme. In addition, the flux computation rendered by the present scheme also preserves the parallel computation feature of the LBM, making the scheme more efficient than conventional finite-difference schemes in the study of large-scale problems. Finally, a comparison between the single-relaxation-time model and the MRT model is also conducted, and the results show that the MRT model is more accurate than the single-relaxation-time model, both in solving the convection-diffusion equation and in computing the flux.

  6. L{sup 2}-stability of the Vlasov-Maxwell-Boltzmann system near global Maxwellians

    Energy Technology Data Exchange (ETDEWEB)

    Ha, Seung-Yeal, E-mail:; Xiao, Qinghua, E-mail: [Department of Mathematical Sciences, Seoul National University, Seoul 151-747 (Korea, Republic of); Xiong, Linjie, E-mail:; Zhao, Huijiang, E-mail: [School of Mathematics and Statistics, Wuhan University, Wuhan 430072 (China)


    We present a L{sup 2}-stability theory of the Vlasov-Maxwell-Boltzmann system for the two-species collisional plasma. We show that in a perturbative regime of a global Maxwellian, the L{sup 2}-distance between two strong solutions can be controlled by that between initial data in a Lipschitz manner. Our stability result extends earlier results [Ha, S.-Y. and Xiao, Q.-H., “A revisiting to the L{sup 2}-stability theory of the Boltzmann equation near global Maxwellians,” (submitted) and Ha, S.-Y., Yang, X.-F., and Yun, S.-B., “L{sup 2} stability theory of the Boltzmann equation near a global Maxwellian,” Arch. Ration. Mech. Anal. 197, 657–688 (2010)] on the L{sup 2}-stability of the Boltzmann equation to the Boltzmann equation coupled with self-consistent external forces. As a direct application of our stability result, we show that classical solutions in Duan et al. [“Optimal large-time behavior of the Vlasov-Maxwell-Boltzmann system in the whole space,” Commun. Pure Appl. Math. 24, 1497–1546 (2011)] and Guo [“The Vlasov-Maxwell-Boltzmann system near Maxwellians,” Invent. Math. 153(3), 593–630 (2003)] satisfy a uniform L{sup 2}-stability estimate. This is the first result on the L{sup 2}-stability of the Boltzmann equation coupled with self-consistent field equations in three dimensions.

  7. A deterministic solution of the first order linear Boltzmann transport equation in the presence of external magnetic fields

    Energy Technology Data Exchange (ETDEWEB)

    St Aubin, J., E-mail:; Keyvanloo, A.; Fallone, B. G. [Department of Medical Physics, Cross Cancer Institute, 11560 University Avenue Northwest, Edmonton, Alberta T6G 1Z2 (Canada); Vassiliev, O. [Department of Medical Physics, Tom Baker Cancer Center, 1331 29 Street Northwest, Calgary, Alberta T2N 4N2 (Canada)


    Purpose: Accurate radiotherapy dose calculation algorithms are essential to any successful radiotherapy program, considering the high level of dose conformity and modulation in many of today’s treatment plans. As technology continues to progress, such as is the case with novel MRI-guided radiotherapy systems, the necessity for dose calculation algorithms to accurately predict delivered dose in increasingly challenging scenarios is vital. To this end, a novel deterministic solution has been developed to the first order linear Boltzmann transport equation which accurately calculates x-ray based radiotherapy doses in the presence of magnetic fields. Methods: The deterministic formalism discussed here with the inclusion of magnetic fields is outlined mathematically using a discrete ordinates angular discretization in an attempt to leverage existing deterministic codes. It is compared against the EGSnrc Monte Carlo code, utilizing the emf-macros addition which calculates the effects of electromagnetic fields. This comparison is performed in an inhomogeneous phantom that was designed to present a challenging calculation for deterministic calculations in 0, 0.6, and 3 T magnetic fields oriented parallel and perpendicular to the radiation beam. The accuracy of the formalism discussed here against Monte Carlo was evaluated with a gamma comparison using a standard 2%/2 mm and a more stringent 1%/1 mm criterion for a standard reference 10 × 10 cm{sup 2} field as well as a smaller 2 × 2 cm{sup 2} field. Results: Greater than 99.8% (94.8%) of all points analyzed passed a 2%/2 mm (1%/1 mm) gamma criterion for all magnetic field strengths and orientations investigated. All dosimetric changes resulting from the inclusion of magnetic fields were accurately calculated using the deterministic formalism. However, despite the algorithm’s high degree of accuracy, it is noticed that this formalism was not unconditionally stable using a discrete ordinate angular discretization

  8. Computation of electron transport and relaxation properties in gases based on improved multi-term approximation of Boltzmann equation (United States)

    Cai, X. J.; Wang, X. X.; Zou, X. B.; Lu, Z. W.


    An understanding of electron kinetics is of importance in various applications of low temperature plasmas. We employ a series of model and real gases to investigate electron transport and relaxation properties based on improved multi-term approximation of the Boltzmann equation. First, a comparison of different methods to calculate the interaction integrals has been carried out; the effects of free parameters, such as vmax, lmax, and the arbitrary temperature Tb, on the convergence of electron transport coefficients are analyzed. Then, the modified attachment model of Ness et al. and SF6 are considered to investigate the effect of attachment on the electron transport properties. The deficiency of the pulsed Townsend technique to measure the electron transport and reaction coefficients in electronegative gases is highlighted when the reduced electric field is small. In order to investigate the effect of external magnetic field on the electron transport properties, Ar plasmas in high power impulse sputtering devices are considered. In the end, the electron relaxation properties of the Reid model under the influence of electric and magnetic fields are demonstrated.

  9. Finite element method for neutron transport. Part IV. A variational principle giving an upper bound for the lowest eigenvalue of the Boltzmann Equation

    Energy Technology Data Exchange (ETDEWEB)

    Ackroyd, R.T.


    A maximum principle for neutron transport in systems with extraneous sources is used with the method of source iteration to suggest a functional for a variational principle for self-sustaining systems. By using the general properties of the leakage and removal operators of the even-parity transport equation the variational principle is shown to give an upper bound to the lowest eigenvalue of the one-speed Boltzmann equation. Thus by making use of the method of Part III for a lower bound, the lowest eigenvalue can be bracketed. The variational principle leads to the finite element equations identical to those arising in the Williams/Galliara finite element formulation of the source-iteration methods, thus showing that the latter method always gives an upper bound to the lowest eigenvalue. Their upper bounds are very close to the exact value for some benchmark calculations.

  10. Numerical solution of the Fokker--Planck equations for a multi-species plasma

    Energy Technology Data Exchange (ETDEWEB)

    Killeen, J.; Mirin, A.A.


    Two numerical models used for studying collisional multispecies plasmas are described. The mathematical model is the Boltzmann kinetic equation with Fokker-Planck collision terms. A one-dimensional code and a two-dimensional code, used for the solution of the time-dependent Fokker-Planck equations for ion and electron distribution functions in velocity space, are described. The required equations and boundary conditions are derived and numerical techniques for their solution are given.

  11. Compressible fluids with Maxwell-type equations, the minimal coupling with electromagnetic field and the Stefan–Boltzmann law

    Energy Technology Data Exchange (ETDEWEB)

    Mendes, Albert C.R., E-mail: [Departamento de Física, Universidade Federal de Juiz de Fora, 36036-330, Juiz de Fora - MG (Brazil); Takakura, Flavio I., E-mail: [Departamento de Física, Universidade Federal de Juiz de Fora, 36036-330, Juiz de Fora - MG (Brazil); Abreu, Everton M.C., E-mail: [Grupo de Física Teórica e Matemática Física, Departamento de Física, Universidade Federal Rural do Rio de Janeiro, 23890-971, Seropédica - RJ (Brazil); Departamento de Física, Universidade Federal de Juiz de Fora, 36036-330, Juiz de Fora - MG (Brazil); Neto, Jorge Ananias, E-mail: [Departamento de Física, Universidade Federal de Juiz de Fora, 36036-330, Juiz de Fora - MG (Brazil)


    In this work we have obtained a higher-derivative Lagrangian for a charged fluid coupled with the electromagnetic fluid and the Dirac’s constraints analysis was discussed. A set of first-class constraints fixed by noncovariant gauge condition were obtained. The path integral formalism was used to obtain the partition function for the corresponding higher-derivative Hamiltonian and the Faddeev–Popov ansatz was used to construct an effective Lagrangian. Through the partition function, a Stefan–Boltzmann type law was obtained. - Highlights: • Higher-derivative Lagrangian for a charged fluid. • Electromagnetic coupling and Dirac’s constraint analysis. • Partition function through path integral formalism. • Stefan–Boltzmann-kind law through the partition function.

  12. Accuracy assessment of the linear Poisson-Boltzmann equation and reparametrization of the OBC generalized Born model for nucleic acids and nucleic acid-protein complexes. (United States)

    Fogolari, Federico; Corazza, Alessandra; Esposito, Gennaro


    The generalized Born model in the Onufriev, Bashford, and Case (Onufriev et al., Proteins: Struct Funct Genet 2004, 55, 383) implementation has emerged as one of the best compromises between accuracy and speed of computation. For simulations of nucleic acids, however, a number of issues should be addressed: (1) the generalized Born model is based on a linear model and the linearization of the reference Poisson-Boltmann equation may be questioned for highly charged systems as nucleic acids; (2) although much attention has been given to potentials, solvation forces could be much less sensitive to linearization than the potentials; and (3) the accuracy of the Onufriev-Bashford-Case (OBC) model for nucleic acids depends on fine tuning of parameters. Here, we show that the linearization of the Poisson Boltzmann equation has mild effects on computed forces, and that with optimal choice of the OBC model parameters, solvation forces, essential for molecular dynamics simulations, agree well with those computed using the reference Poisson-Boltzmann model. © 2015 Wiley Periodicals, Inc.

  13. Ludwig Boltzmann: Atomic genius

    Energy Technology Data Exchange (ETDEWEB)

    Cercignani, C. [Department of Mathematics, Politecnico di Milano (Italy)]. E-mail:


    On the centenary of the death of Ludwig Boltzmann, Carlo Cercignani examines the immense contributions of the man who pioneered our understanding of the atomic nature of matter. The man who first gave a convincing explanation of the irreversibility of the macroscopic world and the symmetry of the laws of physics was the Austrian physicist Ludwig Boltzmann, who tragically committed suicide 100 years ago this month. One of the key figures in the development of the atomic theory of matter, Boltzmann's fame will be forever linked to two fundamental contributions to science. The first was his interpretation of 'entropy' as a mathematically well-defined measure of the disorder of atoms. The second was his derivation of what is now known as the Boltzmann equation, which describes the statistical properties of a gas as made up of molecules. The equation, which described for the first time how a probability can evolve with time, allowed Boltzmann to explain why macroscopic phenomena are irreversible. The key point is that while microscopic objects like atoms can behave reversibly, we never see broken coffee cups reforming because it would involve a long series of highly improbable interactions - and not because it is forbidden by the laws of physics. (U.K.)

  14. On a two-relaxation-time D2Q9 lattice Boltzmann model for the Navier-Stokes equations (United States)

    Zhao, Weifeng; Wang, Liang; Yong, Wen-An


    In this paper, we are concerned with the stability of some lattice kinetic schemes. First, we show that a recently proposed lattice kinetic scheme is a two-relaxation-time model different from those in the literature. Second, we analyze the stability of the model by verifying the Onsager-like relation. In addition, a necessary stability criterion for hyperbolic relaxation systems is adapted to the lattice Boltzmann method. As an application of this criterion, we find some necessary stability conditions for a previously proposed lattice kinetic scheme. Numerical experiments are conducted to validate the necessary stability conditions.

  15. Compressible Fluids with Maxwell-type equations, the minimal coupling with electromagnetic field and the Stefan-Boltzmann law

    CERN Document Server

    Mendes, Albert C R; Abreu, Everton M C; Neto, Jorge Ananias


    In this work we have obtained a higher-derivative Lagrangian for a charged fluid coupled with the electromagnetic fluid and the Dirac's constraints analysis was discussed. A set of first-class constraints fixed by noncovariant gauge condition was obtained. The path integral formalism was used to obtain the partition function for the corresponding higher-derivative Hamiltonian and the Faddeev-Popov ansatz was used to construct an effective Lagrangian. Through the partition function, a Stefan-Boltzmann type law was obtained.

  16. Compressible fluids with Maxwell-type equations, the minimal coupling with electromagnetic field and the Stefan-Boltzmann law (United States)

    Mendes, Albert C. R.; Takakura, Flavio I.; Abreu, Everton M. C.; Neto, Jorge Ananias


    In this work we have obtained a higher-derivative Lagrangian for a charged fluid coupled with the electromagnetic fluid and the Dirac's constraints analysis was discussed. A set of first-class constraints fixed by noncovariant gauge condition were obtained. The path integral formalism was used to obtain the partition function for the corresponding higher-derivative Hamiltonian and the Faddeev-Popov ansatz was used to construct an effective Lagrangian. Through the partition function, a Stefan-Boltzmann type law was obtained.

  17. Influence of non-collisional laser heating on the electron dynamics in dielectric materials

    CERN Document Server

    Barilleau, L; Chimier, B; Geoffroy, G; Tikhonchuk, V


    The electron dynamics in dielectric materials induced by intense femtosecond laser pulses is theoretically addressed. The laser driven temporal evolution of the energy distribution of electrons in the conduction band is described by a kinetic Boltzmann equation. In addition to the collisional processes for energy transfer such as electron-phonon-photon and electron-electron interactions, a non-collisional process for photon absorption in the conduction band is included. It relies on direct transitions between sub-bands of the conduction band through multiphoton absorption. This mechanism is shown to significantly contribute to the laser heating of conduction electrons for large enough laser intensities. It also increases the time required for the electron distribution to reach the equilibrium state as described by the Fermi-Dirac statistics. Quantitative results are provided for quartz irradiated by a femtosecond laser pulse with a wavelength of 800 nm and for intensities in the range of tens of TW/cm$^2$, lo...

  18. Discrete effect on the halfway bounce-back boundary condition of multiple-relaxation-time lattice Boltzmann model for convection-diffusion equations (United States)

    Cui, Shuqi; Hong, Ning; Shi, Baochang; Chai, Zhenhua


    In this paper, we will focus on the multiple-relaxation-time (MRT) lattice Boltzmann model for two-dimensional convection-diffusion equations (CDEs), and analyze the discrete effect on the halfway bounce-back (HBB) boundary condition (or sometimes called bounce-back boundary condition) of the MRT model where three different discrete velocity models are considered. We first present a theoretical analysis on the discrete effect of the HBB boundary condition for the simple problems with a parabolic distribution in the x or y direction, and a numerical slip proportional to the second-order of lattice spacing is observed at the boundary, which means that the MRT model has a second-order convergence rate in space. The theoretical analysis also shows that the numerical slip can be eliminated in the MRT model through tuning the free relaxation parameter corresponding to the second-order moment, while it cannot be removed in the single-relaxation-time model or the Bhatnagar-Gross-Krook model unless the relaxation parameter related to the diffusion coefficient is set to be a special value. We then perform some simulations to confirm our theoretical results, and find that the numerical results are consistent with our theoretical analysis. Finally, we would also like to point out the present analysis can be extended to other boundary conditions of lattice Boltzmann models for CDEs.

  19. Three-dimensional coupled double-distribution-function lattice Boltzmann models for compressible Navier-Stokes equations (United States)

    Qiu, Ruo-Fan; You, Yan-Cheng; Zhu, Cheng-Xiang; Chen, Rong-Qian; Zhu, Jian-Feng


    Two three-dimensional (3D) lattice Boltzmann models in the framework of coupled double-distribution-function approach for compressible flows, in which specific-heat ratio and Prandtl number can be adjustable, are developed in this paper. The main differences between the two models are discrete equilibrium density and total energy distribution function. One is the D3Q25 model obtained from spherical function, and the other is the D3Q27 standard lattice model obtained from Hermite expansions of the corresponding continuous equilibrium distribution functions. The two models are tested by numerical simulations of some typical compressible flows, and their numerical stability and precision are also analysed. The results indicate that the two models are capable for supersonic flows, while the one from Hermite expansions is not suitable for compressible flows with shock waves.

  20. Discontinuous finite element space-angle treatment of the first order linear Boltzmann transport equation with magnetic fields: Application to MRI-guided radiotherapy. (United States)

    St Aubin, J; Keyvanloo, A; Fallone, B G


    The advent of magnetic resonance imaging (MRI) guided radiotherapy systems demands the incorporation of the magnetic field into dose calculation algorithms of treatment planning systems. This is due to the fact that the Lorentz force of the magnetic field perturbs the path of the relativistic electrons, hence altering the dose deposited by them. Building on the previous work, the authors have developed a discontinuous finite element space-angle treatment of the linear Boltzmann transport equation to accurately account for the effects of magnetic fields on radiotherapy doses. The authors present a detailed description of their new formalism and compare its accuracy to geant4 Monte Carlo calculations for magnetic fields parallel and perpendicular to the radiation beam at field strengths of 0.5 and 3 T for an inhomogeneous 3D slab geometry phantom comprising water, bone, and air or lung. The accuracy of the authors' new formalism was determined using a gamma analysis with a 2%/2 mm criterion. Greater than 98.9% of all points analyzed passed the 2%/2 mm gamma criterion for the field strengths and orientations tested. The authors have benchmarked their new formalism against Monte Carlo in a challenging radiation transport problem with a high density material (bone) directly adjacent to a very low density material (dry air at STP) where the effects of the magnetic field dominate collisions. A discontinuous finite element space-angle approach has been proven to be an accurate method for solving the linear Boltzmann transport equation with magnetic fields for cases relevant to MRI guided radiotherapy. The authors have validated the accuracy of this novel technique against geant4, even in cases of strong magnetic field strengths and low density air.

  1. Spectroscopic characterization of an ultrashort-pulse-laser-driven Ar cluster target incorporating both Boltzmann and particle-in-cell models. (United States)

    Sherrill, M E; Abdallah, J; Csanak, G; Dodd, E S; Fukuda, Y; Akahane, Y; Aoyama, M; Inoue, N; Ueda, H; Yamakawa, K; Faenov, A Ya; Magunov, A I; Pikuz, T A; Skobelev, I Yu


    A model that solves simultaneously both the electron and atomic kinetics was used to generate a synthetic He alpha and satellite x-ray spectra to characterize a high intensity ultrashort laser driven Ar cluster target experiment. In particular, level populations were obtained from a detailed collisional-radiative model where collisional rates were computed from a time varying electron distribution function obtained from the solution of the zero-dimensional Boltzmann equation. In addition, a particle-in-cell simulation was used to model the laser interaction with the cluster target and provided the initial electron energy distribution function (EEDF) for the Boltzmann solver. This study suggests that a high density average, high, of 3.2 x 10(20) cm(-3) was held by the system for a time, delta tau, of 5.7 ps, and during this time the plasma was in a highly nonequilibrium state in both the EEDF and the ion level populations.

  2. Polarised photon and flavoured lepton quantum Boltzmann equations in the early universe; Polarisierte Photon- und geflavourte Lepton-Quantenboltzmanngleichungen im fruehen Universum

    Energy Technology Data Exchange (ETDEWEB)

    Fidler, Christian


    Polarisation and Nongaussianity are expected to play a central role in future studies of the cosmic microwave background radiation. Polarisation can be split into a divergence-like E-mode and a curl-like B-mode, of which the later can only be induced by primordial gravitational waves (tensor fluctuations of the metric) at leading order. Nongaussianity is not generated at first order and is directly proportional to the primordial Nongaussianity of inflation. Thus B-mode polarisation and Nongaussianity constrain inflation models directly. While E-mode polarisation has already been detected and is being observed with increasing precision, B-mode polarisation and Nongaussianity remains elusive. The absence of B-mode polarisation when the primordial fluctuations are purely scalar holds, however, only in linear perturbation theory. B-mode polarisation is also generated from scalar sources in second order, which may constitute an important background to the search for primordial gravitational waves. While such an effect would naturally be expected to be relevant at tensor-to-scalar ratios of order 10{sup -5}, which is the size of perturbations in the microwave background, only a full second order calculation can tell whether there are no enhancements. For Nongaussianity the situation is analogous: At second order intrinsic Nongaussianities are induced to the spectrum, which may be an important background to the primordial Nongaussianity. After the full second-order Boltzmann equations for the cosmological evolution of the polarised radiation distribution have become available, I focused on the novel sources to B-mode polarisation that appear in the second-order collision term, which have not been calculated before. In my PHD thesis I developed a numerical code, which solves the second order Boltzmann hierarchy and calculates the C{sub l}{sup BB}-spectrum.

  3. Lattices for the lattice Boltzmann method. (United States)

    Chikatamarla, Shyam S; Karlin, Iliya V


    A recently introduced theory of higher-order lattice Boltzmann models [Chikatamarla and Karlin, Phys. Rev. Lett. 97, 190601 (2006)] is elaborated in detail. A general theory of the construction of lattice Boltzmann models as an approximation to the Boltzmann equation is presented. New lattices are found in all three dimensions and are classified according to their accuracy (degree of approximation of the Boltzmann equation). The numerical stability of these lattices is argued based on the entropy principle. The efficiency and accuracy of many new lattices are demonstrated via simulations in all three dimensions.

  4. Electron attachment coefficient in low E/N regions and a discussion of discharge-instability in KrF laser. ; Analysis by logarithm transformed Boltzmann equation. Tei E/N ryoiki no denshi fuchaku keisu to KrF laser reiki hoden no fuanteisei ni kansuru ichi kosatsu. ; Tai su henkan Boltzmann hoteishiki ni yoru kaiseki

    Energy Technology Data Exchange (ETDEWEB)

    Kawakami, H.; Urabe, J.; Yukimura, K. (Doshisha Univ., Kyoto (Japan))


    In a discharge excitation rare gas halide excima laser, uniform generation and stable maintenance of the excited discharge determines the laser characteristics. In this report, an approximate solution was obtained on the Boltzmann equation (frequently used for the theoretical analysis of this laser) to examine the nature of the solution. By optimizing the conversion of the variables, calculation of an electron swarm parameter in the hitherto uncertain range of the low conversion electric field was made possible, giving a generation mechanism of the uncertainty of the excited dischareg. The results are summarized as below. (1) The Boltzmann equation gives a linear solution for a logarithmic value of an electron energy in the range of low conversion electric field. (2) Time-wise responce ability between the measured voltage, current characteristics of the excitation discharge was clarified and the attachment and ionization coefficients calculated by Boltzmann equation. (3) Dependency of the attachment coefficient on the partial pressure of fluorine and kripton was examined, and the attachment coefficient was found to increase with the increase of the partial pressure for the both cases. 20 refs., 9 figs., 2 tabs.

  5. A Review on the Use of Grid-Based Boltzmann Equation Solvers for Dose Calculation in External Photon Beam Treatment Planning

    Directory of Open Access Journals (Sweden)

    Monica W. K. Kan


    Full Text Available Deterministic linear Boltzmann transport equation (D-LBTE solvers have recently been developed, and one of the latest available software codes, Acuros XB, has been implemented in a commercial treatment planning system for radiotherapy photon beam dose calculation. One of the major limitations of most commercially available model-based algorithms for photon dose calculation is the ability to account for the effect of electron transport. This induces some errors in patient dose calculations, especially near heterogeneous interfaces between low and high density media such as tissue/lung interfaces. D-LBTE solvers have a high potential of producing accurate dose distributions in and near heterogeneous media in the human body. Extensive previous investigations have proved that D-LBTE solvers were able to produce comparable dose calculation accuracy as Monte Carlo methods with a reasonable speed good enough for clinical use. The current paper reviews the dosimetric evaluations of D-LBTE solvers for external beam photon radiotherapy. This content summarizes and discusses dosimetric validations for D-LBTE solvers in both homogeneous and heterogeneous media under different circumstances and also the clinical impact on various diseases due to the conversion of dose calculation from a conventional convolution/superposition algorithm to a recently released D-LBTE solver.

  6. Comparison of the lattice Boltzmann equation and discrete unified gas-kinetic scheme methods for direct numerical simulation of decaying turbulent flows. (United States)

    Wang, Peng; Wang, Lian-Ping; Guo, Zhaoli


    The main objective of this work is to perform a detailed comparison of the lattice Boltzmann equation (LBE) and the recently developed discrete unified gas-kinetic scheme (DUGKS) methods for direct numerical simulation (DNS) of the decaying homogeneous isotropic turbulence and the Kida vortex flow in a periodic box. The flow fields and key statistical quantities computed by both methods are compared with those from the pseudospectral method at both low and moderate Reynolds numbers. The results show that the LBE is more accurate and efficient than the DUGKS, but the latter has a superior numerical stability, particularly for high Reynolds number flows. In addition, we conclude that the DUGKS can adequately resolve the flow when the minimum spatial resolution parameter k_{max}η>3, where k_{max} is the maximum resolved wave number and η is the flow Kolmogorov length. This resolution requirement can be contrasted with the requirements of k_{max}η>1 for the pseudospectral method and k_{max}η>2 for the LBE. It should be emphasized that although more validations should be conducted before the DUGKS can be called a viable tool for DNS of turbulent flows, the present work contributes to the overall assessment of the DUGKS, and it provides a basis for further applications of DUGKS in studying the physics of turbulent flows.

  7. Lattice Boltzmann model for incompressible axisymmetric flows. (United States)

    Chen, Sheng; Tölke, Jonas; Geller, Sebastian; Krafczyk, Manfred


    A lattice Boltzmann model for incompressible axisymmetric flow is proposed in this paper. Unlike previous axisymmetric lattice Boltzmann models, which were based on "primitive-variables" Navier-Stokes equations, the target macroscopic equations of the present model are vorticity-stream-function formulations. Due to the intrinsic features of vorticity-stream-function formulations, the present model is more efficient, more stable, and much simpler than the existing models. The advantages of the present model are validated by numerical experiments.

  8. Boltzmann equation analysis of electrons swarm parameters and properties of excited particle number densities in Xe/Ne plasmas. Laser absorption effect; Xe/Ne plasma chudenshi yuso keisu narabi ni reiki ryushisu mitsudo tokusei no Boltzmann hoteishiki kaiseki. Laser ko kyushu koka

    Energy Technology Data Exchange (ETDEWEB)

    Uchida, S.; Sugawara, H.; Ventzek, P.; Sakai, Y. [Hokkaido University, Sapporo (Japan)


    Xe/Ne plasmas are important for plasma display panels and VUV light sources. However, reactions between electrons and excited particles in the mixtures are so complicated that influence of the reactions on the plasma properties is not understood well. In this work, taking account of reactions through which electrons are produced, such as cumulative and Penning ionization, and of transition between excited levels, the electron and excited particle properties in Xe/Ne plasmas are calculated using the Boltzmann equation. The ionization coefficient and electron drift velocity agreed with experimental data. The influence of laser absorption in Xe/Ne plasmas on the plasma properties is also discussed. 25 refs., 15 figs.

  9. Contribution to the solution of the multigroup Boltzmann equation by the determinist methods and the Monte Carlo method; Contribution a la resolution de l`equation de Bolztmann en multigroupe par les methodes deterministes et Monte-Carlo

    Energy Technology Data Exchange (ETDEWEB)

    Li, M


    In this thesis, two methods for solving the multigroup Boltzmann equation have been studied: the interface-current method and the Monte Carlo method. A new version of interface-current (IC) method has been develop in the TDT code at SERMA, where the currents of interface are represented by piecewise constant functions in the solid angle space. The convergence of this method to the collision probability (CP) method has been tested. Since the tracking technique is used for both the IC and CP methods, it is necessary to normalize he collision probabilities obtained by this technique. Several methods for this object have been studied and implemented in our code, we have compared their performances and chosen the best one as the standard choice. The transfer matrix treatment has been a long-standing difficulty for the multigroup Monte Carlo method: when the cross-sections are converted into multigroup form, important negative parts will appear in the angular transfer laws represented by low-order Legendre polynomials. Several methods based on the preservation of the first moments, such as the discrete angles methods and the equally-probable step function method, have been studied and implemented in the TRIMARAN-II code. Since none of these codes has been satisfactory, a new method, the non equally-probably step function method, has been proposed and realized in our code. The comparisons for these methods have been done in several aspects: the preservation of the moments required, the calculation of a criticality problem and the calculation of a neutron-transfer in water problem. The results have showed that the new method is the best one in all these comparisons, and we have proposed that it should be a standard choice for the multigroup transfer matrix. (author) 76 refs.

  10. Lattice Boltzmann approach for complex nonequilibrium flows. (United States)

    Montessori, A; Prestininzi, P; La Rocca, M; Succi, S


    We present a lattice Boltzmann realization of Grad's extended hydrodynamic approach to nonequilibrium flows. This is achieved by using higher-order isotropic lattices coupled with a higher-order regularization procedure. The method is assessed for flow across parallel plates and three-dimensional flows in porous media, showing excellent agreement of the mass flow with analytical and numerical solutions of the Boltzmann equation across the full range of Knudsen numbers, from the hydrodynamic regime to ballistic motion.

  11. Transverse-momentum spectra and nuclear modification factor using Boltzmann Transport Equation with flow in Pb+Pb collisions at √(s{sub NN}) = 2.76 TeV

    Energy Technology Data Exchange (ETDEWEB)

    Tripathy, Sushanta; Khuntia, Arvind; Tiwari, Swatantra Kumar; Sahoo, Raghunath [Indian Institute of Technology Indore, Discipline of Physics, School of Basic Sciences, Indore (India)


    In the continuation of our previous work, the transverse-momentum (p{sub T}) spectra and nuclear modification factor (R{sub AA}) are derived using the relaxation time approximation of Boltzmann Transport Equation (BTE). The initial p{sub T}-distribution used to describe p + p collisions has been studied with the perturbative-Quantum Chromodynamics (pQCD) inspired power-law distribution, Hagedorn's empirical formula and with the Tsallis non-extensive statistical distribution. The non-extensive Tsallis distribution is observed to describe the complete range of the transverse-momentum spectra. The Boltzmann-Gibbs Blast Wave (BGBW) distribution is used as the equilibrium distribution in the present formalism, to describe the p{sub T}-distribution and nuclear modification factor in nucleus-nucleus collisions. The experimental data for Pb+Pb collisions at √(s{sub NN}) = 2.76 TeV at the Large Hadron Collider at CERN have been analyzed for pions, kaons, protons, K{sup *0} and φ. It is observed that the present formalism while explaining the transverse-momentum spectra up to 5 GeV/c, explains the nuclear modification factor very well up to 8 GeV/c in p{sub T} for all these particles except for protons. R{sub AA} is found to be independent of the degree of non-extensivity, q{sub pp} after p{sub T} ∝ 8 GeV/c. (orig.)

  12. Multiple-Relaxation-Time Lattice Boltzmann Models in 3D (United States)

    dHumieres, Dominique; Ginzburg, Irina; Krafczyk, Manfred; Lallemand, Pierre; Luo, Li-Shi; Bushnell, Dennis M. (Technical Monitor)


    This article provides a concise exposition of the multiple-relaxation-time lattice Boltzmann equation, with examples of fifteen-velocity and nineteen-velocity models in three dimensions. Simulation of a diagonally lid-driven cavity flow in three dimensions at Re=500 and 2000 is performed. The results clearly demonstrate the superior numerical stability of the multiple-relaxation-time lattice Boltzmann equation over the popular lattice Bhatnagar-Gross-Krook equation.

  13. A new Green's function Monte Carlo algorithm for the solution of the two-dimensional nonlinear Poisson–Boltzmann equation: Application to the modeling of the communication breakdown problem in space vehicles during re-entry

    Energy Technology Data Exchange (ETDEWEB)

    Chatterjee, Kausik, E-mail: [Strategic and Military Space Division, Space Dynamics Laboratory, North Logan, UT 84341 (United States); Center for Atmospheric and Space Sciences, Utah State University, Logan, UT 84322 (United States); Roadcap, John R., E-mail: [Air Force Research Laboratory, Kirtland AFB, NM 87117 (United States); Singh, Surendra, E-mail: [Department of Electrical Engineering, The University of Tulsa, Tulsa, OK 74104 (United States)


    The objective of this paper is the exposition of a recently-developed, novel Green's function Monte Carlo (GFMC) algorithm for the solution of nonlinear partial differential equations and its application to the modeling of the plasma sheath region around a cylindrical conducting object, carrying a potential and moving at low speeds through an otherwise neutral medium. The plasma sheath is modeled in equilibrium through the GFMC solution of the nonlinear Poisson–Boltzmann (NPB) equation. The traditional Monte Carlo based approaches for the solution of nonlinear equations are iterative in nature, involving branching stochastic processes which are used to calculate linear functionals of the solution of nonlinear integral equations. Over the last several years, one of the authors of this paper, K. Chatterjee has been developing a philosophically-different approach, where the linearization of the equation of interest is not required and hence there is no need for iteration and the simulation of branching processes. Instead, an approximate expression for the Green's function is obtained using perturbation theory, which is used to formulate the random walk equations within the problem sub-domains where the random walker makes its walks. However, as a trade-off, the dimensions of these sub-domains have to be restricted by the limitations imposed by perturbation theory. The greatest advantage of this approach is the ease and simplicity of parallelization stemming from the lack of the need for iteration, as a result of which the parallelization procedure is identical to the parallelization procedure for the GFMC solution of a linear problem. The application area of interest is in the modeling of the communication breakdown problem during a space vehicle's re-entry into the atmosphere. However, additional application areas are being explored in the modeling of electromagnetic propagation through the atmosphere/ionosphere in UHF/GPS applications.

  14. Finite Boltzmann schemes

    NARCIS (Netherlands)

    Sman, van der R.G.M.


    In the special case of relaxation parameter = 1 lattice Boltzmann schemes for (convection) diffusion and fluid flow are equivalent to finite difference/volume (FD) schemes, and are thus coined finite Boltzmann (FB) schemes. We show that the equivalence is inherent to the homology of the

  15. Collisional and collisionless expansion of Yukawa balls. (United States)

    Piel, Alexander; Goree, John A


    The expansion of Yukawa balls is studied by means of molecular dynamics simulations of collisionless and collisional situations. High computation speed was achieved by using the parallel computing power of graphics processing units. When the radius of the Yukawa ball is large compared to the shielding length, the expansion process starts with the blow-off of the outermost layer. A rarefactive wave subsequently propagates radially inward at the speed of longitudinal phonons. This mechanism is fundamentally different from Coulomb explosions, which employ a self-similar expansion of the entire system. In the collisionless limit, the outer layers carry away most of the available energy. The simulations are compared with analytical estimates. In the collisional case, the expansion process can be described by a nonlinear diffusion equation that is a special case of the porous medium equation.

  16. Boltzmann-Electron Model in Aleph.

    Energy Technology Data Exchange (ETDEWEB)

    Hughes, Thomas Patrick; Hooper, Russell


    We apply the Boltzmann-electron model in the electrostatic, particle-in-cell, finite- element code Aleph to a plasma sheath. By assuming a Boltzmann energy distribution for the electrons, the model eliminates the need to resolve the electron plasma fre- quency, and avoids the numerical "grid instability" that can cause unphysical heating of electrons. This allows much larger timesteps to be used than with kinetic electrons. Ions are treated with the standard PIC algorithm. The Boltzmann-electron model re- quires solution of a nonlinear Poisson equation, for which we use an iterative Newton solver (NOX) from the Trilinos Project. Results for the spatial variation of density and voltage in the plasma sheath agree well with an analytic model

  17. Coulomb collisional relaxation process of ion beams in magnetized plasmas


    Nishimura, Y.


    An orbit following code is developed to calculate ion beam trajectories in magnetized plasmas. The equation of motion (the Newton's equation) is solved including the Lorentz force term and Coulomb collisional relaxation term. Furthermore, a new algorithm is introduced by applying perturbation method regarding the collision term as a small term. The reduction of computation time is suggested.

  18. Generalized Boltzmann formalism for oscillating neutrinos (United States)

    Strack, P.; Burrows, A.


    In the standard approaches to neutrino transport in the simulation of core-collapse supernovas, one will often start from the classical Boltzmann equation for the neutrino’s spatial, temporal, and spectral evolution. For each neutrino species, and its antiparticle, the classical density in phase space, or the associated specific intensity, will be calculated as a function of time. The neutrino radiation is coupled to matter by source and sink terms on the “right-hand side” of the transport equation and together with the equations of hydrodynamics this set of coupled partial differential equations for classical densities describes, in principle, the evolution of core collapse and explosion. However, with the possibility of neutrino oscillations between species, a purely quantum-physical effect, how to generalize this set of Boltzmann equations for classical quantities to reflect oscillation physics has not been clear. To date, the formalisms developed have retained the character of quantum operator physics involving complex quantities and have not been suitable for easy incorporation into standard supernova codes. In this paper, we derive generalized Boltzmann equations for quasiclassical, real-valued phase-space densities that retain all the standard oscillation phenomenology, including the matter-enhanced resonant flavor conversion (Mikheev-Smirnov-Wolfenstein effect), neutrino self-interactions, and the interplay between decohering matter coupling and flavor oscillations. With this formalism, any code(s) that can now handle the solution of the classical Boltzmann or transport equation can easily be generalized to include neutrino oscillations in a quantum-physically consistent fashion.

  19. General structure of quantum collisional models (United States)

    Vacchini, Bassano


    We point to the connection between a recently introduced class of non-Markovian master equations and the general structure of quantum collisional models. The basic construction relies on three basic ingredients: a collection of time dependent completely positive maps, a completely positive trace preserving transformation and a waiting time distribution characterizing a renewal process. The relationship between this construction and a Lindblad dynamics is clarified by expressing the solution of a Lindblad master equation in terms of demixtures over different stochastic trajectories for the statistical operator weighted by suitable probabilities on the trajectory space.

  20. Simulations of a molecular plasma in collisional-radiative nonequilibrium (United States)

    Cambier, Jean-Luc; Moreau, Stephane


    A code for the simulation of nonequilibrium plasmas is being developed, with the capability to couple the plasma fluid-dynamics for a single fluid with a collisional-radiative model, where electronic states are treated as separate species. The model allows for non-Boltzmann distribution of the electronic states. Deviations from the Boltzmann distributions are expected to occur in the rapidly ionizing regime behind a strong shock or in the recombining regime during a fast expansion. This additional step in modeling complexity is expected to yield more accurate predictions of the nonequilibrium state and the radiation spectrum and intensity. An attempt at extending the code to molecular plasma flows is presented. The numerical techniques used, the thermochemical model, and the results of some numerical tests are described.

  1. Weakly Collisional and Collisionless Astrophysical Plasmas

    DEFF Research Database (Denmark)

    Berlok, Thomas

    investigate helium mixing in the weakly collisional intracluster medium of galaxy clusters using Braginskii MHD. Secondly, we present a newly developed Vlasov-fluid code which can be used for studying fully collisionless plasmas such as the solar wind and hot accretions flows. The equations of Braginskii MHD...... are used to study weakly collisional, stratified atmospheres which offer a useful model of the intracluster medium of galaxy clusters. Using linear theory and computer simulations, we study instabilities that feed off thermal and compositional gradients. We find that these instabilities lead to vigorous...... mixing of the composition and discuss the potential consequences for X-ray observations of galaxy clusters. Collisionless plasmas can be subject to microscale velocity-space instabilities which are not well-described by Braginskii MHD. In contrast, Vlasov-fluid theory captures all the kinetic phenomena...

  2. Gli atomi di Boltzmann

    CERN Document Server

    Lindley, David


    Ludwig Boltzmann (1844-1906) è il fisico e matematico austriaco che negli ultimi decenni dell'Ottocento e ancora ai primi del Novecento lottò contro l'opinione dominante tra gli scienziati dell'epoca per affermare la teoria atomica della materia. È noto come con Albert Einstein e fino a oggi la fisica si sia sviluppata e abbia celebrato i propri trionfi lungo le linee anticipate da Boltzmann. La controversia con Mach non riguardava soltanto l'esistenza degli atomi, ma l'intero modo di fare fisica che Boltzmann non riteneva di dover limitare allo studio di quantità misurabili, introducendo invece spiegazioni più elaborate basate su ipotesi più ampie.

  3. Entropy a la Boltzmann

    Indian Academy of Sciences (India)

    Home; Journals; Resonance – Journal of Science Education; Volume 6; Issue 9. Entropy à la Boltzmann. Jayanta K Bhattacharjee. General Article Volume 6 Issue 9 September 2001 pp 19-34. Fulltext. Click here to view fulltext PDF. Permanent link: Author Affiliations.

  4. Ludluig Eduard Boltzmann

    Indian Academy of Sciences (India)

    Page 1. CPMGlKAlBGE-340/2001. Licenced to post WPP(E) No.6. Resonance - September 2001. Ludluig Eduard Boltzmann. (1844 - 19(6). Registered with Registrar of Newspapers in India vide Regn. No. 66273196. ISSN 0971-8044.

  5. Lattice Boltzmann Large Eddy Simulation Model of MHD

    CERN Document Server

    Flint, Christopher


    The work of Ansumali \\textit{et al.}\\cite{Ansumali} is extended to Two Dimensional Magnetohydrodynamic (MHD) turbulence in which energy is cascaded to small spatial scales and thus requires subgrid modeling. Applying large eddy simulation (LES) modeling of the macroscopic fluid equations results in the need to apply ad-hoc closure schemes. LES is applied to a suitable mesoscopic lattice Boltzmann representation from which one can recover the MHD equations in the long wavelength, long time scale Chapman-Enskog limit (i.e., the Knudsen limit). Thus on first performing filter width expansions on the lattice Boltzmann equations followed by the standard small Knudsen expansion on the filtered lattice Boltzmann system results in a closed set of MHD turbulence equations provided we enforce the physical constraint that the subgrid effects first enter the dynamics at the transport time scales. In particular, a multi-time relaxation collision operator is considered for the density distribution function and a single rel...

  6. An advanced time-dependent collisional-radiative model of helium plasma discharges (United States)

    Claustre, J.; Boukandou-Mombo, C.; Margot, J.; Matte, J.-P.; Vidal, F.


    A new spatially averaged time-dependent collisional-radiative model for helium plasmas, coupled to the electron Boltzmann equation (EBE), has been developed. Its main novelties are: (1) full time dependence for both the multi-species kinetics and the EBE. It is shown that this is necessary to correctly simulate discharges where the parameters vary on nanoseconds-microsecond timescales. (2) All electron processes are accounted for accurately. In particular, for the various ionization and recombination processes, free electrons are added or removed at the appropriate energy, with the appropriate interpolation on the energy grid. (3) The energy dependence of the electron loss by ambipolar diffusion is taken into account approximately. (4) All of the processes which are known to be important in helium discharges for pressure P≤slant 760 Torr are included, and 42 energy levels up to n = 6, where n is the main quantum number, are taken into account. Atomic and molecular ions, as well as excimers, are also included. (5) The gas temperature is calculated self-consistently. The model is validated through comparisons with known numerical steady-state results of Santos et al (2014 J. Phys. D. 47 265201) which they compared to their experimental results, and good agreement is obtained for their measured quantities. It is then applied to post-discharge decay cases with very short power decay times. The time evolution of the population densities and reaction rates are analyzed in detail with emphasis on the observed large increase of the metastable density.

  7. The exact form of the Bohm criterion for a collisional plasma

    CERN Document Server

    Tsankov, Tsanko Vaskov


    A long-standing debate in the literature about the kinetic form of the Bohm criterion is resolved for plasmas with single positive ion species when transport is dominated by charge exchange collisions. The solution of the Boltzmann equation for the ions gives the exact form free of any divergence and contains an additional term that is not included in the classical result. This term includes collisional and geometric effects and leads to a noticeable correction. Further, the question is addressed whether the space charge argument at the bottom of the Bohm criterion can actually lead to a meaningful definition of the transition point between bulk and sheath. The analysis is supported by a numerical model and experiments, showing excellent agreement throughout. As a novelty in diagnostics, the theoretical results allow from the ion velocity distribution function (IVDF), measured at the wall, a reconstruction of the IVDF and the electric field at any point in the plasma. This property is used to reconstruct non-...

  8. Collisional Growth of Planetesimals (United States)

    Schroeter, Thomas; Nyffenegger, Oliver; Benz, Willy


    Motivation ---------- In the current planet formation paradigm, planets form through collisions. While the size of the primordial planetesimals is not yet established, it is recognized that this collision cascade plays an crucial role not only in determining the growth rate of the bodies but also in determining their internal structure as well as bulk chemical composition. In the case of giant gaseous planets, the nucleated instability scenario begins with the formation of critical cores of order 10 Earth masses through this very process as well. Hence, the process of collisional growth underpins the early formation of all planets massive or not. The most natural and physically appropriate approach for studying these processes is to perform N-body simulations. Unfortunately, simulating the collisional dynamics of a very large number of bodies (several hundreds of millions) over very long timescales (hundred million orbits) turns out to be computationally prohibitive. Therefore, this approach remains for the moment limited to the late stages of formation when the number of bodies has become tractable. Statistical approaches while allowing treating an arbitrary number of bodies do not provide individual collision histories and therefore cannot address some of the most important issues related to the internal structure of young planets. By introducing an orbit averaging method based on a Monte Carlo technique that allows integrating the system using time steps much longer than an orbital period, we are in a position to follow the individual collision history of several tens of millions of bodies over long evolution times. Hence, this method effectively bridges the gap between the early small planetesimals and the large embryos for which the evolution can be followed using an N-body approach. Approach -------- The method is based on an orbit averaging Monte Carlo process. The essential advantage of the method is to allow for time steps that are not dictated by the

  9. FLYCHK Collisional-Radiative Code (United States)

    SRD 160 FLYCHK Collisional-Radiative Code (Web, free access)   FLYCHK provides a capability to generate atomic level populations and charge state distributions for low-Z to mid-Z elements under NLTE conditions.

  10. Collisional Effect On Magnetosonic Solitons In A Dusty Plasma Slab ...

    African Journals Online (AJOL)

    An analytical investigation of collisional effect on magnetosonic solitons in a dusty plasma slab is presented. We have derived and presented solutions of nonlinear magetohydrodynamic equations for a warm dusty magnetoplasma. It is observed that, our work could be considered a general case for magnetosonic solutions ...

  11. Ludwig Edward Boltzmann


    Rajasekar, S.; Athavan, N.


    In this manuscript we present a brief life history of Ludwig Edward Boltzmann and his achivements. Particularly, we discuss his H-theorem, his work on entropy and statistical interpretation of second-law of thermodynamics. We point out his some other contributions in physics, characteristics of his work, his strong support on atomism, character of his personality and relationship with his students and final part of his life.

  12. Mathematical model of Boltzmann's sigmoidal equation applicable ...

    Indian Academy of Sciences (India)

    Ingeniería-Química, COARA—Universidad Autónoma de San Luis Potosí, Matehuala, San Luis Potosí, Mexico; Instituto Politécnico Nacional, CICATA Legaria, Calzada Legaria No. 694, Colonia Irrigación, 11500 Ciudad de México, Mexico; Departamento de Ingeniería Agrícola, DICIVA, Universidad de Guanajuato, Campus ...

  13. Mathematical model of Boltzmann's sigmoidal equation applicable ...

    Indian Academy of Sciences (India)


    Aug 18, 2017 ... Abstract. In this work, we present the stoichiometric behaviour of Ba2+ and Sr2+ when they are deposited to make a solid solution of barium strontium titanate. Bax Sr1−x TiO3 (BST) thin films of nanometric order on a quartz substrate were obtained by means of in-situ RF-magnetron co-sputtering at 495.

  14. Galilean-invariant lattice-Boltzmann models with H theorem. (United States)

    Boghosian, Bruce M; Love, Peter J; Coveney, Peter V; Karlin, Iliya V; Succi, Sauro; Yepez, Jeffrey


    We demonstrate that the requirement of Galilean invariance determines the choice of H function for a wide class of entropic lattice-Boltzmann models for the incompressible Navier-Stokes equations. The required H function has the form of the Burg entropy for D=2, and of a Tsallis entropy with q=1-(2/D) for D>2, where D is the number of spatial dimensions. We use this observation to construct a fully explicit, unconditionally stable, Galilean-invariant, lattice-Boltzmann model for the incompressible Navier-Stokes equations, for which attainable Reynolds number is limited only by grid resolution.

  15. Nonequilibrium phenomena in QCD and BEC. Boltzmann and beyond

    Energy Technology Data Exchange (ETDEWEB)

    Stockamp, T.


    In chapter 2 we chose the real time formalism to discuss some basic principles in quantum field theory at finite temperature. This enables us to derive the quantum Boltzmann equation from the Schwinger-Dyson series. We then shortly introduce the basic concepts of QCD which are needed to understand the physics of QGP formation. After a detailed account on the bottom-up scenario we show the consistency of this approach by a diagramatical analysis of the relevant Boltzmann collision integrals. Chapter 3 deals with BEC dynamics out of equilibrium. After an introduction to the fundamental theoretical tool - namely the Gross-Pitaevskii equation - we focus on a generalization to finite temperature developed by Zaremba, Nikuni and Griffin (ZNG). These authors use a Boltzmann equation to describe the interactions between condensed and excited atoms and manage in this way to describe condensate growth. We then turn to a discussion on the 2PI effective action and derive equations of motion for a relativistic scalar field theory. In the nonrelativistic limit these equations are shown to coincide with the ZNG theory when a quasiparticle approximation is applied. Finally, we perform a numerical analysis of the full 2PI equations. These remain valid even at strong coupling and far from equilibrium, and thus go far beyond Boltzmann's approach. For simplicity, we limit ourselves to a homogeneous system and present the first 3+1 dimensional study of condensate melting. (orig.)

  16. Calculation of collisional mixing (United States)

    Koponen, I.; Hautala, M.


    Collisional mixing of markers is calculated by splitting the calculation into two parts. Relocation cross sections have been calculated using a realistic potential in a Monte Carlo simulation. The cross sections are used in the computation of marker relocation. The cumulative effect of successive relocations is assumed to be an uncorrelated transport process and it is treated as a weighted random walk. Matrix relocation was not included in the calculations. The results from this two-step simulation model are compared with analytical models. A fit to the simulated differential relocation cross sections has been found which makes the numerical integration of the Bothe formula feasible. The influence of primaries has been treated in this way. When all the recoils are included the relocation profiles are nearly Gaussian and the Pearson IV distributions yield acceptable profiles in the studied cases. The approximations and cut-off procedures which cause the major uncertainties in calculations are pointed out. The choice of the cut-off energy is shown to be the source of the largest uncertainty whereas the mathematical approximations can be used with good accuracy. The methods are used to study the broadening of a Pt marker in Si mixed by 300 keV Xe ions, broadening of a Ti marker in Al mixed by 300 keV Xe ions and broadening of a Ti marker in Hf mixed by 750 keV Kr ions. The fluence in each case is 2 × 10 16{ions}/{cm 2}. The calculated averages of half widths at half maximum vary between 11-18, 9-12 and 10-15 nm, respectively, depending on the cut-off energy and the mixing efficiencies vary between 11-29, 6-11 and 6-14 {Å5}/{eV}, respectively. The broadenings of Pt in Si and Ti in Al are about two times smaller than the measured values and the broadening of Ti in Hf is in agreement with the measured values.

  17. Poisson-Boltzmann versus Size-Modified Poisson-Boltzmann Electrostatics Applied to Lipid Bilayers. (United States)

    Wang, Nuo; Zhou, Shenggao; Kekenes-Huskey, Peter M; Li, Bo; McCammon, J Andrew


    Mean-field methods, such as the Poisson-Boltzmann equation (PBE), are often used to calculate the electrostatic properties of molecular systems. In the past two decades, an enhancement of the PBE, the size-modified Poisson-Boltzmann equation (SMPBE), has been reported. Here, the PBE and the SMPBE are reevaluated for realistic molecular systems, namely, lipid bilayers, under eight different sets of input parameters. The SMPBE appears to reproduce the molecular dynamics simulation results better than the PBE only under specific parameter sets, but in general, it performs no better than the Stern layer correction of the PBE. These results emphasize the need for careful discussions of the accuracy of mean-field calculations on realistic systems with respect to the choice of parameters and call for reconsideration of the cost-efficiency and the significance of the current SMPBE formulation.

  18. Essentially Entropic Lattice Boltzmann Model (United States)

    Atif, Mohammad; Kolluru, Praveen Kumar; Thantanapally, Chakradhar; Ansumali, Santosh


    The entropic lattice Boltzmann model (ELBM), a discrete space-time kinetic theory for hydrodynamics, ensures nonlinear stability via the discrete time version of the second law of thermodynamics (the H theorem). Compliance with the H theorem is numerically enforced in this methodology and involves a search for the maximal discrete path length corresponding to the zero dissipation state by iteratively solving a nonlinear equation. We demonstrate that an exact solution for the path length can be obtained by assuming a natural criterion of negative entropy change, thereby reducing the problem to solving an inequality. This inequality is solved by creating a new framework for construction of Padé approximants via quadrature on appropriate convex function. This exact solution also resolves the issue of indeterminacy in case of nonexistence of the entropic involution step. Since our formulation is devoid of complex mathematical library functions, the computational cost is drastically reduced. To illustrate this, we have simulated a model setup of flow over the NACA-0012 airfoil at a Reynolds number of 2.88 ×106.

  19. Lattice Boltzmann model for numerical relativity. (United States)

    Ilseven, E; Mendoza, M


    In the Z4 formulation, Einstein equations are written as a set of flux conservative first-order hyperbolic equations that resemble fluid dynamics equations. Based on this formulation, we construct a lattice Boltzmann model for numerical relativity and validate it with well-established tests, also known as "apples with apples." Furthermore, we find that by increasing the relaxation time, we gain stability at the cost of losing accuracy, and by decreasing the lattice spacings while keeping a constant numerical diffusivity, the accuracy and stability of our simulations improve. Finally, in order to show the potential of our approach, a linear scaling law for parallelization with respect to number of CPU cores is demonstrated. Our model represents the first step in using lattice kinetic theory to solve gravitational problems.

  20. Accuracy analysis of a 2D Poisson-Vlasov PIC solver and estimates of the collisional effects in space charge dynamics

    CERN Document Server

    Bazzani, A; Franchi, A; Rambaldi, S; Turchetti, G


    We analyze the accuracy of a 2D Poisson-Vlasov PIC integrator, taking the KV as a reference solution for a FODO cell. The particle evolution is symplectic and the Poisson solver is based on FFT. The numerical error, evaluated by comparing the moments of the distribution and the electric field with the exact solution, shows a linear growth. This effect can be modeled by a white noise in the envelope equations for the KV beam. In order to investigate the collisional effects we have integrated the Hamilton's equations for N charged macro-particles with a hard-core r/sub H/ reducing the computational complexity to N/sup 3/2/. In the constant focusing case we observed that a KV beam, matched or mismatched relaxes to the Maxwell-Boltzmann self consistent distribution on a time interval, which depends on r/sub H/ and has a finite limit, for r/sub H/ to 0. A fully 3D PIC code for short bunches was developed for the ADS linac design at LNL (Italy). A 3D particle-core model, based on Langevin's equations with the drift...

  1. Training Restricted Boltzmann Machines

    DEFF Research Database (Denmark)

    Fischer, Asja

    Restricted Boltzmann machines (RBMs) are probabilistic graphical models that can also be interpreted as stochastic neural networks. Training RBMs is known to be challenging. Computing the likelihood of the model parameters or its gradient is in general computationally intensive. Thus, training re...... of the applied sampling procedure and I will introduce a transition operator that leads to faster mixing. Finally, a different parametrisation of RBMs will be discussed that leads to better learning results and more robustness against changes in the data representation....... relies on sampling based approximations of the log-likelihood gradient. I will present an empirical and theoretical analysis of the bias of these approximations and show that the approximation error can lead to a distortion of the learning process. The bias decreases with increasing mixing rate...

  2. Reduction of collisional-radiative models for transient, atomic plasmas (United States)

    Abrantes, Richard June; Karagozian, Ann; Bilyeu, David; Le, Hai


    Interactions between plasmas and any radiation field, whether by lasers or plasma emissions, introduce many computational challenges. One of these computational challenges involves resolving the atomic physics, which can influence other physical phenomena in the radiated system. In this work, a collisional-radiative (CR) model with reduction capabilities is developed to capture the atomic physics at a reduced computational cost. Although the model is made with any element in mind, the model is currently supplemented by LANL's argon database, which includes the relevant collisional and radiative processes for all of the ionic stages. Using the detailed data set as the true solution, reduction mechanisms in the form of Boltzmann grouping, uniform grouping, and quasi-steady-state (QSS), are implemented to compare against the true solution. Effects on the transient plasma stemming from the grouping methods are compared. Distribution A: Approved for public release; unlimited distribution, PA (Public Affairs) Clearance Number 17449. This work was supported by the Air Force Office of Scientific Research (AFOSR), Grant Number 17RQCOR463 (Dr. Jason Marshall).

  3. Aplicação da equação de Poisson-Boltzmann ao cálculo de propriedades dependentes do pH em proteínas Aplications of the Poisson-Boltzmann equation to the calculation of pH-dependent properties in proteins

    Directory of Open Access Journals (Sweden)

    Thereza A. Soares


    Full Text Available The ability of biomolecules to catalyze chemical reactions is due chiefly to their sensitivity to variations of the pH in the surrounding environment. The reason for this is that they are made up of chemical groups whose ionization states are modulated by pH changes that are of the order of 0.4 units. The determination of the protonation states of such chemical groups as a function of conformation of the biomolecule and the pH of the environment can be useful in the elucidation of important biological processes from enzymatic catalysis to protein folding and molecular recognition. In the past 15 years, the theory of Poisson-Boltzmann has been successfully used to estimate the pKa of ionizable sites in proteins yielding results, which may differ by 0.1 unit from the experimental values. In this study, we review the theory of Poisson-Boltzmann under the perspective of its application to the calculation of pKa in proteins.

  4. Lattice gas cellular automata and lattice Boltzmann models an introduction

    CERN Document Server

    Wolf-Gladrow, Dieter A


    Lattice-gas cellular automata (LGCA) and lattice Boltzmann models (LBM) are relatively new and promising methods for the numerical solution of nonlinear partial differential equations. The book provides an introduction for graduate students and researchers. Working knowledge of calculus is required and experience in PDEs and fluid dynamics is recommended. Some peculiarities of cellular automata are outlined in Chapter 2. The properties of various LGCA and special coding techniques are discussed in Chapter 3. Concepts from statistical mechanics (Chapter 4) provide the necessary theoretical background for LGCA and LBM. The properties of lattice Boltzmann models and a method for their construction are presented in Chapter 5.

  5. Large Time Behavior of the Vlasov-Poisson-Boltzmann System

    Directory of Open Access Journals (Sweden)

    Li Li


    Full Text Available The motion of dilute charged particles can be modeled by Vlasov-Poisson-Boltzmann system. We study the large time stability of the VPB system. To be precise, we prove that when time goes to infinity, the solution of VPB system tends to global Maxwellian state in a rate Ot−∞, by using a method developed for Boltzmann equation without force in the work of Desvillettes and Villani (2005. The improvement of the present paper is the removal of condition on parameter λ as in the work of Li (2008.

  6. Comparison of the Nernst-Planck model and the Poisson-Boltzmann model for electroosmotic flows in microchannels. (United States)

    Park, H M; Lee, J S; Kim, T W


    In the analysis of electroosmotic flows, the internal electric potential is usually modeled by the Poisson-Boltzmann equation. The Poisson-Boltzmann equation is derived from the assumption of thermodynamic equilibrium where the ionic distributions are not affected by fluid flows. Although this is a reasonable assumption for steady electroosmotic flows through straight microchannels, there are some important cases where convective transport of ions has nontrivial effects. In these cases, it is necessary to adopt the Nernst-Planck equation instead of the Poisson-Boltzmann equation to model the internal electric field. In the present work, the predictions of the Nernst-Planck equation are compared with those of the Poisson-Boltzmann equation for electroosmotic flows in various microchannels where the convective transport of ions is not negligible.

  7. Lattice-Boltzmann simulation of laser interaction with weakly ionized helium plasmas. (United States)

    Li, Huayu; Ki, Hyungson


    This paper presents a lattice Boltzmann method for laser interaction with weakly ionized plasmas considering electron impact ionization and three-body recombination. To simulate with physical properties of plasmas, the authors' previous work on the rescaling of variables is employed and the electromagnetic fields are calculated from the Maxwell equations by using the finite-difference time-domain method. To calculate temperature fields, energy equations are derived separately from the Boltzmann equations. In this way, we attempt to solve the full governing equations for plasma dynamics. With the developed model, the continuous-wave CO2 laser interaction with helium is simulated successfully.

  8. Simple Navier’s slip boundary condition for the non-Newtonian Lattice Boltzmann fluid dynamics solver

    DEFF Research Database (Denmark)

    Svec, Oldrich; Skoček, Jan


    The ability of the Lattice Boltzmann method, as the fluid dynamics solver, to properly simulate macroscopic Navier’s slip boundary condition is investigated. An approximate equation relating the Lattice Boltzmann variable slip boundary condition with the macroscopic Navier’s slip boundary condition...

  9. Collisional-radiative model: a plasma spectroscopy theory for experimentalists

    Energy Technology Data Exchange (ETDEWEB)

    Fujimoto, Takashi [Kyoto Univ. (Japan); Sawada, Keiji


    The rate equation describing the population n(p) of an excited (and the ground state) level p of ions immersed in plasma is shown. In 1962, the method of quasi-steady state solution (collisional-radiative model) was proposed. Its idea is explained. The coupled differential equations reduce to a set of coupled linear equations for excited levels. The solution of these coupled equations is presented. The equations giving the ionization and recombination of this system of ions under consideration are described in terms of the effective rate coefficients. The collisional-radiative ionization and recombination rate coefficients are expressed in terms of the population coefficients for p > 1. As for ionizing plasma, the excited level populations, the populations, the population distribution among the excited levels, two regimes of the excited levels, the dominant flows of electrons among the levels and so on are shown. As for recombining plasma, the excited level populations, the population distribution among the excited levels, the dominant flows of electrons and so on are shown. Ionization balance plasma may be considered. (K.I.)

  10. Physics of Collisional Plasmas Introduction to High-Frequency Discharges

    CERN Document Server

    Moisan, Michel


    The Physics of Collisional Plasmas deals with the plasma physics of interest to laboratory research and industrial applications, such as lighting, fabrication of microelectronics, destruction of greenhouse gases. Its emphasis is on explaining the physical mechanisms, rather than the detailed mathematical description and theoretical analysis. At the introductory level, it is important to convey the characteristic physical phenomena of plasmas, before addressing the ultimate formalism of kinetic theory, with its microscopic, statistical mechanics approach. To this aim, this text translates the physical phenomena into more tractable equations, using the hydrodynamic model; this considers the plasma as a fluid, in which the macroscopic physical parameters are the statistical averages of the microscopic (individual) parameters. This book is an introduction to the physics of collisional plasmas, as opposed to plasmas in space. It is intended for graduate students in physics and engineering . The first chapter intr...

  11. Temperature based Restricted Boltzmann Machines (United States)

    Li, Guoqi; Deng, Lei; Xu, Yi; Wen, Changyun; Wang, Wei; Pei, Jing; Shi, Luping


    Restricted Boltzmann machines (RBMs), which apply graphical models to learning probability distribution over a set of inputs, have attracted much attention recently since being proposed as building blocks of multi-layer learning systems called deep belief networks (DBNs). Note that temperature is a key factor of the Boltzmann distribution that RBMs originate from. However, none of existing schemes have considered the impact of temperature in the graphical model of DBNs. In this work, we propose temperature based restricted Boltzmann machines (TRBMs) which reveals that temperature is an essential parameter controlling the selectivity of the firing neurons in the hidden layers. We theoretically prove that the effect of temperature can be adjusted by setting the parameter of the sharpness of the logistic function in the proposed TRBMs. The performance of RBMs can be improved by adjusting the temperature parameter of TRBMs. This work provides a comprehensive insights into the deep belief networks and deep learning architectures from a physical point of view.

  12. Drift-Diffusion Equation

    Directory of Open Access Journals (Sweden)

    K. Banoo


    equation in the discrete momentum space. This is shown to be similar to the conventional drift-diffusion equation except that it is a more rigorous solution to the Boltzmann equation because the current and carrier densities are resolved into M×1 vectors, where M is the number of modes in the discrete momentum space. The mobility and diffusion coefficient become M×M matrices which connect the M momentum space modes. This approach is demonstrated by simulating electron transport in bulk silicon.

  13. Two Dimensional Lattice Boltzmann Method for Cavity Flow Simulation


    Panjit MUSIK; Krisanadej JAROENSUTASINEE


    This paper presents a simulation of incompressible viscous flow within a two-dimensional square cavity. The objective is to develop a method originated from Lattice Gas (cellular) Automata (LGA), which utilises discrete lattice as well as discrete time and can be parallelised easily. Lattice Boltzmann Method (LBM), known as discrete Lattice kinetics which provide an alternative for solving the Navier–Stokes equations and are generally used for fluid simulation, is chosen for the study. A spec...

  14. A double-distribution-function lattice Boltzmann method for bed-load sediment transport


    Cai, Li; Xu, Wenjing; Luo, Xiaoyu


    The governing equations of bed-load sediment transport are the shallow water equations and the Exner equation. To embody the advantages of the lattice Boltzmann method (e.g., simplicity, efficiency), the three-velocity (D1Q3) and five-velocity (D1Q5) double-distribution-function lattice Boltzmann models (DDF-LBMs), which can present the numerical solution for one-dimensional bed-load sediment transport, are proposed here based on the quasi-steady approach. The so-called DDF-LBM means we use t...

  15. A lattice Boltzmann coupled to finite volumes method for solving phase change problems

    Directory of Open Access Journals (Sweden)

    El Ganaoui Mohammed


    Full Text Available A numerical scheme coupling lattice Boltzmann and finite volumes approaches has been developed and qualified for test cases of phase change problems. In this work, the coupled partial differential equations of momentum conservation equations are solved with a non uniform lattice Boltzmann method. The energy equation is discretized by using a finite volume method. Simulations show the ability of this developed hybrid method to model the effects of convection, and to predict transfers. Benchmarking is operated both for conductive and convective situation dominating solid/liquid transition. Comparisons are achieved with respect to available analytical solutions and experimental results.

  16. Extended lattice Boltzmann scheme for droplet combustion. (United States)

    Ashna, Mostafa; Rahimian, Mohammad Hassan; Fakhari, Abbas


    The available lattice Boltzmann (LB) models for combustion or phase change are focused on either single-phase flow combustion or two-phase flow with evaporation assuming a constant density for both liquid and gas phases. To pave the way towards simulation of spray combustion, we propose a two-phase LB method for modeling combustion of liquid fuel droplets. We develop an LB scheme to model phase change and combustion by taking into account the density variation in the gas phase and accounting for the chemical reaction based on the Cahn-Hilliard free-energy approach. Evaporation of liquid fuel is modeled by adding a source term, which is due to the divergence of the velocity field being nontrivial, in the continuity equation. The low-Mach-number approximation in the governing Navier-Stokes and energy equations is used to incorporate source terms due to heat release from chemical reactions, density variation, and nonluminous radiative heat loss. Additionally, the conservation equation for chemical species is formulated by including a source term due to chemical reaction. To validate the model, we consider the combustion of n-heptane and n-butanol droplets in stagnant air using overall single-step reactions. The diameter history and flame standoff ratio obtained from the proposed LB method are found to be in good agreement with available numerical and experimental data. The present LB scheme is believed to be a promising approach for modeling spray combustion.

  17. Weakly nonlinear electron plasma waves in collisional plasmas

    DEFF Research Database (Denmark)

    Pecseli, H. L.; Rasmussen, J. Juul; Tagare, S. G.


    The nonlinear evolution of a high frequency plasma wave in a weakly magnetized, collisional plasma is considered. In addition to the ponderomotive-force-nonlinearity the nonlinearity due to the heating of the electrons is taken into account. A set of nonlinear equations including the effect...... of a constantly maintained pump wave is derived and a general dispersion relation describing the modulation of the high frequency wave due to different low frequency responses is obtained. Particular attention is devoted to a purely growing modulation. The relative importance of the ponderomotive force...

  18. Collisional-radiative nonequilibrium in partially ionized atomic nitrogen (United States)

    Kunc, J. A.; Soon, W. H.


    A nonlinear collisional-radiative model for determination of nonequilibrium production of electrons, excited atoms, and bound-bound, dielectronic and continuum line intensities in stationary partially ionized atomic nitrogen is presented. Populations of 14 atomic levels and line intensities are calculated in plasma with T(e) = 8000-15,000 K and N(t) = 10 to the 12th - 10 to the 18th/cu cm. Transport of radiation is included by coupling the rate equations of production of the electrons and excited atoms with the radiation escape factors, which are not constant but depend on plasma conditions.

  19. A lattice Boltzmann model for multiphase flows with moving contact line and variable density (United States)

    Huang, Jizu; Wang, Xiao-Ping


    In this paper, we develop an efficient lattice Boltzmann model for the two-phase moving contact line problem with variable density. The Navier-Stokes and Cahn-Hilliard equations are recovered from the lattice Boltzmann model proposed by Fakhari and Rahimian [5]. To improve numerical stability, we present a semi-implicit lattice Boltzmann method together with a mixed finite difference scheme. In order to describe the behavior of the contact line motion on the boundary, we incorporate the generalized Navier boundary condition [25] by the nonequilibrium extrapolation method [8]. The proposed method is easy to implement and retains the advantage of the standard lattice Boltzmann method. Numerical tests are carried out to verify the proposed method. Our numerical results show that the present approach is able to model two-phase flows with variable density and moving contact line.

  20. A collisional-radiative model of iron vapour in a thermal arc plasma (United States)

    Baeva, M.; Uhrlandt, D.; Murphy, A. B.


    A collisional-radiative model for the ground state and fifty effective excited levels of atomic iron, and one level for singly-ionized iron, is set up for technological plasmas. Attention is focused on the population of excited states of atomic iron as a result of excitation, de-excitation, ionization, recombination and spontaneous emission. Effective rate coefficients for ionization and recombination, required in non-equilibrium plasma transport models, are also obtained. The collisional-radiative model is applied to a thermal arc plasma. Input parameters for the collisional-radiative model are provided by a magnetohydrodynamic simulation of a gas-metal welding arc, in which local thermodynamic equilibrium is assumed and the treatment of the transport of metal vapour is based on combined diffusion coefficients. The results clearly identify the conditions in the arc, under which the atomic state distribution satisfies the Boltzmann distribution, with an excitation temperature equal to the plasma temperature. These conditions are met in the central part of the arc, even though a local temperature minimum occurs here. This provides assurance that diagnostic methods based on local thermodynamic equilibrium, in particular those of optical emission spectroscopy, are reliable here. In contrast, deviations from the equilibrium atomic-state distribution are obtained in the near-electrode and arc fringe regions. As a consequence, the temperatures determined from the ratio of line intensities and number densities obtained from the emission coefficient in these regions are questionable. In this situation, the collisional-radiative model can be used as a diagnostic tool to assist in the interpretation of spectroscopic measurements.

  1. Rovibrationally Resolved Time-Dependent Collisional-Radiative Model of Molecular Hydrogen and Its Application to a Fusion Detached Plasma

    Directory of Open Access Journals (Sweden)

    Keiji Sawada


    Full Text Available A novel rovibrationally resolved collisional-radiative model of molecular hydrogen that includes 4,133 rovibrational levels for electronic states whose united atom principal quantum number is below six is developed. The rovibrational X 1 Σ g + population distribution in a SlimCS fusion demo detached divertor plasma is investigated by solving the model time dependently with an initial 300 K Boltzmann distribution. The effective reaction rate coefficients of molecular assisted recombination and of other processes in which atomic hydrogen is produced are calculated using the obtained time-dependent population distribution.

  2. Collisional stripping of planetary crusts (United States)

    Carter, Philip J.; Leinhardt, Zoë M.; Elliott, Tim; Stewart, Sarah T.; Walter, Michael J.


    Geochemical studies of planetary accretion and evolution have invoked various degrees of collisional erosion to explain differences in bulk composition between planets and chondrites. Here we undertake a full, dynamical evaluation of 'crustal stripping' during accretion and its key geochemical consequences. Crusts are expected to contain a significant fraction of planetary budgets of incompatible elements, which include the major heat producing nuclides. We present smoothed particle hydrodynamics simulations of collisions between differentiated rocky planetesimals and planetary embryos. We find that the crust is preferentially lost relative to the mantle during impacts, and we have developed a scaling law based on these simulations that approximates the mass of crust that remains in the largest remnant. Using this scaling law and a recent set of N-body simulations of terrestrial planet formation, we have estimated the maximum effect of crustal stripping on incompatible element abundances during the accretion of planetary embryos. We find that on average approximately one third of the initial crust is stripped from embryos as they accrete, which leads to a reduction of ∼20% in the budgets of the heat producing elements if the stripped crust does not reaccrete. Erosion of crusts can lead to non-chondritic ratios of incompatible elements, but the magnitude of this effect depends sensitively on the details of the crust-forming melting process on the planetesimals. The Lu/Hf system is fractionated for a wide range of crustal formation scenarios. Using eucrites (the products of planetesimal silicate melting, thought to represent the crust of Vesta) as a guide to the Lu/Hf of planetesimal crust partially lost during accretion, we predict the Earth could evolve to a superchondritic 176Hf/177Hf (3-5 parts per ten thousand) at present day. Such values are in keeping with compositional estimates of the bulk Earth. Stripping of planetary crusts during accretion can lead to

  3. Thermodynamic stabilities of the generalized Boltzmann entropies (United States)

    Wada, Tatsuaki


    We consider the thermodynamic stability conditions (TSC) on the Boltzmann entropies generalized by Tsallis’ q- and Kaniadakis’ κ-deformed logarithmic functions. It is shown that the corresponding TSCs are not necessarily equivalent to the concavity of the generalized Boltzmann entropies with respect to internal energy. Nevertheless, both the TSCs are equivalent to the positivity of standard heat capacity.

  4. Collisionality dependent transport in TCV SOL plasmas

    DEFF Research Database (Denmark)

    Garcia, Odd Erik; Pitts, R.A.; Horacek, J.


    and radial transport increase with plasma collisionality. Such a collisionality dependence is consistent with a recent theory for radial blob motion, which suggests that filamentary structures become electrically disconnected from the target sheaths at large collisionality and thus experience less sheath......Results are presented from probe measurements in the low field side scrape-off layer (SOL) region of TCV during plasma current scan experiments. It is shown that with decreasing plasma current the radial particle density profile becomes broader and the fluctuation levels and turbulence driven...... radial particle flux increase. In the far SOL the fluctuations exhibit a high degree of statistical similarity and the particle density and flux at the wall radius scale inversely with the plasma current. Together with previous TCV density scan experiments, this indicates that plasma fluctuations...

  5. Simulations of ion transport in a collisional radio-frequency plasma sheath. (United States)

    Dai, Zhong-Ling; Wang, You-Nian


    A hybrid theoretical model, capable of describing the characteristics of a collisional sheath driven by a sinusoidal current source and determining the energy and angular distributions of ions incident onto the substrate, is proposed. The model consists of one-dimensional time-dependent fluid equations coupled with the Poisson equation determining spatiotemporal evolution of the sheath and the Monte Carlo simulation predicting the energy and angular distributions of ions striking the electrode, in which charge-exchange collisions between ions and neutrals are included. Additionally, an equivalent circuit model in conjunction with the fluid equations is adopted to self-consistently determine the relationship between the instantaneous potential at a rf-biased electrode and the sheath thickness. It is found that the collisional effects would influence the height of the energy peaks in the ion energy distributions and the ion angular distributions.

  6. Coherent collisional spin dynamics in optical lattices. (United States)

    Widera, Artur; Gerbier, Fabrice; Fölling, Simon; Gericke, Tatjana; Mandel, Olaf; Bloch, Immanuel


    We report on the observation of coherent, purely collisionally driven spin dynamics of neutral atoms in an optical lattice. For high lattice depths, atom pairs confined to the same lattice site show weakly damped Rabi-type oscillations between two-particle Zeeman states of equal magnetization, induced by spin-changing collisions. Moreover, measurement of the oscillation frequency allows for precise determination of the spin-changing collisional coupling strengths, which are directly related to fundamental scattering lengths describing interatomic collisions at ultracold temperatures.

  7. Three-dimensional lattice Boltzmann model for compressible flows. (United States)

    Sun, Chenghai; Hsu, Andrew T


    A three-dimensional compressible lattice Boltzmann model is formulated on a cubic lattice. A very large particle-velocity set is incorporated in order to enable a greater variation in the mean velocity. Meanwhile, the support set of the equilibrium distribution has only six directions. Therefore, this model can efficiently handle flows over a wide range of Mach numbers and capture shock waves. Due to the simple form of the equilibrium distribution, the fourth-order velocity tensors are not involved in the formulation. Unlike the standard lattice Boltzmann model, no special treatment is required for the homogeneity of fourth-order velocity tensors on square lattices. The Navier-Stokes equations were recovered, using the Chapman-Enskog method from the Bhatnagar-Gross-Krook (BGK) lattice Boltzmann equation. The second-order discretization error of the fluctuation velocity in the macroscopic conservation equation was eliminated by means of a modified collision invariant. The model is suitable for both viscous and inviscid compressible flows with or without shocks. Since the present scheme deals only with the equilibrium distribution that depends only on fluid density, velocity, and internal energy, boundary conditions on curved wall are easily implemented by an extrapolation of macroscopic variables. To verify the scheme for inviscid flows, we have successfully simulated a three-dimensional shock-wave propagation in a box and a normal shock of Mach number 10 over a wedge. As an application to viscous flows, we have simulated a flat plate boundary layer flow, flow over a cylinder, and a transonic flow over a NACA0012 airfoil cascade.

  8. Multi-component lattice-Boltzmann model with interparticle interaction

    CERN Document Server

    Shan, X; Shan, Xiaowen; Doolen, Gary


    Abstract: A previously proposed [X. Shan and H. Chen, Phys. Rev. E {\\bf 47}, 1815, (1993)] lattice Boltzmann model for simulating fluids with multiple components and interparticle forces is described in detail. Macroscopic equations governing the motion of each component are derived by using Chapman-Enskog method. The mutual diffusivity in a binary mixture is calculated analytically and confirmed by numerical simulation. The diffusivity is generally a function of the concentrations of the two components but independent of the fluid velocity so that the diffusion is Galilean invariant. The analytically calculated shear kinematic viscosity of this model is also confirmed numerically.

  9. A large eddy lattice Boltzmann simulation of magnetohydrodynamic turbulence (United States)

    Flint, Christopher; Vahala, George


    Large eddy simulations (LES) of a lattice Boltzmann magnetohydrodynamic (LB-MHD) model are performed for the unstable magnetized Kelvin-Helmholtz jet instability. This algorithm is an extension of Ansumali et al. [1] to MHD in which one performs first an expansion in the filter width on the kinetic equations followed by the usual low Knudsen number expansion. These two perturbation operations do not commute. Closure is achieved by invoking the physical constraint that subgrid effects occur at transport time scales. The simulations are in very good agreement with direct numerical simulations.

  10. Moving charged particles in lattice Boltzmann-based electrokinetics (United States)

    Kuron, Michael; Rempfer, Georg; Schornbaum, Florian; Bauer, Martin; Godenschwager, Christian; Holm, Christian; de Graaf, Joost


    The motion of ionic solutes and charged particles under the influence of an electric field and the ensuing hydrodynamic flow of the underlying solvent is ubiquitous in aqueous colloidal suspensions. The physics of such systems is described by a coupled set of differential equations, along with boundary conditions, collectively referred to as the electrokinetic equations. Capuani et al. [J. Chem. Phys. 121, 973 (2004)] introduced a lattice-based method for solving this system of equations, which builds upon the lattice Boltzmann algorithm for the simulation of hydrodynamic flow and exploits computational locality. However, thus far, a description of how to incorporate moving boundary conditions into the Capuani scheme has been lacking. Moving boundary conditions are needed to simulate multiple arbitrarily moving colloids. In this paper, we detail how to introduce such a particle coupling scheme, based on an analogue to the moving boundary method for the pure lattice Boltzmann solver. The key ingredients in our method are mass and charge conservation for the solute species and a partial-volume smoothing of the solute fluxes to minimize discretization artifacts. We demonstrate our algorithm's effectiveness by simulating the electrophoresis of charged spheres in an external field; for a single sphere we compare to the equivalent electro-osmotic (co-moving) problem. Our method's efficiency and ease of implementation should prove beneficial to future simulations of the dynamics in a wide range of complex nanoscopic and colloidal systems that were previously inaccessible to lattice-based continuum algorithms.

  11. Lattice-Boltzmann-based two-phase thermal model for simulating phase change

    NARCIS (Netherlands)

    Kamali, M.R.; Gillissen, J.J.J.; Van den Akker, H.E.A.; Sundaresan, S.


    A lattice Boltzmann (LB) method is presented for solving the energy conservation equation in two phases when the phase change effects are included in the model. This approach employs multiple distribution functions, one for a pseudotemperature scalar variable and the rest for the various species. A

  12. Thermal cascaded lattice Boltzmann method

    CERN Document Server

    Fei, Linlin


    In this paper, a thermal cascaded lattice Boltzmann method (TCLBM) is developed in combination with the double-distribution-function (DDF) approach. A density distribution function relaxed by the cascaded scheme is employed to solve the flow field, and a total energy distribution function relaxed by the BGK scheme is used to solve temperature field, where two distribution functions are coupled naturally. The forcing terms are incorporated by means of central moments, which is consistent with the previous force scheme [Premnath \\emph{et al.}, Phys. Rev. E \\textbf{80}, 036702 (2009)] but the derivation is more intelligible and the evolution process is simpler. In the method, the viscous heat dissipation and compression work are taken into account, the Prandtl number and specific-heat ratio are adjustable, the external force is considered directly without the Boussinesq assumption, and the low-Mach number compressible flows can also be simulated. The forcing scheme is tested by simulating a steady Taylor-Green f...

  13. Spectral-Lagrangian methods for collisional models of non-equilibrium statistical states (United States)

    Gamba, Irene M.; Tharkabhushanam, Sri Harsha


    We propose a new spectral Lagrangian based deterministic solver for the non-linear Boltzmann transport equation (BTE) in d-dimensions for variable hard sphere (VHS) collision kernels with conservative or non-conservative binary interactions. The method is based on symmetries of the Fourier transform of the collision integral, where the complexity in its computation is reduced to a separate integral over the unit sphere S. The conservation of moments is enforced by Lagrangian constraints. The resulting scheme, implemented in free space, is very versatile and adjusts in a very simple manner to several cases that involve energy dissipation due to local micro-reversibility (inelastic interactions) or elastic models of slowing down process. Our simulations are benchmarked with available exact self-similar solutions, exact moment equations and analytical estimates for the homogeneous Boltzmann equation, both for elastic and inelastic VHS interactions. Benchmarking of the simulations involves the selection of a time self-similar rescaling of the numerical distribution function which is performed using the continuous spectrum of the equation for Maxwell molecules as studied first in Bobylev et al. [A.V. Bobylev, C. Cercignani, G. Toscani, Proof of an asymptotic property of self-similar solutions of the Boltzmann equation for granular materials, Journal of Statistical Physics 111 (2003) 403-417] and generalized to a wide range of related models in Bobylev et al. [A.V. Bobylev, C. Cercignani, I.M. Gamba, On the self-similar asymptotics for generalized non-linear kinetic Maxwell models, Communication in Mathematical Physics, in press. URL: ]. The method also produces accurate results in the case of inelastic diffusive Boltzmann equations for hard spheres (inelastic collisions under thermal bath), where overpopulated non-Gaussian exponential tails have been conjectured in computations by stochastic methods [T.V. Noije, M. Ernst, Velocity distributions in homogeneously cooling

  14. Collisional disruptions of rotating targets (United States)

    Ševeček, Pavel; Broz, Miroslav


    Collisions are key processes in the evolution of the Main Asteroid Belt and impact events - i.e. target fragmentation and gravitational reaccumulation - are commonly studied by numerical simulations, namely by SPH and N-body methods. In our work, we extend the previous studies by assuming rotating targets and we study the dependence of resulting size-distributions on the pre-impact rotation of the target. To obtain stable initial conditions, it is also necessary to include the self-gravity already in the fragmentation phase which was previously neglected.To tackle this problem, we developed an SPH code, accelerated by SSE/AVX instruction sets and parallelized. The code solves the standard set of hydrodynamic equations, using the Tillotson equation of state, von Mises criterion for plastic yielding and scalar Grady-Kipp model for fragmentation. We further modified the velocity gradient by a correction tensor (Schäfer et al. 2007) to ensure a first-order conservation of the total angular momentum. As the intact target is a spherical body, its gravity can be approximated by a potential of a homogeneous sphere, making it easy to set up initial conditions. This is however infeasible for later stages of the disruption; to this point, we included the Barnes-Hut algorithm to compute the gravitational accelerations, using a multipole expansion of distant particles up to hexadecapole order.We tested the code carefully, comparing the results to our previous computations obtained with the SPH5 code (Benz and Asphaug 1994). Finally, we ran a set of simulations and we discuss the difference between the synthetic families created by rotating and static targets.

  15. Dielectric boundary force in numerical Poisson-Boltzmann methods: Theory and numerical strategies (United States)

    Cai, Qin; Ye, Xiang; Wang, Jun; Luo, Ray


    Continuum modeling of electrostatic interactions based upon the numerical solutions of the Poisson-Boltzmann equation has been widely adopted in biomolecular applications. To extend their applications to molecular dynamics and energy minimization, robust and efficient methodologies to compute solvation forces must be developed. In this study, we have first reviewed the theory for the computation of dielectric boundary force based on the definition of the Maxwell stress tensor. This is followed by a new formulation of the dielectric boundary force suitable for the finite-difference Poisson-Boltzmann methods. We have validated the new formulation with idealized analytical systems and realistic molecular systems.

  16. Boltzmann's H-theorem and the assumption of molecular chaos

    Energy Technology Data Exchange (ETDEWEB)

    Boozer, A D, E-mail: [Department of Physics, University of New Mexico, Albuquerque, NM 87131 (United States)


    We describe a simple dynamical model of a one-dimensional ideal gas and use computer simulations of the model to illustrate two fundamental results of kinetic theory: the Boltzmann transport equation and the Boltzmann H-theorem. Although the model is time-reversal invariant, both results predict that the behaviour of the gas is time-asymmetric. We show that the assumption of molecular chaos is a necessary condition, but not a sufficient condition, for such time-asymmetric results to correctly describe the model, and we use computer simulations to investigate the conditions under which the assumption of molecular chaos holds.

  17. Boltzmann Transport in Hybrid PIC HET Modeling (United States)


    Paper 3. DATES COVERED (From - To) July 2015-July 2015 4. TITLE AND SUBTITLE Boltzmann transport in hybrid PIC HET modeling 5a. CONTRACT NUMBER In...reproduce experimentally observed mobility trends derived from HPHall, a workhorse hybrid- PIC HET simulation code. 15. SUBJECT TERMS 16. SECURITY...Std. 239.18 Boltzmann transport in hybrid PIC HET modeling IEPC-2015- /ISTS-2015-b- Presented at Joint Conference of 30th International

  18. Student understanding of the Boltzmann factor

    Directory of Open Access Journals (Sweden)

    Trevor I. Smith


    Full Text Available [This paper is part of the Focused Collection on Upper Division Physics Courses.] We present results of our investigation into student understanding of the physical significance and utility of the Boltzmann factor in several simple models. We identify various justifications, both correct and incorrect, that students use when answering written questions that require application of the Boltzmann factor. Results from written data as well as teaching interviews suggest that many students can neither recognize situations in which the Boltzmann factor is applicable nor articulate the physical significance of the Boltzmann factor as an expression for multiplicity, a fundamental quantity of statistical mechanics. The specific student difficulties seen in the written data led us to develop a guided-inquiry tutorial activity, centered around the derivation of the Boltzmann factor, for use in undergraduate statistical mechanics courses. We report on the development process of our tutorial, including data from teaching interviews and classroom observations of student discussions about the Boltzmann factor and its derivation during the tutorial development process. This additional information informed modifications that improved students’ abilities to complete the tutorial during the allowed class time without sacrificing the effectiveness as we have measured it. These data also show an increase in students’ appreciation of the origin and significance of the Boltzmann factor during the student discussions. Our findings provide evidence that working in groups to better understand the physical origins of the canonical probability distribution helps students gain a better understanding of when the Boltzmann factor is applicable and how to use it appropriately in answering relevant questions.

  19. PB-AM: An open-source, fully analytical linear poisson-boltzmann solver

    Energy Technology Data Exchange (ETDEWEB)

    Felberg, Lisa E. [Department of Chemical and Biomolecular Engineering, University of California Berkeley, Berkeley California 94720; Brookes, David H. [Department of Chemistry, University of California Berkeley, Berkeley California 94720; Yap, Eng-Hui [Department of Systems and Computational Biology, Albert Einstein College of Medicine, Bronx New York 10461; Jurrus, Elizabeth [Division of Computational and Statistical Analytics, Pacific Northwest National Laboratory, Richland Washington 99352; Scientific Computing and Imaging Institute, University of Utah, Salt Lake City Utah 84112; Baker, Nathan A. [Advanced Computing, Mathematics, and Data Division, Pacific Northwest National Laboratory, Richland Washington 99352; Division of Applied Mathematics, Brown University, Providence Rhode Island 02912; Head-Gordon, Teresa [Department of Chemical and Biomolecular Engineering, University of California Berkeley, Berkeley California 94720; Department of Chemistry, University of California Berkeley, Berkeley California 94720; Department of Bioengineering, University of California Berkeley, Berkeley California 94720; Chemical Sciences Division, Lawrence Berkeley National Labs, Berkeley California 94720


    We present the open source distributed software package Poisson-Boltzmann Analytical Method (PB-AM), a fully analytical solution to the linearized Poisson Boltzmann equation. The PB-AM software package includes the generation of outputs files appropriate for visualization using VMD, a Brownian dynamics scheme that uses periodic boundary conditions to simulate dynamics, the ability to specify docking criteria, and offers two different kinetics schemes to evaluate biomolecular association rate constants. Given that PB-AM defines mutual polarization completely and accurately, it can be refactored as a many-body expansion to explore 2- and 3-body polarization. Additionally, the software has been integrated into the Adaptive Poisson-Boltzmann Solver (APBS) software package to make it more accessible to a larger group of scientists, educators and students that are more familiar with the APBS framework.

  20. Simulating condensation on microstructured surfaces using Lattice Boltzmann Method (United States)

    Alexeev, Alexander; Vasyliv, Yaroslav


    We simulate a single component fluid condensing on 2D structured surfaces with different wettability. To simulate the two phase fluid, we use the athermal Lattice Boltzmann Method (LBM) driven by a pseudopotential force. The pseudopotential force results in a non-ideal equation of state (EOS) which permits liquid-vapor phase change. To account for thermal effects, the athermal LBM is coupled to a finite volume discretization of the temperature evolution equation obtained using a thermal energy rate balance for the specific internal energy. We use the developed model to probe the effect of surface structure and surface wettability on the condensation rate in order to identify microstructure topographies promoting condensation. Financial support is acknowledged from Kimberly-Clark.

  1. Computational Aeroacoustics Using the Generalized Lattice Boltzmann Equation Project (United States)

    National Aeronautics and Space Administration — The research proposed targets airframe noise (AFN) prediction and reduction. AFN originates from complex interactions of turbulent flow with airframe components that...

  2. Collisional transfer of population and orientation in NaK. (United States)

    Wolfe, C M; Ashman, S; Bai, J; Beser, B; Ahmed, E H; Lyyra, A M; Huennekens, J


    Collisional satellite lines with |ΔJ| ≤ 58 have been identified in recent polarization spectroscopy V-type optical-optical double resonance (OODR) excitation spectra of the Rb(2) molecule [H. Salami et al., Phys. Rev. A 80, 022515 (2009)]. Observation of these satellite lines clearly requires a transfer of population from the rotational level directly excited by the pump laser to a neighboring level in a collision of the molecule with an atomic perturber. However to be observed in polarization spectroscopy, the collision must also partially preserve the angular momentum orientation, which is at least somewhat surprising given the extremely large values of ΔJ that were observed. In the present work, we used the two-step OODR fluorescence and polarization spectroscopy techniques to obtain quantitative information on the transfer of population and orientation in rotationally inelastic collisions of the NaK molecules prepared in the 2(A)(1)Σ(+)(v' = 16, J' = 30) rovibrational level with argon and potassium perturbers. A rate equation model was used to study the intensities of these satellite lines as a function of argon pressure and heat pipe oven temperature, in order to separate the collisional effects of argon and potassium atoms. Using a fit of this rate equation model to the data, we found that collisions of NaK molecules with potassium atoms are more likely to transfer population and destroy orientation than collisions with argon atoms. Collisions with argon atoms show a strong propensity for population transfer with ΔJ = even. Conversely, collisions with potassium atoms do not show this ΔJ = even propensity, but do show a propensity for ΔJ = positive compared to ΔJ = negative, for this particular initial state. The density matrix equations of motion have also been solved numerically in order to test the approximations used in the rate equation model and to calculate fluorescence and polarization spectroscopy line shapes. In addition, we have measured

  3. A Truly Second-Order and Unconditionally Stable Thermal Lattice Boltzmann Method

    Directory of Open Access Journals (Sweden)

    Zhen Chen


    Full Text Available An unconditionally stable thermal lattice Boltzmann method (USTLBM is proposed in this paper for simulating incompressible thermal flows. In USTLBM, solutions to the macroscopic governing equations that are recovered from lattice Boltzmann equation (LBE through Chapman–Enskog (C-E expansion analysis are resolved in a predictor–corrector scheme and reconstructed within lattice Boltzmann framework. The development of USTLBM is inspired by the recently proposed simplified thermal lattice Boltzmann method (STLBM. Comparing with STLBM which can only achieve the first-order of accuracy in time, the present USTLBM ensures the second-order of accuracy both in space and in time. Meanwhile, all merits of STLBM are maintained by USTLBM. Specifically, USTLBM directly updates macroscopic variables rather than distribution functions, which greatly saves virtual memories and facilitates implementation of physical boundary conditions. Through von Neumann stability analysis, it can be theoretically proven that USTLBM is unconditionally stable. It is also shown in numerical tests that, comparing to STLBM, lower numerical error can be expected in USTLBM at the same mesh resolution. Four typical numerical examples are presented to demonstrate the robustness of USTLBM and its flexibility on non-uniform and body-fitted meshes.

  4. Accurate determination of the Boltzmann constant by Doppler spectroscopy: Towards a new definition of the kelvin

    CERN Document Server

    Darquié, Benoît; Sow, Papa Lat Tabara; Lemarchand, Cyril; Triki, Meriam; Tokunaga, Sean; Bordé, Christian J; Chardonnet, Christian; Daussy, Christophe


    Accurate molecular spectroscopy in the mid-infrared region allows precision measurements of fundamental constants. For instance, measuring the linewidth of an isolated Doppler-broadened absorption line of ammonia around 10 $\\mu$m enables a determination of the Boltzmann constant k B. We report on our latest measurements. By fitting this lineshape to several models which include Dicke narrowing or speed-dependent collisional effects, we find that a determination of k B with an uncertainty of a few ppm is reachable. This is comparable to the best current uncertainty obtained using acoustic methods and would make a significant contribution to any new value of k B determined by the CODATA. Furthermore, having multiple independent measurements at these accuracies opens the possibility of defining the kelvin by fixing k B, an exciting prospect considering the upcoming redefinition of the International System of Units.

  5. Polyelectrolyte Microcapsules: Ion Distributions from a Poisson-Boltzmann Model (United States)

    Tang, Qiyun; Denton, Alan R.; Rozairo, Damith; Croll, Andrew B.


    Recent experiments have shown that polystyrene-polyacrylic-acid-polystyrene (PS-PAA-PS) triblock copolymers in a solvent mixture of water and toluene can self-assemble into spherical microcapsules. Suspended in water, the microcapsules have a toluene core surrounded by an elastomer triblock shell. The longer, hydrophilic PAA blocks remain near the outer surface of the shell, becoming charged through dissociation of OH functional groups in water, while the shorter, hydrophobic PS blocks form a networked (glass or gel) structure. Within a mean-field Poisson-Boltzmann theory, we model these polyelectrolyte microcapsules as spherical charged shells, assuming different dielectric constants inside and outside the capsule. By numerically solving the nonlinear Poisson-Boltzmann equation, we calculate the radial distribution of anions and cations and the osmotic pressure within the shell as a function of salt concentration. Our predictions, which can be tested by comparison with experiments, may guide the design of microcapsules for practical applications, such as drug delivery. This work was supported by the National Science Foundation under Grant No. DMR-1106331.

  6. Element Free Lattice Boltzmann Method for Fluid-Flow Problems

    Energy Technology Data Exchange (ETDEWEB)

    Jo, Jong Chull; Roh, Kyung Wan; Yune, Young Gill; Kim, Hho Jhung [Korea Institute of Nuclear Safety, Daejeon (Korea, Republic of); Kwon, Young Kwon [US Naval Postgraduate School, New York (United States)


    The Lattice Boltzmann Method (LBM) has been developed for application to thermal-fluid problems. Most of the those studies considered a regular shape of lattice or mesh like square and cubic grids. In order to apply the LBM to more practical cases, it is necessary to be able to solve complex or irregular shapes of problem domains. Some techniques were based on the finite element method. Generally, the finite element method is very powerful for solving two or three-dimensional complex or irregular shapes of domains using the iso-parametric element formulation which is based on a mathematical mapping from a regular shape of element in an imaginary domain to a more general and irregular shape of element in the physical domain. In addition, the element free technique is also quite useful to analyze a complex shape of domain because there is no need to divide a domain by a compatible finite element mesh. This paper presents a new finite element and element free formulations for the lattice Boltzmann equation using the general weighted residual technique. Then, a series of validation examples are presented.

  7. Generalized Boltzmann solution for non-equilibrium flows and the computation of flowfields of binary gas mixture

    Directory of Open Access Journals (Sweden)

    Baoguo Wang


    Full Text Available Hypersonic flows about space vehicles produce flowfields in thermodynamic non-equilibrium with the local Knudsen numbers Kn which may lie in all the three regimes: continuum, transition and rarefied. Continuum flows can be modeled accurately by solving the Navier–Stokes (NS equations; however, the flows in transition and rarefied regimes require a kinetic approach such as the direct simulation Monte Carlo (DSMC method or the solution of the Boltzmann equation. The Boltzmann equation and the general solution approach, using the splitting method, will be introduced in this paper. Details of the method used for solving both the classical Boltzmann equation (CBE and the generalized Boltzmann equation (GBE are also provided. The gas mixture discussed in this paper may consist of both monoatomic and diatomic gases. In particular, the method is applied to simulate two of the three primary constituents of air (N2, O2, and Ar in a binary mixture at 1:1 density ratio at Mach 2 and 5, with gases in translational, rotational and vibrational non-equilibrium.

  8. Longitudinal and transversal current in collisional plasma, generated by two transversal electromagnetic waves

    CERN Document Server

    Latyshev, A V


    From kinetic Vlasov equation for collisional plasmas distribution function in square-law approximation on sizes of intensivities of electric fields is received. The known integral of collisions of relaxation type, so-called BGK (Bhatnagar, Gross, Krook) integral of collisions is considered. The formula for calculation electric current at any temperature (any degree of degeneration of electronic gas) is deduced. This formula contains an one-dimensional quadrature. It is shown, that the nonlinearity account leads to occurrence the longitudinal electric current directed along a wave vector. This longitudinal current is orthogonal to a known transversal classical current, received at the linear analysis. When frequency of collisions tends to the zero, all received results for collisional plasmas pass in corresponding formulas for collisionless plasmas. The case of small values of wave number is considered. It is shown, that the received quantity of longitudinal current at aspiration of frequency of collisions to ...

  9. Astrophysical Gyrokinetics: Kinetic and Fluid Turbulent Cascades In Magentized Weakly Collisional Plasmas

    Energy Technology Data Exchange (ETDEWEB)

    Schekochihin, A. A.; Cowley, S. C.; Dorland, W.; Hammett, G. W.; Howes, G. G.; Quataert, E.; Tatsuno, T.


    This paper presents a theoretical framework for understanding plasma turbulence in astrophysical plasmas. It is motivated by observations of electromagnetic and density fluctuations in the solar wind, interstellar medium and galaxy clusters, as well as by models of particle heating in accretion disks. All of these plasmas and many others have turbulentmotions at weakly collisional and collisionless scales. The paper focuses on turbulence in a strong mean magnetic field. The key assumptions are that the turbulent fluctuations are small compared to the mean field, spatially anisotropic with respect to it and that their frequency is low compared to the ion cyclotron frequency. The turbulence is assumed to be forced at some system-specific outer scale. The energy injected at this scale has to be dissipated into heat, which ultimately cannot be accomplished without collisions. A kinetic cascade develops that brings the energy to collisional scales both in space and velocity. The nature of the kinetic cascade in various scale ranges depends on the physics of plasma fluctuations that exist there. There are four special scales that separate physically distinct regimes: the electron and ion gyroscales, the mean free path and the electron diffusion scale. In each of the scale ranges separated by these scales, the fully kinetic problem is systematically reduced to a more physically transparent and computationally tractable system of equations, which are derived in a rigorous way. In the "inertial range" above the ion gyroscale, the kinetic cascade separates into two parts: a cascade of Alfvenic fluctuations and a passive cascade of density and magnetic-fieldstrength fluctuations. The former are governed by the Reduced Magnetohydrodynamic (RMHD) equations at both the collisional and collisionless scales; the latter obey a linear kinetic equation along the (moving) field lines associated with the Alfvenic component (in the collisional limit, these compressive fluctuations

  10. Multispeed models in off-lattice Boltzmann simulations

    NARCIS (Netherlands)

    Bardow, A.; Karlin, I.V.; Gusev, A.A.


    The lattice Boltzmann method is a highly promising approach to the simulation of complex flows. Here, we realize recently proposed multispeed lattice Boltzmann models [S. Chikatamarla et al., Phys. Rev. Lett. 97 190601 (2006)] by exploiting the flexibility offered by off-lattice Boltzmann methods.

  11. A Nonlinear Evolution Equation in an Ordered Space, Arising from Kinetic Theory

    CERN Document Server

    Grünfeld, C P


    We investigate the Cauchy problem for a nonlinear evolution equation, formulated in an abstract Lebesgue space, as a generalization of various Boltzmann kinetic models. Our main result provides sufficient conditions for the existence, uniqueness, and positivity of global in time solutions. The proof is based on ideas behind a well-known monotonicity method, originally developed within the existence theory of the classical Boltzmann equation in $L^1$. Our application examples concern Smoluchowski's coagulation equation, a Povzner-like equation with dissipative collisions, and a Boltzmann model with chemical reactions.

  12. A Stochastic Sharpening Method for the Propagation of Phase Boundaries in Multiphase Lattice Boltzmann Simulations

    KAUST Repository

    Reis, T.


    Existing lattice Boltzmann models that have been designed to recover a macroscopic description of immiscible liquids are only able to make predictions that are quantitatively correct when the interface that exists between the fluids is smeared over several nodal points. Attempts to minimise the thickness of this interface generally leads to a phenomenon known as lattice pinning, the precise cause of which is not well understood. This spurious behaviour is remarkably similar to that associated with the numerical simulation of hyperbolic partial differential equations coupled with a stiff source term. Inspired by the seminal work in this field, we derive a lattice Boltzmann implementation of a model equation used to investigate such peculiarities. This implementation is extended to different spacial discretisations in one and two dimensions. We shown that the inclusion of a quasi-random threshold dramatically delays the onset of pinning and facetting.

  13. Heat flux viscosity in collisional magnetized plasmas

    Energy Technology Data Exchange (ETDEWEB)

    Liu, C., E-mail: [Princeton University, Princeton, New Jersey 08544 (United States); Fox, W. [Princeton Plasma Physics Laboratory, Princeton, New Jersey 08543 (United States); Bhattacharjee, A. [Princeton University, Princeton, New Jersey 08544 (United States); Princeton Plasma Physics Laboratory, Princeton, New Jersey 08543 (United States)


    Momentum transport in collisional magnetized plasmas due to gradients in the heat flux, a “heat flux viscosity,” is demonstrated. Even though no net particle flux is associated with a heat flux, in a plasma there can still be momentum transport owing to the velocity dependence of the Coulomb collision frequency, analogous to the thermal force. This heat-flux viscosity may play an important role in numerous plasma environments, in particular, in strongly driven high-energy-density plasma, where strong heat flux can dominate over ordinary plasma flows. The heat flux viscosity can influence the dynamics of the magnetic field in plasmas through the generalized Ohm's law and may therefore play an important role as a dissipation mechanism allowing magnetic field line reconnection. The heat flux viscosity is calculated directly using the finite-difference method of Epperlein and Haines [Phys. Fluids 29, 1029 (1986)], which is shown to be more accurate than Braginskii's method [S. I. Braginskii, Rev. Plasma Phys. 1, 205 (1965)], and confirmed with one-dimensional collisional particle-in-cell simulations. The resulting transport coefficients are tabulated for ease of application.

  14. Poisson-Boltzmann Calculations: van der Waals or Molecular Surface? (United States)

    Pang, Xiaodong; Zhou, Huan-Xiang


    The Poisson-Boltzmann equation is widely used for modeling the electrostatics of biomolecules, but the calculation results are sensitive to the choice of the boundary between the low solute dielectric and the high solvent dielectric. The default choice for the dielectric boundary has been the molecular surface, but the use of the van der Waals surface has also been advocated. Here we review recent studies in which the two choices are tested against experimental results and explicit-solvent calculations. The assignment of the solvent high dielectric constant to interstitial voids in the solute is often used as a criticism against the van der Waals surface. However, this assignment may not be as unrealistic as previously thought, since hydrogen exchange and other NMR experiments have firmly established that all interior parts of proteins are transiently accessible to the solvent. PMID:23293674

  15. Ionic size effects on the Poisson-Boltzmann theory. (United States)

    Colla, Thiago; Nunes Lopes, Lucas; Dos Santos, Alexandre P


    In this paper, we develop a simple theory to study the effects of ionic size on ionic distributions around a charged spherical particle. We include a correction to the regular Poisson-Boltzmann equation in order to take into account the size of ions in a mean-field regime. The results are compared with Monte Carlo simulations and a density functional theory based on the fundamental measure approach and a second-order bulk expansion which accounts for electrostatic correlations. The agreement is very good even for multivalent ions. Our results show that the theory can be applied with very good accuracy in the description of ions with highly effective ionic radii and low concentration, interacting with a colloid or a nanoparticle in an electrolyte solution.

  16. Chemical-potential-based lattice Boltzmann method for nonideal fluids (United States)

    Wen, Binghai; Zhou, Xuan; He, Bing; Zhang, Chaoying; Fang, Haiping


    Chemical potential, as an important thermodynamic quantity, has been popularly used in thermodynamic modeling for complex systems, especially for those involving the phase transitions and chemical reactions. Here we present a chemical-potential-based multiphase lattice Boltzmann model, in which the nonideal force is directly evaluated by a chemical potential. The numerical computation is more efficient than the pressure-tensor-based model [Wen et al. Europhys. Lett. 112, 44002 (2015), 10.1209/0295-5075/112/44002] because the calculations of the pressure tensor and its divergence are avoided. We have derived several chemical potentials of the popular equations of state from the free-energy density function. The theoretical analyses and numerical results support that the present model satisfies thermodynamics and Galilean invariance. An effective chemical-potential boundary condition is also implemented to investigate the wettability of a solid surface, and the contact angle can be linearly tuned by the surface chemical potential.

  17. Quantum Heat Engine and Negative Boltzmann Temperature (United States)

    Xi, Jing-Yi; Quan, Hai-Tao


    To clarify the ambiguity on negative Boltzmann temperature in literature, we study the Carnot and the Otto cycle with one of the heat reservoirs at the negative Boltzmann temperature based on a canonical ensemble description. The work extraction, entropy production and the efficiency of these cycles are explored. Conditions for constructing and properties of these thermodynamic cycles are elucidated. We find that the apparent “violation” of the second law of thermodynamics in these cycles are due to the fact that the traditional definition of thermodynamic efficiency is inappropriate in this situation. When properly understanding the efficiency and the adiabatic processes, in which the system crosses over “absolute ZERO” in a limit sense, the Carnot cycle with one of the heat reservoirs at a negative Boltzmann temperature can be understood straightforwardly, and it contradicts neither the second nor the third law of thermodynamics. Hence, negative Boltzmann temperature is a consistent concept in thermodynamics. We use a two-level system and an Ising spin system to illustrate our central results. Support from the National Science Foundation of China under Grants Nos. 11375012, 11534002, and The Recruitment Program of Global Youth Experts of China

  18. Dynamically adaptive Lattice Boltzmann simulation of shallow water flows with the Peano framework

    KAUST Repository

    Neumann, Philipp


    © 2014 Elsevier Inc. All rights reserved. We present a dynamically adaptive Lattice Boltzmann (LB) implementation for solving the shallow water equations (SWEs). Our implementation extends an existing LB component of the Peano framework. We revise the modular design with respect to the incorporation of new simulation aspects and LB models. The basic SWE-LB implementation is validated in different breaking dam scenarios. We further provide a numerical study on stability of the MRT collision operator used in our simulations.

  19. Collisional effects in weakly collisional plasmas: nonlinear electrostatic waves and recurrence phenomena (United States)

    Camporeale, E.; Pezzi, O.; Valentini, F.


    The longstanding problem of collisions in plasmas is a very fascinating and huge topic in plasma physics. The 'natural' operator that describes the Coulombian interactions between charged particles is the Landau (LAN) integral operator. The LAN operator is a nonlinear, integro-differential and Fokker-Planck type operator which satisfies the H theorem for the entropy growth. Due to its nonlinear nature and multi-dimensionality, any approach to the solution of the Landau integral is almost prohibitive. Therefore collisions are usually modeled by simplified collisional operators. Here collisional effects are modeled by i) the one-dimensional Lenard-Bernstein (LB) operator and ii) the three-dimensional Dougherty (DG) operator. In the first case i), by focusing on a 1D-1V phase space, we study recurrence effects in a weakly collisional plasma, being collisions modeled by the LB operator. By decomposing the linear Vlasov-Poisson system in the Fourier-Hermite space, the recurrence problem is investigated in the linear regime of the damping of a Langmuir wave and of the onset of the bump-on-tail instability. The analysis is then confirmed and extended to the nonlinear regime through a Eulerian collisional Vlasov-Poisson code. Despite being routinely used, an artificial collisionality is not in general a viable way of preventing recurrence in numerical simulations. Moreover, recursive phenomena affect both the linear exponential growth and the nonlinear saturation of a linear instability by producing a fake growth in the electric field, thus showing that, although the filamentation is usually associated with low amplitude fluctuations contexts, it can occur also in nonlinear phenomena. On the other hand ii), the effects of electron-electron collisions on the propagation of nonlinear electrostatic waves are shown by means of Eulerian simulations in a 1D-3V (one dimension in physical space, three dimensions in velocity space) phase space. The nonlinear regime of the symmetric

  20. From the N-body problem to Euler equations (United States)

    Lykov, A. A.; Malyshev, V. A.


    This paper contains a rigorous mathematical example of direct derivation of the system of Euler hydrodynamic equations from the Hamiltonian equations for an N point particle system as N → ∞. "Direct" means that the following standard tools are not used in the proof: stochastic dynamics, thermodynamics, Boltzmann kinetic equations, and the correlation functions approach due to Bogolyubov.

  1. A generalized Poisson and Poisson-Boltzmann solver for electrostatic environments

    Energy Technology Data Exchange (ETDEWEB)

    Fisicaro, G., E-mail:; Goedecker, S. [Department of Physics, University of Basel, Klingelbergstrasse 82, 4056 Basel (Switzerland); Genovese, L. [University of Grenoble Alpes, CEA, INAC-SP2M, L-Sim, F-38000 Grenoble (France); Andreussi, O. [Institute of Computational Science, Università della Svizzera Italiana, Via Giuseppe Buffi 13, CH-6904 Lugano (Switzerland); Theory and Simulations of Materials (THEOS) and National Centre for Computational Design and Discovery of Novel Materials (MARVEL), École Polytechnique Fédérale de Lausanne, Station 12, CH-1015 Lausanne (Switzerland); Marzari, N. [Theory and Simulations of Materials (THEOS) and National Centre for Computational Design and Discovery of Novel Materials (MARVEL), École Polytechnique Fédérale de Lausanne, Station 12, CH-1015 Lausanne (Switzerland)


    The computational study of chemical reactions in complex, wet environments is critical for applications in many fields. It is often essential to study chemical reactions in the presence of applied electrochemical potentials, taking into account the non-trivial electrostatic screening coming from the solvent and the electrolytes. As a consequence, the electrostatic potential has to be found by solving the generalized Poisson and the Poisson-Boltzmann equations for neutral and ionic solutions, respectively. In the present work, solvers for both problems have been developed. A preconditioned conjugate gradient method has been implemented for the solution of the generalized Poisson equation and the linear regime of the Poisson-Boltzmann, allowing to solve iteratively the minimization problem with some ten iterations of the ordinary Poisson equation solver. In addition, a self-consistent procedure enables us to solve the non-linear Poisson-Boltzmann problem. Both solvers exhibit very high accuracy and parallel efficiency and allow for the treatment of periodic, free, and slab boundary conditions. The solver has been integrated into the BigDFT and Quantum-ESPRESSO electronic-structure packages and will be released as an independent program, suitable for integration in other codes.

  2. Nonlinear magnetic reconnection in low collisionality plasmas

    Energy Technology Data Exchange (ETDEWEB)

    Ottaviani, M. [Commission of the European Communities, Abingdon (United Kingdom). JET Joint Undertaking; Porcelli, F. [Politecnico di Torino, Turin (Italy)


    The magnetic reconnection in collisionless regimes, where electron inertia is responsible for the decoupling of the plasma motion from that of the field lines, is discussed. Since the linear theory of m=1 modes breaks down for very small magnetic island widths, a non linear analysis is called for. Thus, the behaviour of a collisionless, 2-D fluid slab model in the limit {rho}/d -> 0, is analyzed. The main result is that, when the island size is larger than the linear layer but smaller than the equilibrium scale length, the reconnection rate exhibits a quasi-explosive time behaviour, during which a current density sub-layer narrower than the skin depth is formed. It is believed that the inclusion of the electron initial term in Ohm`s law opens the possibility to understand the rapidity of relaxation process observed in low collisionality plasmas. 7 refs., 6 figs.

  3. Magnetosonic shock wave in collisional pair-ion plasma

    Energy Technology Data Exchange (ETDEWEB)

    Adak, Ashish, E-mail:; Khan, Manoranjan, E-mail: [Department of Instrumentation Science, Jadavpur University, Kolkata 700 032 (India); Sikdar, Arnab, E-mail: [Department of Mathematics, Calcutta Institute of Engineering and Management, 24/1A Chandi Ghosh Road, Kolkata 700040 (India); Ghosh, Samiran, E-mail: [Department of Applied Mathematics, University of Calcutta, 92, Acharya Prafulla Chandra Road, Kolkata 700 009 (India)


    Nonlinear propagation of magnetosonic shock wave has been studied in collisional magnetized pair-ion plasma. The masses of both ions are same but the temperatures are slightly different. Two fluid model has been taken to describe the model. Two different modes of the magnetosonic wave have been obtained. The dynamics of the nonlinear magnetosonic wave is governed by the Korteweg-de Vries Burgers' equation. It has been shown that the ion-ion collision is the source of dissipation that causes the Burgers' term which is responsible for the shock structures in equal mass pair-ion plasma. The numerical investigations reveal that the magnetosonic wave exhibits both oscillatory and monotonic shock structures depending on the strength of the dissipation. The nonlinear wave exhibited the oscillatory shock wave for strong magnetic field (weak dissipation) and monotonic shock wave for weak magnetic field (strong dissipation). The results have been discussed in the context of the fullerene pair-ion plasma experiments.

  4. On pressure balance in a low collisionality tokamak scrape-off layer (United States)

    Churchill, R. M.; Chang, C. S.; Hager, R.


    Understanding the physics governing the scrape-off layer is necessary in order to reliably predict machine and operation critical quantities, such as the heat flux width at the divertor, plasma-wall interaction, material migration, effect of divertor condition on the pedestal profile, detachment of the divertor plasma, etc. Recent simulation results using the axisymmetric gyrokinetic code XGCa suggest that in a lower ion collisionality near scrape-off layer, where the plasma is highly non-Maxwellian, the fluid form of the momentum equation is not conserved between the low-field side (LFS) midplane and divertor. Taking care to include neutral friction and a Chew-Goldberger-Low (CGL) form of the pressure tensor (i.e. only the dominant diagonal terms) does not resolve the imbalance. Using the full kinetic distribution function in the XGC gyrokinetic code, we explore the effect of off-diagonal pressure tensor terms, to determine their effect in the momentum balance in the scrape-off layer. We also explore other simulations with higher ion collisionality, to begin to study the effect of ion collisionality versus proximity to the separatrix (flux surfaces closer to the separatrix can be more influenced by e.g. X-point loss).

  5. Stability of Nonlinear Wave Patterns to the Bipolar Vlasov-Poisson-Boltzmann System (United States)

    Li, Hailiang; Wang, Yi; Yang, Tong; Zhong, Mingying


    The main purpose of the present paper is to investigate the nonlinear stability of viscous shock waves and rarefaction waves for the bipolar Vlasov-Poisson-Boltzmann (VPB) system. To this end, motivated by the micro-macro decomposition to the Boltzmann equation in Liu and Yu (Commun Math Phys 246:133-179, 2004) and Liu et al. (Physica D 188:178-192, 2004), we first set up a new micro-macro decomposition around the local Maxwellian related to the bipolar VPB system and give a unified framework to study the nonlinear stability of the basic wave patterns to the system. Then, as applications of this new decomposition, the time-asymptotic stability of the two typical nonlinear wave patterns, viscous shock waves and rarefaction waves are proved for the 1D bipolar VPB system. More precisely, it is first proved that the linear superposition of two Boltzmann shock profiles in the first and third characteristic fields is nonlinearly stable to the 1D bipolar VPB system up to some suitable shifts without the zero macroscopic mass conditions on the initial perturbations. Then the time-asymptotic stability of the rarefaction wave fan to compressible Euler equations is proved for the 1D bipolar VPB system. These two results are concerned with the nonlinear stability of wave patterns for Boltzmann equation coupled with additional (electric) forces, which together with spectral analysis made in Li et al. (Indiana Univ Math J 65(2):665-725, 2016) sheds light on understanding the complicated dynamic behaviors around the wave patterns in the transportation of charged particles under the binary collisions, mutual interactions, and the effect of the electrostatic potential forces.

  6. Lattice-Boltzmann simulations of droplet evaporation

    KAUST Repository

    Ledesma-Aguilar, Rodrigo


    © the Partner Organisations 2014. We study the utility and validity of lattice-Boltzmann (LB) simulations to explore droplet evaporation driven by a concentration gradient. Using a binary-fluid lattice-Boltzmann algorithm based on Cahn-Hilliard dynamics, we study the evaporation of planar films and 3D sessile droplets from smooth solid surfaces. Our results show that LB simulations accurately reproduce the classical regime of quasi-static dynamics. Beyond this limit, we show that the algorithm can be used to explore regimes where the evaporative and diffusive timescales are not widely separated, and to include the effect of boundaries of prescribed driving concentration. We illustrate the method by considering the evaporation of a droplet from a solid surface that is chemically patterned with hydrophilic and hydrophobic stripes. This journal is

  7. Nonequilibrium thermodynamics of restricted Boltzmann machines (United States)

    Salazar, Domingos S. P.


    In this work, we analyze the nonequilibrium thermodynamics of a class of neural networks known as restricted Boltzmann machines (RBMs) in the context of unsupervised learning. We show how the network is described as a discrete Markov process and how the detailed balance condition and the Maxwell-Boltzmann equilibrium distribution are sufficient conditions for a complete thermodynamics description, including nonequilibrium fluctuation theorems. Numerical simulations in a fully trained RBM are performed and the heat exchange fluctuation theorem is verified with excellent agreement to the theory. We observe how the contrastive divergence functional, mostly used in unsupervised learning of RBMs, is closely related to nonequilibrium thermodynamic quantities. We also use the framework to interpret the estimation of the partition function of RBMs with the annealed importance sampling method from a thermodynamics standpoint. Finally, we argue that unsupervised learning of RBMs is equivalent to a work protocol in a system driven by the laws of thermodynamics in the absence of labeled data.

  8. Nonlinear Interaction of Elliptical Laser Beam with Collisional Plasma: Effect of Linear Absorption (United States)

    Walia, Keshav; Kaur, Sarabjit


    In the present work, nonlinear interaction of elliptical laser beam with collisional plasma is studied by using paraxial ray approximation. Nonlinear differential equations for the beam width parameters of semi-major axis and semi-minor axis of elliptical laser beam have been set up and solved numerically to study the variation of beam width parameters with normalized distance of propagation. Effects of variation in absorption coefficient and plasma density on the beam width parameters are also analyzed. It is observed from the analysis that extent of self-focusing of beam increases with increase/decrease in plasma density/absorption coefficient.

  9. Immersed Boundary-Lattice Boltzmann Method Using Two Relaxation Times

    Directory of Open Access Journals (Sweden)

    Kosuke Hayashi


    Full Text Available An immersed boundary-lattice Boltzmann method (IB-LBM using a two-relaxation time model (TRT is proposed. The collision operator in the lattice Boltzmann equation is modeled using two relaxation times. One of them is used to set the fluid viscosity and the other is for numerical stability and accuracy. A direct-forcing method is utilized for treatment of immersed boundary. A multi-direct forcing method is also implemented to precisely satisfy the boundary conditions at the immersed boundary. Circular Couette flows between a stationary cylinder and a rotating cylinder are simulated for validation of the proposed method. The method is also validated through simulations of circular and spherical falling particles. Effects of the functional forms of the direct-forcing term and the smoothed-delta function, which interpolates the fluid velocity to the immersed boundary and distributes the forcing term to fixed Eulerian grid points, are also examined. As a result, the following conclusions are obtained: (1 the proposed method does not cause non-physical velocity distribution in circular Couette flows even at high relaxation times, whereas the single-relaxation time (SRT model causes a large non-physical velocity distortion at a high relaxation time, (2 the multi-direct forcing reduces the errors in the velocity profile of a circular Couette flow at a high relaxation time, (3 the two-point delta function is better than the four-point delta function at low relaxation times, but worse at high relaxation times, (4 the functional form of the direct-forcing term does not affect predictions, and (5 circular and spherical particles falling in liquids are well predicted by using the proposed method both for two-dimensional and three-dimensional cases.

  10. Simulating anomalous transport and multiphase segregation in porous media with the Lattice Boltzmann Method (United States)

    Matin, Rastin; Hernandez, Anier; Misztal, Marek; Mathiesen, Joachim


    Many hydrodynamic phenomena ranging from flows at micron scale in porous media, large Reynolds numbers flows, non-Newtonian and multiphase flows have been simulated on computers using the lattice Boltzmann (LB) method. By solving the Lattice Boltzmann Equation on unstructured meshes in three dimensions, we have developed methods to efficiently model the fluid flow in real rock samples. We use this model to study the spatio-temporal statistics of the velocity field inside three-dimensional real geometries and investigate its relation to the, in general, anomalous transport of passive tracers for a wide range of Peclet and Reynolds numbers. We extend this model by free-energy based method, which allows us to simulate binary systems with large-density ratios in a thermodynamically consistent way and track the interface explicitly. In this presentation we will present our recent results on both anomalous transport and multiphase segregation.

  11. Simulating Anomalous Dispersion and Multiphase Segregation in Porous Media with the Lattice Boltzmann Method (United States)

    Matin, Rastin; Misztal, Marek K.; Hernandez-Garcia, Anier; Mathiesen, Joachim


    Many hydrodynamic phenomena such as flows at micron scale in porous media, large Reynolds numbers flows, non-Newtonian and multiphase flows have been simulated numerically using the lattice Boltzmann method. By solving the Lattice Boltzmann Equation on three-dimensional unstructured meshes, we efficiently model single-phase fluid flow in real rock samples. We use the flow field to estimate the permeability and further investigate the anomalous dispersion of passive tracers in porous media. By extending our single-phase model with a free-energy based method, we are able to simulate binary systems with moderate density ratios in a thermodynamically consistent way. In this presentation we will present our recent results on both anomalous transport and multiphase segregation.

  12. Hydrodynamic Force Evaluation by Momentum Exchange Method in Lattice Boltzmann Simulations

    Directory of Open Access Journals (Sweden)

    Binghai Wen


    Full Text Available As a native scheme to evaluate hydrodynamic force in the lattice Boltzmann method, the momentum exchange method has some excellent features, such as simplicity, accuracy, high efficiency and easy parallelization. Especially, it is independent of boundary geometry, preventing from solving the Navier–Stokes equations on complex boundary geometries in the boundary-integral methods. We review the origination and main developments of the momentum exchange method in lattice Boltzmann simulations. Then several practical techniques to fill newborn fluid nodes are discussed for the simulations of fluid-structure interactions. Finally, some representative applications show the wide applicability of the momentum exchange method, such as movements of rigid particles, interactions of deformation particles, particle suspensions in turbulent flow and multiphase flow, etc.

  13. Moment-based boundary conditions for lattice Boltzmann simulations of natural convection in cavities

    KAUST Repository

    Allen, Rebecca


    We study a multiple relaxation time lattice Boltzmann model for natural convection with moment-based boundary conditions. The unknown primary variables of the algorithm at a boundary are found by imposing conditions directly upon hydrodynamic moments, which are then translated into conditions for the discrete velocity distribution functions. The method is formulated so that it is consistent with the second order implementation of the discrete velocity Boltzmann equations for fluid flow and temperature. Natural convection in square cavities is studied for Rayleigh numbers ranging from 103 to 108. An excellent agreement with benchmark data is observed and the flow fields are shown to converge with second order accuracy. Copyright © 2016 Inderscience Enterprises Ltd.

  14. Magnetic field generation via parametric instabilities in collisional plasmas (United States)

    Katoh, K.


    A formalism for the parametric instability with low-frequency magnetic field generation in a collisional plasma is developed. The spontaneous low-frequency magnetic field results from a resonant decay instability of an intense pump wave into a magneto-static wave which appears only in a collisional plasma. The growth rate for a resonant decay instability of an intense electromagnetic wave into a Langmuir wave and a magneto-static wave is obtained.

  15. Convergence of Convective-Diffusive Lattice Boltzmann Methods

    CERN Document Server

    Elton, B H; Levermore, C D; Elton, Bracy H.; Rodrigue, Garry H.


    Lattice Boltzmann methods are numerical schemes derived as a kinetic approximation of an underlying lattice gas. A numerical convergence theory for nonlinear convective-diffusive lattice Boltzmann methods is established. Convergence, consistency, and stability are defined through truncated Hilbert expansions. In this setting it is shown that consistency and stability imply convergence. Monotone lattice Boltzmann methods are defined and shown to be stable, hence convergent when consistent. Examples of diffusive and convective-diffusive lattice Boltzmann methods that are both consistent and monotone are presented.

  16. Statistical equilibrium equations for trace elements in stellar atmospheres


    Kubat, Jiri


    The conditions of thermodynamic equilibrium, local thermodynamic equilibrium, and statistical equilibrium are discussed in detail. The equations of statistical equilibrium and the supplementary equations are shown together with the expressions for radiative and collisional rates with the emphasize on the solution for trace elements.

  17. Large eddy simulation of rotating turbulent flows and heat transfer by the lattice Boltzmann method (United States)

    Liou, Tong-Miin; Wang, Chun-Sheng


    Due to its advantage in parallel efficiency and wall treatment over conventional Navier-Stokes equation-based methods, the lattice Boltzmann method (LBM) has emerged as an efficient tool in simulating turbulent heat and fluid flows. To properly simulate the rotating turbulent flow and heat transfer, which plays a pivotal role in tremendous engineering devices such as gas turbines, wind turbines, centrifugal compressors, and rotary machines, the lattice Boltzmann equations must be reformulated in a rotating coordinate. In this study, a single-rotating reference frame (SRF) formulation of the Boltzmann equations is newly proposed combined with a subgrid scale model for the large eddy simulation of rotating turbulent flows and heat transfer. The subgrid scale closure is modeled by a shear-improved Smagorinsky model. Since the strain rates are also locally determined by the non-equilibrium part of the distribution function, the calculation process is entirely local. The pressure-driven turbulent channel flow with spanwise rotation and heat transfer is used for validating the approach. The Reynolds number characterized by the friction velocity and channel half height is fixed at 194, whereas the rotation number in terms of the friction velocity and channel height ranges from 0 to 3.0. A working fluid of air is chosen, which corresponds to a Prandtl number of 0.71. Calculated results are demonstrated in terms of mean velocity, Reynolds stress, root mean square (RMS) velocity fluctuations, mean temperature, RMS temperature fluctuations, and turbulent heat flux. Good agreement is found between the present LBM predictions and previous direct numerical simulation data obtained by solving the conventional Navier-Stokes equations, which confirms the capability of the proposed SRF LBM and subgrid scale relaxation time formulation for the computation of rotating turbulent flows and heat transfer.

  18. Thermalization via collisional and non-collisional mechanisms in ultracold neutral plasmas (United States)

    Witte, Craig; Roberts, Jacob


    Many ultracold neutral plasmas (UCPs) are formed with non-uniform electron and ion densities. They are also formed in a way that the initial electron velocity distribution is not in thermal equilibrium. We present the results of a numerical simulation that compares the electron velocity distribution evolution after UCP formation between uniform and non-uniform density UCPs. We find three distinct thermalization time periods for the electron velocity: a rapid thermalization on the order of the electron plasma frequency timescale where position variations lead to velocity randomization; a slower second phase where non-collisional effects play a role in thermalization as evidenced by differences between thermalization rates in uniform density and non-uniform density plasmas; and an even slower third phase where the highest velocity portion of the electron thermal distribution equilibrates primarily via collisional mechanisms. These mechanisms are relevant for understanding the establishment of equilibrium in the electron component of UCPs in experimentally relevant conditions. This work supported by the AFOSR.

  19. Privacy-preserving restricted boltzmann machine. (United States)

    Li, Yu; Zhang, Yuan; Ji, Yue


    With the arrival of the big data era, it is predicted that distributed data mining will lead to an information technology revolution. To motivate different institutes to collaborate with each other, the crucial issue is to eliminate their concerns regarding data privacy. In this paper, we propose a privacy-preserving method for training a restricted boltzmann machine (RBM). The RBM can be got without revealing their private data to each other when using our privacy-preserving method. We provide a correctness and efficiency analysis of our algorithms. The comparative experiment shows that the accuracy is very close to the original RBM model.

  20. Privacy-Preserving Restricted Boltzmann Machine

    Directory of Open Access Journals (Sweden)

    Yu Li


    Full Text Available With the arrival of the big data era, it is predicted that distributed data mining will lead to an information technology revolution. To motivate different institutes to collaborate with each other, the crucial issue is to eliminate their concerns regarding data privacy. In this paper, we propose a privacy-preserving method for training a restricted boltzmann machine (RBM. The RBM can be got without revealing their private data to each other when using our privacy-preserving method. We provide a correctness and efficiency analysis of our algorithms. The comparative experiment shows that the accuracy is very close to the original RBM model.

  1. The Approach to Equilibrium: Detailed Balance and the Master Equation (United States)

    Alexander, Millard H.; Hall, Gregory E.; Dagdigian, Paul J.


    The approach to the equilibrium (Boltzmann) distribution of populations of internal states of a molecule is governed by inelastic collisions in the gas phase and with surfaces. The set of differential equations governing the time evolution of the internal state populations is commonly called the master equation. An analytic solution to the master…

  2. Lattice Boltzmann Methods for Fluid Structure Interaction (United States)


    Approved By: Knox T. Millsaps, Professor & Chair, Dept. of Mechanical & Aerospace Engineering Approved By: Doug Moses, Vice Provost for Academic Affairs...similar procedure including the Burgers Equation [10], the Korteweg-de Vries equation [11], the Brinkman equation [12] and the Schrödinger equation [13...method for a two-dimensional viscous Burgers equation: Computational results.,” in Supercomputing, pp. 242–252, 1991. [11] G. Yan and J. Zhang, “A higher

  3. Boltzmann and Einstein: Statistics and dynamics–An unsolved ...

    Indian Academy of Sciences (India)

    The struggle of Boltzmann with the proper description of the behavior of classical macroscopic bodies in equilibrium in terms of the properties of the particles out of which they consist will be sketched. He used both a dynamical and a statistical method. However, Einstein strongly disagreed with Boltzmann's statistical method ...

  4. Boltzmann and Einstein: Statistics and dynamics – An unsolved ...

    Indian Academy of Sciences (India)

    Boltzmann and Einstein: Statistics and dynamics –. An unsolved problem. E G D COHEN. The Rockefeller University, 1230 York Avenue, New York, NY 10021, USA. E-mail: Abstract. The struggle of Boltzmann with the proper description of the behavior of classical macroscopic bodies in equilibrium in ...

  5. Regularization of Grad’s 13 -Moment-Equations in Kinetic Gas Theory (United States)


    core, Boltzmann equation, can be found in many text books like Cercignani (1988), Cercignani (2000), Chapman and Cowling (1970) or Vincenti and Kruger...3776–3786. Cercignani , C. (1988). The Boltzmann Equation and its Applications. Applied Mathe- matical Sciences. Springer, New York. RTO-EN-AVT-194 10...45 REFERENCES REFERENCES Cercignani , C. (2000). Rarefied Gas Dynamics: From Basic Concepts to Actual Calcula- tions. Texts in Applied Mathematics

  6. The Vlasov-Poisson-Boltzmann System for a Disparate Mass Binary Mixture (United States)

    Duan, Renjun; Liu, Shuangqian


    The Vlasov-Poisson-Boltzmann system is often used to govern the motion of plasmas consisting of electrons and ions with disparate masses when collisions of charged particles are described by the two-component Boltzmann collision operator. The perturbation theory of the system around global Maxwellians recently has been well established in Guo (Commun Pure Appl Math 55:1104-1135, 2002). It should be more interesting to further study the existence and stability of nontrivial large time asymptotic profiles for the system even with slab symmetry in space, particularly understanding the effect of the self-consistent potential on the non-trivial long-term dynamics of the binary system. In this paper, we consider the problem in the setting of rarefaction waves. The analytical tool is based on the macro-micro decomposition introduced in Liu et al. (Physica D 188(3-4):178-192, 2004) that we have been able to develop for the case of the two-component Boltzmann equations around local bi-Maxwellians. Our focus is to explore how the disparate masses and charges of particles play a role in the analysis of the approach of the complex coupling system time-asymptotically toward a non-constant equilibrium state whose macroscopic quantities satisfy the quasineutral nonisentropic Euler system.

  7. Effects of nanoparticles on melting process with phase-change using the lattice Boltzmann method

    Directory of Open Access Journals (Sweden)

    Ahmed M. Ibrahem

    Full Text Available In this work, the problem of nanoparticles dispersion effects on coupled heat transfer and solid-liquid phase change has been studied. The lattice Boltzmann method (LBM enthalpy-based is employed. The collision model of lattice Bhatnagar-Gross-Krook (LBGK is used to solve the problem of 1D melting by conduction. On the other hand, we use the model of multi-distribution functions (MDF to calculate the density, the velocity and the temperature for the problem of 2D melting by free convection, associated with different boundary conditions. In these simulations, the volume fractions of copper nanoparticles (0–2% added to water-base fluid and Rayleigh numbers of 103–105. We use the Chapman-Enskog expansion to derive the governing macroscopic quantities from the mesoscopic lattice Boltzmann equation. The results obtained by these models have been compared to an analytical solution or other numerical methods. The effects of nanoparticles on conduction and natural convection during the melting process have been investigated. Moreover, the influences of nanoparticles on moving of the phase change front, the thermal conductivity and the latent heat of fusion are also studied. Keywords: Lattice Boltzmann method, Nanofluids, Conduction melting, Convection melting, BGK collision model

  8. Immersed boundary lattice Boltzmann model based on multiple relaxation times (United States)

    Lu, Jianhua; Han, Haifeng; Shi, Baochang; Guo, Zhaoli


    As an alterative version of the lattice Boltzmann models, the multiple relaxation time (MRT) lattice Boltzmann model introduces much less numerical boundary slip than the single relaxation time (SRT) lattice Boltzmann model if some special relationship between the relaxation time parameters is chosen. On the other hand, most current versions of the immersed boundary lattice Boltzmann method, which was first introduced by Feng and improved by many other authors, suffer from numerical boundary slip as has been investigated by Le and Zhang. To reduce such a numerical boundary slip, an immerse boundary lattice Boltzmann model based on multiple relaxation times is proposed in this paper. A special formula is given between two relaxation time parameters in the model. A rigorous analysis and the numerical experiments carried out show that the numerical boundary slip reduces dramatically by using the present model compared to the single-relaxation-time-based model.

  9. Exploring cluster Monte Carlo updates with Boltzmann machines (United States)

    Wang, Lei


    Boltzmann machines are physics informed generative models with broad applications in machine learning. They model the probability distribution of an input data set with latent variables and generate new samples accordingly. Applying the Boltzmann machines back to physics, they are ideal recommender systems to accelerate the Monte Carlo simulation of physical systems due to their flexibility and effectiveness. More intriguingly, we show that the generative sampling of the Boltzmann machines can even give different cluster Monte Carlo algorithms. The latent representation of the Boltzmann machines can be designed to mediate complex interactions and identify clusters of the physical system. We demonstrate these findings with concrete examples of the classical Ising model with and without four-spin plaquette interactions. In the future, automatic searches in the algorithm space parametrized by Boltzmann machines may discover more innovative Monte Carlo updates.

  10. Collisional interaction between metastable neon atoms

    Energy Technology Data Exchange (ETDEWEB)

    Drunen, Wouter Johannes van


    In this thesis, the study of cold gases of neon atoms in different metastable states is described. It contains measurements of the collisional parameters for both the 3s[3/2]{sub 2} and the 3s'[1/2]{sub 0} metastable state and the dependence of the inelastic loss on external fields. Furthermore, the investigation of frequency dependent laser-induced collisions, and the possibility to excite photoassociation resonances is presented. For the measurements described here, neon atoms have been confined in a magnetooptical trap, in a magnetostatic trap, or in an optical dipole trap, respectively. By laser cooling inside the magnetic trap, atomic samples with more than 95 percent occupation of the magnetic substate m{sub J} = +2 could be prepared. They have a typical temperature of 0.5 mK, central densities up to 10{sup 11} cm{sup -3}, and a central phase-space density of up to 2.2.10{sup -7}. After loading the optical dipole trap from the magnetic trap, 2.5.10{sup 6} atoms with typical temperatures of 0.1 mK, and central densities up to 5.10{sup 10} cm{sup -3} were trapped. By evaporative cooling of the atoms in the magnetic trap we could increase the phase-space density by a factor of 200 to 5.10{sup -5}. Investigating the frequency dependence of laser-induced collisions did not reveal an experimental signature for the excitation of photoassociation resonances. For the {sup 3}D{sub 3} line a frequency dependence of laser enhanced Penning ionization was observed. Measurement of the two-body loss coefficient as function of the magnetic field showed a field dependence of the inelastic loss. These losses increase towards both small and large offset fields. The implementation of an optical dipole trap allowed us to trap the {sup 3}P{sub 0} metastable state. From the trap loss measurements we determined the two-body loss coefficient of the {sup 3}P{sub 0} metastable state for both bosonic isotopes {sup 20}Ne and {sup 22}Ne. For {sup 20}Ne we obtained {beta}=6{sup +5}{sub

  11. A mass-conserving multiphase lattice Boltzmann model for simulation of multiphase flows (United States)

    Niu, Xiao-Dong; Li, You; Ma, Yi-Ren; Chen, Mu-Feng; Li, Xiang; Li, Qiao-Zhong


    In this study, a mass-conserving multiphase lattice Boltzmann (LB) model is proposed for simulating the multiphase flows. The proposed model developed in the present study is to improve the model of Shao et al. ["Free-energy-based lattice Boltzmann model for simulation of multiphase flows with density contrast," Phys. Rev. E 89, 033309 (2014)] by introducing a mass correction term in the lattice Boltzmann model for the interface. The model of Shao et al. [(the improved Zheng-Shu-Chew (Z-S-C model)] correctly considers the effect of the local density variation in momentum equation and has an obvious improvement over the Zheng-Shu-Chew (Z-S-C) model ["A lattice Boltzmann model for multiphase flows with large density ratio," J. Comput. Phys. 218(1), 353-371 (2006)] in terms of solution accuracy. However, due to the physical diffusion and numerical dissipation, the total mass of each fluid phase cannot be conserved correctly. To solve this problem, a mass correction term, which is similar to the one proposed by Wang et al. ["A mass-conserved diffuse interface method and its application for incompressible multiphase flows with large density ratio," J. Comput. Phys. 290, 336-351 (2015)], is introduced into the lattice Boltzmann equation for the interface to compensate the mass losses or offset the mass increase. Meanwhile, to implement the wetting boundary condition and the contact angle, a geometric formulation and a local force are incorporated into the present mass-conserving LB model. The proposed model is validated by verifying the Laplace law, simulating both one and two aligned droplets splashing onto a liquid film, droplets standing on an ideal wall, droplets with different wettability splashing onto smooth wax, and bubbles rising under buoyancy. Numerical results show that the proposed model can correctly simulate multiphase flows. It was found that the mass is well-conserved in all cases considered by the model developed in the present study. The developed

  12. Modelling viscoacoustic wave propagation with the lattice Boltzmann method. (United States)

    Xia, Muming; Wang, Shucheng; Zhou, Hui; Shan, Xiaowen; Chen, Hanming; Li, Qingqing; Zhang, Qingchen


    In this paper, the lattice Boltzmann method (LBM) is employed to simulate wave propagation in viscous media. LBM is a kind of microscopic method for modelling waves through tracking the evolution states of a large number of discrete particles. By choosing different relaxation times in LBM experiments and using spectrum ratio method, we can reveal the relationship between the quality factor Q and the parameter τ in LBM. A two-dimensional (2D) homogeneous model and a two-layered model are tested in the numerical experiments, and the LBM results are compared against the reference solution of the viscoacoustic equations based on the Kelvin-Voigt model calculated by finite difference method (FDM). The wavefields and amplitude spectra obtained by LBM coincide with those by FDM, which demonstrates the capability of the LBM with one relaxation time. The new scheme is relatively simple and efficient to implement compared with the traditional lattice methods. In addition, through a mass of experiments, we find that the relaxation time of LBM has a quantitative relationship with Q. Such a novel scheme offers an alternative forward modelling kernel for seismic inversion and a new model to describe the underground media.

  13. Lattice Boltzmann heat transfer model for permeable voxels (United States)

    Pereira, Gerald G.; Wu, Bisheng; Ahmed, Shakil


    We develop a gray-scale lattice Boltzmann (LB) model to study fluid flow combined with heat transfer for flow through porous media where voxels may be partially solid (or void). Heat transfer in rocks may lead to deformation, which in turn can modulate the fluid flow and so has significant contribution to rock permeability. The LB temperature field is compared to a finite difference solution of the continuum partial differential equations for fluid flow in a channel. Excellent quantitative agreement is found for both Poiseuille channel flow and Brinkman flow. The LB model is then applied to sample porous media such as packed beds and also more realistic sandstone rock sample, and both the convective and diffusive regimes are recovered when varying the thermal diffusivity. It is found that while the rock permeability can be comparatively small (order milli-Darcy), the temperature field can show significant variation depending on the thermal convection of the fluid. This LB method has significant advantages over other numerical methods such as finite and boundary element methods in dealing with coupled fluid flow and heat transfer in rocks which have irregular and nonsmooth pore spaces.

  14. High-order weighted essentially nonoscillatory finite-difference formulation of the lattice Boltzmann method in generalized curvilinear coordinates. (United States)

    Hejranfar, Kazem; Saadat, Mohammad Hossein; Taheri, Sina


    In this work, a high-order weighted essentially nonoscillatory (WENO) finite-difference lattice Boltzmann method (WENOLBM) is developed and assessed for an accurate simulation of incompressible flows. To handle curved geometries with nonuniform grids, the incompressible form of the discrete Boltzmann equation with the Bhatnagar-Gross-Krook (BGK) approximation is transformed into the generalized curvilinear coordinates and the spatial derivatives of the resulting lattice Boltzmann equation in the computational plane are solved using the fifth-order WENO scheme. The first-order implicit-explicit Runge-Kutta scheme and also the fourth-order Runge-Kutta explicit time integrating scheme are adopted for the discretization of the temporal term. To examine the accuracy and performance of the present solution procedure based on the WENOLBM developed, different benchmark test cases are simulated as follows: unsteady Taylor-Green vortex, unsteady doubly periodic shear layer flow, steady flow in a two-dimensional (2D) cavity, steady cylindrical Couette flow, steady flow over a 2D circular cylinder, and steady and unsteady flows over a NACA0012 hydrofoil at different flow conditions. Results of the present solution are compared with the existing numerical and experimental results which show good agreement. To show the efficiency and accuracy of the solution methodology, the results are also compared with the developed second-order central-difference finite-volume lattice Boltzmann method and the compact finite-difference lattice Boltzmann method. It is shown that the present numerical scheme is robust, efficient, and accurate for solving steady and unsteady incompressible flows even at high Reynolds number flows.

  15. First-principle description of collisional gyrokinetic turbulence in tokamak plasmas

    Energy Technology Data Exchange (ETDEWEB)

    Dif-Pradalier, G


    This dissertation starts in chapter 1 with a comprehensive introduction to nuclear fusion, its basic physics, goals and means. It especially defines the concept of a fusion plasma and some of its essential physical properties. The following chapter 2 discusses some fundamental concepts of statistical physics. It introduces the kinetic and the fluid frameworks, compares them and highlights their respective strengths and limitations. The end of the chapter is dedicated to the fluid theory. It presents two new sets of closure relations for fluid equations which retain important pieces of physics, relevant in the weakly collisional tokamak regimes: collective resonances which lead to Landau damping and entropy production. Nonetheless, since the evolution of the turbulence is intrinsically nonlinear and deeply influenced by velocity space effects, a kinetic collisional description is most relevant. First focusing on the kinetic aspect, chapter 3 introduces the so-called gyrokinetic framework along with the numerical solver - the GYSELA code - which will be used throughout this dissertation. Very generically, code solving is an initial value problem. The impact on turbulent nonlinear evolution of out of equilibrium initial conditions is discussed while studying transient flows, self-organizing dynamics and memory effects due to initial conditions. This dissertation introduces an operational definition, now of routine use in the GYSELA code, for the initial state and concludes on the special importance of the accurate calculation of the radial electric field. The GYSELA framework is further extended in chapter 4 to describe Coulomb collisions. The implementation of a collision operator acting on the full distribution function is presented. Its successful confrontation to collisional theory (neoclassical theory) is also shown. GYSELA is now part of the few gyrokinetic codes which can self-consistently address the interplay between turbulence and collisions. While

  16. Free surface entropic lattice Boltzmann simulations of film condensation on vertical hydrophilic plates

    DEFF Research Database (Denmark)

    Hygum, Morten Arnfeldt; Karlin, Iliya; Popok, Vladimir


    A model for vapor condensation on vertical hydrophilic surfaces is developed using the entropic lattice Boltzmann method extended with a free surface formulation of the evaporation–condensation problem. The model is validated with the steady liquid film formation on a flat vertical wall....... It is shown that the model is in a good agreement with the classical Nusselt equations for the laminar flow regime. Comparisons of the present model with other empirical models also demonstrate good agreement beyond the laminar regime. This allows the film condensation modeling at high film Reynolds numbers...

  17. Calculation of drag and torque coefficients by time-independent lattice-Boltzmann method. (United States)

    Ding, E J


    A method is developed to calculate the drag and torque coefficients of an isolated particle in a Stokes flow. The method is based on solving the time-independent lattice-Boltzmann equation. The advantage of this method is that the algorithm is easy to code, the method can be applied to any shape of the particle without complicated implementation, and the computational cost is independent of the shape of the particle. This method is validated and shown to be accurate by comparing with analytical solutions for certain problems.

  18. Impurity Transport in a Mixed-Collisionality Stellarator Plasma. (United States)

    Helander, P; Newton, S L; Mollén, A; Smith, H M


    A potential threat to the performance of magnetically confined fusion plasmas is the problem of impurity accumulation, which causes the concentration of highly charged impurity ions to rise uncontrollably in the center of the plasma and spoil the energy confinement by excessive radiation. It has long been thought that the collisional transport of impurities in stellarators always leads to such an accumulation (if the electric field points inwards, which is usually the case), whereas tokamaks, being axisymmetric, can benefit from "temperature screening," i.e., an outward flux of impurities driven by the temperature gradient. Here it is shown, using analytical techniques supported by results from a new numerical code, that such screening can arise in stellarator plasmas, too, and indeed does so in one of the most relevant operating regimes, where the impurities are highly collisional while the bulk plasma is at low collisionality.

  19. Magnetic susceptibility and Landau diamagnetism of quantum collisional plasma (United States)

    Latyshev, A. V.; Yushkanov, A. A.


    Quantum collisional plasma with an arbitrary degree of degeneracy of the electron gas is considered. Using the exact expression for the transverse electric conductivity of quantum collisional plasma, the magnetic susceptibility is described using the kinetic approach and a formula for calculating Landau diamagnetism is derived. Quantum Maxwellian plasma is considered as a special case. To this end, in the formulas derived, the limit is taken for the chemical potential tending to minus infinity. The properties of the magnetic susceptibility of quantum plasma are compared to those of degenerate and Maxwellian plasmas.

  20. Boltzmann babies in the proper time measure

    Energy Technology Data Exchange (ETDEWEB)

    Bousso, Raphael; Bousso, Raphael; Freivogel, Ben; Yang, I-Sheng


    After commenting briefly on the role of the typicality assumption in science, we advocate a phenomenological approach to the cosmological measure problem. Like any other theory, a measure should be simple, general, well defined, and consistent with observation. This allows us to proceed by elimination. As an example, we consider the proper time cutoff on a geodesic congruence. It predicts that typical observers are quantum fluctuations in the early universe, or Boltzmann babies. We sharpen this well-known youngness problem by taking into account the expansion and open spatial geometry of pocket universes. Moreover, we relate the youngness problem directly to the probability distribution for observables, such as the temperature of the cosmic background radiation. We consider a number of modifications of the proper time measure, but find none that would make it compatible with observation.

  1. Adaptive filtering for the lattice Boltzmann method (United States)

    Marié, Simon; Gloerfelt, Xavier


    In this study, a new selective filtering technique is proposed for the Lattice Boltzmann Method. This technique is based on an adaptive implementation of the selective filter coefficient σ. The proposed model makes the latter coefficient dependent on the shear stress in order to restrict the use of the spatial filtering technique in sheared stress region where numerical instabilities may occur. Different parameters are tested on 2D test-cases sensitive to numerical stability and on a 3D decaying Taylor-Green vortex. The results are compared to the classical static filtering technique and to the use of a standard subgrid-scale model and give significant improvements in particular for low-order filter consistent with the LBM stencil.

  2. The Lattice Boltzmann method principles and practice

    CERN Document Server

    Krüger, Timm; Kuzmin, Alexandr; Shardt, Orest; Silva, Goncalo; Viggen, Erlend Magnus


    This book is an introduction to the theory, practice, and implementation of the Lattice Boltzmann (LB) method, a powerful computational fluid dynamics method that is steadily gaining attention due to its simplicity, scalability, extensibility, and simple handling of complex geometries. The book contains chapters on the method's background, fundamental theory, advanced extensions, and implementation. To aid beginners, the most essential paragraphs in each chapter are highlighted, and the introductory chapters on various LB topics are front-loaded with special "in a nutshell" sections that condense the chapter's most important practical results. Together, these sections can be used to quickly get up and running with the method. Exercises are integrated throughout the text, and frequently asked questions about the method are dealt with in a special section at the beginning. In the book itself and through its web page, readers can find example codes showing how the LB method can be implemented efficiently on a va...

  3. Study on lattice Boltzmann method/large eddy simulation and its application at high Reynolds number flow

    Directory of Open Access Journals (Sweden)

    Haiqing Si


    Full Text Available Lattice Boltzmann method combined with large eddy simulation is developed in the article to simulate fluid flow at high Reynolds numbers. A subgrid model is used as a large eddy simulation model in the numerical simulation for high Reynolds flow. The idea of subgrid model is based on an assumption to include the physical effects that the unresolved motion has on the resolved fluid motion. It takes a simple form of eddy viscosity models for the Reynolds stress. Lift and drag evaluation in the lattice Boltzmann equation takes momentum-exchange method for curved body surface. First of all, the present numerical method is validated at low Reynolds numbers. Second, the developed lattice Boltzmann method/large eddy simulation method is performed to solve flow problems at high Reynolds numbers. Some detailed quantitative comparisons are implemented to show the effectiveness of the present method. It is demonstrated that lattice Boltzmann method combined with large eddy simulation model can efficiently simulate high Reynolds numbers’ flows.

  4. Fully implicit kinetic modelling of collisional plasmas

    Energy Technology Data Exchange (ETDEWEB)

    Mousseau, Vincent Andrew [Univ. of Idaho, Moscow, ID (United States)


    This dissertation describes a numerical technique, Matrix-Free Newton Krylov, for solving a simplified Vlasov-Fokker-Planck equation. This method is both deterministic and fully implicit, and may not have been a viable option before current developments in numerical methods. Results are presented that indicate the efficiency of the Matrix-Free Newton Krylov method for these fully-coupled, nonlinear integro-differential equations. The use and requirement for advanced differencing is also shown. To this end, implementations of Chang-Cooper differencing and flux limited Quadratic Upstream Interpolation for Convective Kinematics (QUICK) are presented. Results are given for a fully kinetic ion-electron problem with a self consistent electric field calculated from the ion and electron distribution functions. This numerical method, including advanced differencing, provides accurate solutions, which quickly converge on workstation class machines. It is demonstrated that efficient steady-state solutions can be achieved to the non-linear integro-differential equation, obtaining quadratic convergence, without incurring the large memory requirements of an integral operator. Model problems are presented which simulate plasma impinging on a plate with both high and low neutral particle recycling typical of a divertor in a Tokamak device. These model problems demonstrate the performance of the new solution method.

  5. Development of an Innovative Algorithm for Aerodynamics-Structure Interaction Using Lattice Boltzmann Method (United States)

    Mei, Ren-Wei; Shyy, Wei; Yu, Da-Zhi; Luo, Li-Shi; Rudy, David (Technical Monitor)


    The lattice Boltzmann equation (LBE) is a kinetic formulation which offers an alternative computational method capable of solving fluid dynamics for various systems. Major advantages of the method are owing to the fact that the solution for the particle distribution functions is explicit, easy to implement, and the algorithm is natural to parallelize. In this final report, we summarize the works accomplished in the past three years. Since most works have been published, the technical details can be found in the literature. Brief summary will be provided in this report. In this project, a second-order accurate treatment of boundary condition in the LBE method is developed for a curved boundary and tested successfully in various 2-D and 3-D configurations. To evaluate the aerodynamic force on a body in the context of LBE method, several force evaluation schemes have been investigated. A simple momentum exchange method is shown to give reliable and accurate values for the force on a body in both 2-D and 3-D cases. Various 3-D LBE models have been assessed in terms of efficiency, accuracy, and robustness. In general, accurate 3-D results can be obtained using LBE methods. The 3-D 19-bit model is found to be the best one among the 15-bit, 19-bit, and 27-bit LBE models. To achieve desired grid resolution and to accommodate the far field boundary conditions in aerodynamics computations, a multi-block LBE method is developed by dividing the flow field into various blocks each having constant lattice spacing. Substantial contribution to the LBE method is also made through the development of a new, generalized lattice Boltzmann equation constructed in the moment space in order to improve the computational stability, detailed theoretical analysis on the stability, dispersion, and dissipation characteristics of the LBE method, and computational studies of high Reynolds number flows with singular gradients. Finally, a finite difference-based lattice Boltzmann method is

  6. Theory of sheath in a collisional multi-component plasma

    Indian Academy of Sciences (India)

    Abstract. The aim of this brief report is to study the behaviour of sheath structure in a multi- component plasma with dust-neutral collisions. The plasma consists of electrons, ions, micron size negatively charged dust particles and neutrals. The sheath-edge potential and sheath width are cal- culated for collisionally ...

  7. Theory of sheath in a collisional multi-component plasma

    Indian Academy of Sciences (India)

    The aim of this brief report is to study the behaviour of sheath structure in a multicomponent plasma with dust-neutral collisions. The plasma consists of electrons, ions, micron size negatively charged dust particles and neutrals. The sheath-edge potential and sheath width are calculated for collisionally dominated sheath.

  8. Electrostatic sheath at the boundary of a collisional dusty plasma

    Indian Academy of Sciences (India)

    Moreover, the collisions of the dust charged grains with the neutrals are ex- pected to exhibit novel features. Keywords. ... of continuity and motion. On the inertial time scale of the cold but .... However, in a highly collisional atmosphere the drag force on the dust charged grains due to collisions becomes sufficiently large and ...

  9. An Efficient Statistical Method to Compute Molecular Collisional Rate Coefficients (United States)

    Loreau, Jérôme; Lique, François; Faure, Alexandre


    Our knowledge about the “cold” universe often relies on molecular spectra. A general property of such spectra is that the energy level populations are rarely at local thermodynamic equilibrium. Solving the radiative transfer thus requires the availability of collisional rate coefficients with the main colliding partners over the temperature range ∼10–1000 K. These rate coefficients are notoriously difficult to measure and expensive to compute. In particular, very few reliable collisional data exist for inelastic collisions involving reactive radicals or ions. In this Letter, we explore the use of a fast quantum statistical method to determine molecular collisional excitation rate coefficients. The method is benchmarked against accurate (but costly) rigid-rotor close-coupling calculations. For collisions proceeding through the formation of a strongly bound complex, the method is found to be highly satisfactory up to room temperature. Its accuracy decreases with decreasing potential well depth and with increasing temperature, as expected. This new method opens the way to the determination of accurate inelastic collisional data involving key reactive species such as {{{H}}}3+, H2O+, and H3O+ for which exact quantum calculations are currently not feasible.

  10. Collisional Losses from a Light-Force Atom Trap (United States)

    Sesko, D.; Walker, T.; Monroe, C.; Gallagher, A.; Wieman, C.

    We have studied the collisional loss rates for very cold cesium atoms held in a spontaneous-force optical trap. In contrast with previous work, we find that collisions involving excitation by the trapping light fields are the dominant loss mechanism. We also find that hyperfine-changing collisions between atoms in the ground state can be significant under some circumstances.

  11. Numerical study of active control of mixing in electro-osmotic flows by temperature difference using lattice Boltzmann methods. (United States)

    Alizadeh, A; Wang, J K; Pooyan, S; Mirbozorgi, S A; Wang, M


    In this paper, the effect of temperature difference between inlet flow and walls on the electro-osmotic flow through a two-dimensional microchannel is investigated. The main objective is to study the effect of temperature variations on the distribution of ions and consequently internal electric potential field, electric body force, and velocity fields in an electro-osmotic flow. We assume constant temperature and zeta potential on walls and use the mean temperature of each cross section to characterize the Boltzmann ion distribution across the channel. Based on these assumptions, the multiphysical transports are still able to be described by the classical Poisson-Boltzmann model. In this work, the Navier-Stokes equation for fluid flow, the Poisson-Boltzmann equation for ion distribution, and the energy equation for heat transfer are solved by a couple lattice Boltzmann method. The modeling results indicate that the temperature difference between walls and the inlet solution may lead to two symmetrical vortices at the entrance region of the microchannel which is appropriate for mixing enhancements. The advantage of this phenomenon for active control of mixing in electro-osmotic flow is the manageability of the vortex scale without extra efforts. For instance, the effective domain of this pattern could broaden by the following modulations: decreasing the external electric potential field, decreasing the electric double layer thickness, or increasing the temperature difference between inlet flow and walls. This work may provide a novel strategy for design or optimization of microsystems. Copyright © 2013 Elsevier Inc. All rights reserved.

  12. Testing the Maxwell-Boltzmann distribution using Brownian particles

    National Research Council Canada - National Science Library

    Mo, Jianyong; Simha, Akarsh; Kheifets, Simon; Raizen, Mark G


    .... We provide a direct verification of a modified Maxwell-Boltzmann velocity distribution and modified energy equipartition theorem that account for the kinetic energy of the liquid displaced by the particle...

  13. Lattice Boltzmann method fundamentals and engineering applications with computer codes

    CERN Document Server

    Mohamad, A A


    Introducing the Lattice Boltzmann Method in a readable manner, this book provides detailed examples with complete computer codes. It avoids the most complicated mathematics and physics without scarifying the basic fundamentals of the method.

  14. Reinforcement Learning via Boltzmann Machines Implemented on Quantum Annealers (United States)

    Levit, A.; Crawford, D.; Ronagh, P.; Oberoi, J.; Ghadermarzy, N.


    Recent studies have suggested promising application of quantum annealing devices as Boltzmann distribution samplers. We exploit this idea to develop a reinforcement learning framework using Boltzmann machines implemented on quantum annealing systems. We apply our method to find optimal policies in Markov decision processes. The evolution of the reinforcement learning algorithms relies on training a Neural Network. We use Quantum Monte-Carlo simulation of an associated network of qubits to provide experimental evidence of suitability of our approach.

  15. [Welding arc temperature field measurements based on Boltzmann spectrometry]. (United States)

    Si, Hong; Hua, Xue-Ming; Zhang, Wang; Li, Fang; Xiao, Xiao


    Arc plasma, as non-uniform plasma, has complicated energy and mass transport processes in its internal, so plasma temperature measurement is of great significance. Compared with absolute spectral line intensity method and standard temperature method, Boltzmann plot measuring is more accurate and convenient. Based on the Boltzmann theory, the present paper calculates the temperature distribution of the plasma and analyzes the principle of lines selection by real time scanning the space of the TIG are measurements.

  16. Thermohydrodynamics of an evaporating droplet studied using a multiphase lattice Boltzmann method. (United States)

    Zarghami, Ahad; Van den Akker, Harry E A


    In this paper, the thermohydrodynamics of an evaporating droplet is investigated by using a single-component pseudopotential lattice Boltzmann model. The phase change is applied to the model by adding source terms to the thermal lattice Boltzmann equation in such a way that the macroscopic energy equation of multiphase flows is recovered. In order to gain an exhaustive understanding of the complex hydrodynamics during evaporation, a single droplet is selected as a case study. At first, some tests for a stationary (non-)evaporating droplet are carried out to validate the method. Then the model is used to study the thermohydrodynamics of a falling evaporating droplet. The results show that the model is capable of reproducing the flow dynamics and transport phenomena of a stationary evaporating droplet quite well. Of course, a moving droplet evaporates faster than a stationary one due to the convective transport. Our study shows that our single-component model for simulating a moving evaporating droplet is limited to low Reynolds numbers.

  17. An improved lattice Boltzmann scheme for multiphase fluid with multi-range interactions

    Energy Technology Data Exchange (ETDEWEB)

    Maquignon, Nicolas; Duchateau, Julien; Roussel, Gilles; Rousselle, François; Renaud, Christophe [Laboratoire Informatique Signal et Image de la Côte d' Opale, 50 rue Ferdinand Buisson, 62100 Calais (France); Université du Littoral Côte d' Opale, 1 place de l' Yser, 59140, Dunkerque (France); Association INNOCOLD, MREI 1, 145 (France)


    Modeling of fluids with liquid to gas phase transition has become important for understanding many environmental or industrial processes. Such simulations need new techniques, because traditional solvers are often limited. The Lattice Boltzmann Model (LBM) allows simulate complex fluids, because its mesoscopic nature gives possibility to incorporate additional physics in comparison to usual methods. In this work, an improved lattice Boltzmann model for phase transition flow will be introduced. First, the state of art for Shan and Chen (SC) type of LBM will be reminded. Then, link to real thermodynamics will be established with Maxwell equal areas construction. Convergence to isothermal liquid vapor equilibrium will be shown and discussed. Inclusion of an equation of state for real fluid and better incorporation of force term is presented. Multi-range interactions have been used for SC model, but it hasn't been yet applied to real fluid with non-ideal equation of state. In this work, we evaluate this model when it is applied to real liquid-vapor equilibrium. We show that important differences are found for evaluation of gas density. In order to recover thermodynamic consistency, we use a new scheme for calculation of force term, which is a combination of multi range model and numerical weighting used by Gong and Cheng. We show the superiority of our new model by studying convergence to equilibrium values over a large temperature range. We prove that spurious velocities remaining at equilibrium are decreased.

  18. Two-relaxation-time lattice Boltzmann method for the anisotropic dispersive Henry problem (United States)

    Servan-Camas, Borja; Tsai, Frank T.-C.


    This study develops a lattice Boltzmann method (LBM) with a two-relaxation-time collision operator (TRT) to cope with anisotropic heterogeneous hydraulic conductivity and anisotropic velocity-dependent hydrodynamic dispersion in the saltwater intrusion problem. The directional-speed-of-sound technique is further developed to address anisotropic hydraulic conductivity and dispersion tensors. Forcing terms are introduced in the LBM to correct numerical errors that arise during the recovery procedure and to describe the sink/source terms in the flow and transport equations. In order to facilitate the LBM implementation, the forcing terms are combined with the equilibrium distribution functions (EDFs) to create pseudo-EDFs. This study performs linear stability analysis and derives LBM stability domains to solve the anisotropic advection-dispersion equation. The stability domains are used to select the time step at which the lattice Boltzmann method provides stable solutions to the numerical examples. The LBM was implemented for the anisotropic dispersive Henry problem with high ratios of longitudinal to transverse dispersivities, and the results compared well to the solutions in the work of Abarca et al. (2007).

  19. Galilean-invariant lattice-Boltzmann simulation of liquid-vapor interface dynamics. (United States)

    Kalarakis, A N; Burganos, V N; Payatakes, A C


    A two-dimensional two-phase lattice-Boltzmann model is presented and used for the study of interfacial phenomena under static and flow conditions. The model is based on the nonideal lattice-Boltzmann model proposed originally by Swift, Osborn, and Yeomans [Phys. Rev. Lett. 75, 830 (1995)] and makes it possible to couple a prescribed equation of state with the pressure tensor at the interface and the excess free-energy density formalism. The characteristic feature of the present model is that Galilean invariance is restored in the presence of interfaces without sacrificing any of the merits of the original model and, hence, the Navier-Stokes equation is adequately (to second order) recovered. The fluid properties can be prescribed in a thermodynamically consistent manner, which remains accurate at states close to the critical point. The model is first validated through static equilibrium tests and then applied to flow systems. It is shown that the simulator can reproduce some known two-phase flow configurations, like the motion of deformable droplets under the action of an external flow field. The simulator can also capture some interesting events during jet breakup and can be useful for the parametric study of the process in the two-dimensional case.

  20. Thermohydrodynamics of an evaporating droplet studied using a multiphase lattice Boltzmann method (United States)

    Zarghami, Ahad; Van den Akker, Harry E. A.


    In this paper, the thermohydrodynamics of an evaporating droplet is investigated by using a single-component pseudopotential lattice Boltzmann model. The phase change is applied to the model by adding source terms to the thermal lattice Boltzmann equation in such a way that the macroscopic energy equation of multiphase flows is recovered. In order to gain an exhaustive understanding of the complex hydrodynamics during evaporation, a single droplet is selected as a case study. At first, some tests for a stationary (non-)evaporating droplet are carried out to validate the method. Then the model is used to study the thermohydrodynamics of a falling evaporating droplet. The results show that the model is capable of reproducing the flow dynamics and transport phenomena of a stationary evaporating droplet quite well. Of course, a moving droplet evaporates faster than a stationary one due to the convective transport. Our study shows that our single-component model for simulating a moving evaporating droplet is limited to low Reynolds numbers.

  1. Simulations of viscoelastic fluids using a coupled lattice Boltzmann method: Transition states of elastic instabilities

    Directory of Open Access Journals (Sweden)

    Jin Su


    Full Text Available Elastic instabilities could happen in viscoelastic flows as the Weissenberg number is enlarged, and this phenomenon makes the numerical simulation of viscoelastic fluids more difficult. In this study, we introduce a coupled lattice Boltzmann method to solve the equations of viscoelastic fluids, which has a great capability of simulating the high Weissenberg number problem. Different from some traditional methods, two kinds of distribution functions are defined respectively for the evolution of the momentum and stress tensor equations. We mainly aim to investigate some key factors of the symmetry-breaking transition induced by elastic instability of viscoelastic fluids using this numerical coupled lattice Boltzmann method. In the results, we firstly find that the ratio of kinematical viscosity has an important influence on the transition of the elastic instability; the transition between the single stationary and cycling dominant vortex can be controlled via changing the ratio of kinematical viscosity in a periodic extensional flow. Finally, we can also observe a new transition state of instability for the flow showing the banded structure at higher Weissenberg number.

  2. Three-dimensional simplified and unconditionally stable lattice Boltzmann method for incompressible isothermal and thermal flows (United States)

    Chen, Z.; Shu, C.; Tan, D.


    In this paper, a three-dimensional simplified and unconditionally stable lattice Boltzmann method (3D-USLBM) is proposed for simulating incompressible isothermal/thermal flows. This method is developed by reconstructing solutions to the macroscopic governing equations recovered from the lattice Boltzmann equation and resolved in a predictor-corrector scheme. The final formulations of 3D-USLBM only involve the equilibrium and the non-equilibrium distribution functions. Among them, the former is calculated from the macroscopic variables and the latter is evaluated from the difference between two equilibrium distribution functions at different locations and time levels. Thus, 3D-USLBM directly tracks the evolution of macroscopic variables, which yields lower cost in virtual memory and facilitates the implementation of physical boundary conditions. A von Neumann stability analysis was performed on the present method to theoretically prove its unconditional stability. By imposing a regular Lagrange interpolation algorithm, this method can be flexibly extended to a non-uniform Cartesian mesh or body-fitted mesh with curved boundaries. Four numerical tests, that is, plane Poiseuille flow, 3D lid-driven cavity flow and 3D natural convection in a cubic cavity, and concentric annulus, were conducted to verify the stability, accuracy, and flexibility of the presented method.

  3. Discrete Boltzmann modeling of Rayleigh-Taylor instability in two-component compressible flows (United States)

    Lin, Chuandong; Xu, Aiguo; Zhang, Guangcai; Luo, Kai Hong; Li, Yingjun


    A discrete Boltzmann model (DBM) is proposed to probe the Rayleigh-Taylor instability (RTI) in two-component compressible flows. Each species has a flexible specific-heat ratio and is described by one discrete Boltzmann equation (DBE). Independent discrete velocities are adopted for the two DBEs. The collision and force terms in the DBE account for the molecular collision and external force, respectively. Two types of force terms are exploited. In addition to recovering the modified Navier-Stokes equations in the hydrodynamic limit, the DBM has the capability of capturing detailed nonequilibrium effects. Furthermore, we use the DBM to investigate the dynamic process of the RTI. The invariants of tensors for nonequilibrium effects are presented and studied. For low Reynolds numbers, both global nonequilibrium manifestations and the growth rate of the entropy of mixing show three stages (i.e., the reducing, increasing, and then decreasing trends) in the evolution of the RTI. On the other hand, the early reducing tendency is suppressed and even eliminated for high Reynolds numbers. Relevant physical mechanisms are analyzed and discussed.


    Directory of Open Access Journals (Sweden)

    Tuti Purwoningsih


    Full Text Available The migration of neutron is arranged by some probability distributions such as probability of spread distribution, probability of distance distribution, probability of energy distribution and probability of flux distribution. One application of these pattern distributions is modelling the reaction between neutron and elements which compose the tissue related to the absorption of neutron in brain cancer tissues. This article explores computation analysis of pattern of distribution of neutron flux in a reactor system. Variables were the amount of neutron simulated and the depth of cylindrical reactor system. Simulations showed that 20-120 minutes was needed in executing 100,000 neutrons to build the distribution pattern of neutrons flux. This pattern was also depended on the depth of the system. In all depths, the peak of neutron flux distribution pattern was in the 3rd bin. Comparison between this simulations and experiment results in literatures showed that by analyzing the simulation of the distribution of neutron flux, a Poisson distribution which follows the Maxwell-Boltzmann was resulted. Perpindahan neutron diatur dengan beberapa peluang distribusi, seperti peluang distribusi sudut hamburan, peluang distribusi jarak perpindahan, peluang distribusi energi transfer, serta peluang distribusi fluks neutron. Salah satu aplikasi dari pola distribusi ini adalah pemodelan reaksi antara neutron dengan elemen-elemen penyusun jaringan yang terkait dengan serapan neutron dan dosis yang terserap oleh jaringan tumor otak pada terapi BNCT (Boron Neutron Capture Therapy. Dalam penelitian ini dibahas analisis komputasi tentang pola distribusi fluks neutron dalam suatu sistem reaktor. Variabel dalam penelitian ini adalah banyaknya neutron yang disimulasikan, serta kedalaman sistem reaktor yang dalam penelitian ini menggunakan sistem reaktor berbentuk silinder. Hasil simulasi menunjukkan bahwa dengan neutron sebanyak 100.000 diperlukan waktu eksekusi sekitar

  5. Effect of gas mixing on physical properties of warm collisional helicon plasmas (United States)

    Kabir, M.; Niknam, A. R.


    The effect of inert gas mixing on the physical properties of a helicon plasma source with a Nagoya type III antenna is analytically investigated by taking into account the thermal and collisional effects. The dielectric permittivity tensor of this mixed gas plasma is obtained by using the Bhatnagar-Gross- Krook kinetic theory. Considering the dielectric tensor of mixed gas plasma and solving the electromagnetic field equations, the profiles of electromagnetic fields and plasma resistance are plotted and discussed. The results show that the plasma resistance peaks decrease with increasing Xe fraction in Ar-Xe plasma, and increase with the He fraction in Ar-He plasma. It is also shown that by increasing the xenon filling fraction, the electromagnetic field amplitudes are lowered, and by increasing the helium filling fraction, they are increased.

  6. A study of the propagation of ulf electromagnetic fields in collisional, inhomogeneous, magnetized plasmas

    Energy Technology Data Exchange (ETDEWEB)

    Borovsky, J.E.


    The propagation of ultralow-frequency (ulf) electromagnetic signals (Alfven and magnetosonic waves) in collisional, inhomogeneous, magnetized plasmas is analyzed by numerical simulation. The problem is formulated from a Maxwell-equation orbit-theory approach rather than from a magnetohydrodynamic point of view, and the problem is numerically treated in a fully time-dependent manner. Boundary-value-problem behavior is distinguished from initial-value-problem behavior. The propagation of two-dimensional small-amplitude electromagnetic disturbances in plasmas with spatially dependent densities and in plasmas with spatially dependent conductivities is numerically simulated, and when possible, the simulations are compared with theory. Changes in the plasma density lead to changes in the signal speed and to reflections; collisions lead to changes in the signal speed, to reflections, and to attenuations. Theoretical descriptions based upon discontinuities in the media are generally incorrect in predicting the amplitudes of signals reflected from plasma inhomogeneities. 19 refs., 16 figs.

  7. Is Poisson-Boltzmann theory insufficient for protein folding simulations? (United States)

    Lwin, Thu Zar; Zhou, Ruhong; Luo, Ray


    The Poisson-Boltzmann theory has been widely used in the studies of energetics and conformations of biological macromolecules. Recently, introduction of the efficient generalized Born approximation has greatly extended its applicability to areas such as protein folding simulations where highly efficient computation is crucial. However, limitations have been found in the folding simulations of a well-studied beta hairpin with several generalized Born implementations and different force fields. These studies have raised the question whether the underlining Poisson-Boltzmann theory, on which the generalized Born model is calibrated, is adequate in the treatment of polar interactions for the challenging protein folding simulations. To address the question whether the Poisson-Boltzmann theory in the current formalism might be insufficient, we directly tested our efficient numerical Poisson-Boltzmann implementation in the beta-hairpin folding simulation. Good agreement between simulation and experiment was found for the beta-hairpin equilibrium structures when the numerical Poisson-Boltzmann solvent and a recently improved generalized Born solvent were used. In addition simulated thermodynamic properties also agree well with experiment in both solvents. Finally, an overall agreement on the beta-hairpin folding mechanism was found between the current and previous studies. Thus, our simulations indicate that previously observed limitations are most likely due to imperfect calibration in previous generalized Born models but not due to the limitation of the Poisson-Boltzmann theory.

  8. Collisional rates for rotational transitions in H2CO and their application (United States)

    Sharma, Monika; Sharma, M. K.; Verma, U. P.; Chandra, Suresh


    Formaldehyde (H2CO) has always been of great importance for physicists. To analyze its spectrum collisional rate coefficients are required. Their computation is quite tedious job. We have calculated collisional rate coefficients for rotational transitions between 23 levels of each of the ortho and para species of H2CO for kinetic temperatures 10, 20, 30, 40, and 50 K. The scattering problem is analyzed with the help of the computer code MOLSCAT where the colliding partner is taken as the He atom. The required potential for interaction between H2CO and He is calculated with the help of the software GAUSSIAN 2003 where the coupled-cluster CCSD(T) method and cc-pVTZ basis set are used. The Basis Set Superposition Errors (BSSE) are accounted for. The wave functions for the asymmetric top molecule H2CO are expressed in terms of the Wigner D-functions and the expansion coefficients gJτK are obtained. For the interaction potential obtained with the help of GAUSSIAN 2003, MOLSCAT is used to derive the parameters q(L,M,M‧|E) as a function of energy E of the colliding partner. After averaging the parameters q(L,M,M‧|E) over the Maxwellian distribution, the parameters Q(L,M,M‧|T) as a function of the kinetic temperature T in the cloud are obtained. The results are compared with the available data. We have also calculated radiative transition probabilities (Einstein A-coefficients) for transitions between 23 rotational levels for each of the ortho and para species of H2CO. Finally, for ortho-H2CO, we have solved a set of 23 statistical equilibrium equations coupled with 39 equations of radiative transfer and discussed anomalous absorption of the 111-110 transition of H2CO at 4.830 GHz.

  9. Dynamic permeability of porous media by the lattice Boltzmann method (United States)

    Adler, P.; Pazdniakou, A.


    The main objective of our work is to determine the dynamic permeability of three dimensional porous media by means of the Lattice Boltzmann method (LBM). The Navier-Stokes equation can be numerically solved by LBM which is widely used to address various fluid dynamics problems. Space is discretized by a three-dimensional cubic lattice and time is discretized as well. The generally accepted notation for lattice Boltzmann models is DdQq where D stands for space dimension and Q for the number of discrete velocities. The present model is denoted by D3Q19. Moreover, the Two Relaxation Times variant of the Multi Relaxation Times model is implemented. Bounce back boundary conditions are used on the solid-fluid interfaces. The porous medium is spatially periodic. Reconstructed media were used; they are obtained by imposing a porosity and a correlation function characterized by a correlation length. Real samples can be obtained by MicroCT. In contrast with other previous contributions, the dynamic permeability K(omega) which is a complex number, is derived by imposing an oscillating body force of pulsation omega on the unit cell and by deriving the amplitude and the phase shift of the resulting time dependent seepage velocity. The influence of two limiting parameters, namely the Knudsen number Kn and the discretization for high frequencies, on K(omega) is carefully studied for the first time. Kn is proportional to nu/(cs H) where nu is the kinematic viscosity, cs the speed of sound in the fluid and H a characteristic length scale of the porous medium. Several porous media such as the classical plane Poiseuille flow and the reconstructed media are used to show that it is only for small enough values of Kn that reliable results are obtained. Otherwise, the data depend on Kn and may even be totally unphysical. However, it should be noticed that the limiting value of Kn could not be derived in general since it depends very much on the structure of the medium. Problems occur at

  10. Phase-field lattice Boltzmann modeling of boiling using a sharp-interface energy solver (United States)

    Mohammadi-Shad, Mahmood; Lee, Taehun


    The main objective of this paper is to extend an isothermal incompressible two-phase lattice Boltzmann equation method to model liquid-vapor phase change problems using a sharp-interface energy solver. Two discrete particle distribution functions, one for the continuity equation and the other for the pressure evolution and momentum equations, are considered in the current model. The sharp-interface macroscopic internal energy equation is discretized with an isotropic finite difference method to find temperature distribution in the system. The mass flow generated at liquid-vapor phase interface is embedded in the pressure evolution equation. The sharp-interface treatment of internal energy equation helps to find the interfacial mass flow rate accurately where no free parameter is needed in the calculations. The proposed model is verified against available theoretical solutions of the two-phase Stefan problem and the two-phase sucking interface problem, with which our simulation results are in good agreement. The liquid droplet evaporation in a superheated vapor, the vapor bubble growth in a superheated liquid, and the vapor bubble rising in a superheated liquid are analyzed and underlying physical characteristics are discussed in detail. The model is successfully tested for the liquid-vapor phase change with large density ratio up to 1000.

  11. AFMPB: An Adaptive Fast Multipole Poisson-Boltzmann Solver for Calculating Electrostatics in Biomolecular Systems (United States)

    Lu, Benzhuo; Cheng, Xiaolin; Huang, Jingfang; McCammon, J. Andrew


    A Fortran program package is introduced for rapid evaluation of the electrostatic potentials and forces in biomolecular systems modeled by the linearized Poisson-Boltzmann equation. The numerical solver utilizes a well-conditioned boundary integral equation (BIE) formulation, a node-patch discretization scheme, a Krylov subspace iterative solver package with reverse communication protocols, and an adaptive new version of fast multipole method in which the exponential expansions are used to diagonalize the multipole to local translations. The program and its full description, as well as several closely related libraries and utility tools are available at and a mirror site at This paper is a brief summary of the program: the algorithms, the implementation and the usage. PMID:20532187

  12. Coupling lattice Boltzmann model for simulation of thermal flows on standard lattices

    CERN Document Server

    Li, Q; He, Y L; Gao, Y J; Tao, W Q


    In this paper, a coupling lattice Boltzmann (LB) model for simulating thermal flows on the standard D2Q9 lattice is developed in the framework of the double-distribution-function (DDF) approach in which the viscous heat dissipation and compression work are considered. In the model, a density distribution function is used to simulate the flow field, while a total energy distribution function is employed to simulate the temperature field. The discrete equilibrium density and total energy distribution functions are obtained from the Hermite expansions of the corresponding continuous equilibrium distribution functions. The pressure given by the equation of state of perfect gases is recovered in the macroscopic momentum and energy equations. The coupling between the momentum and energy transports makes the model applicable for general thermal flows such as non-Boussinesq flows, while the existing DDF LB models on standard lattices are usually limited to Boussinesq flows in which the temperature variation is small....

  13. Analysis of the absorbing layers for the weakly-compressible lattice Boltzmann schemes

    CERN Document Server

    Xu, Hui


    It has been demonstrated that Lattice Boltzmann schemes (LBSs) are very efficient for Computational AeroAcoustics (CAA). In order to handle the issue of absorbing acoustic boundary conditions for LBS, three kinds of damping terms are proposed and added into the right hand sides of the governing equations of LBS. From the classical theory, these terms play an important role to absorb and minimize the acoustic wave reflections from computational boundaries. Meanwhile, the corresponding macroscopic equations with the damping terms are recovered for analyzing the macroscopic behaviors of the these damping terms and determining the critical absorbing strength. Further, in order to detect the dissipation and dispersion behaviors, the linearized LBS with the damping terms is derived and analyzed. The dispersive and dissipative properties are explored in the wave-number spaces via the Von Neumann analysis. The related damping strength critical values and the optimal absorbing term are addressed. Finally, some benchma...

  14. Modeling and simulation of ocean wave propagation using lattice Boltzmann method (United States)

    Nuraiman, Dian


    In this paper, we present on modeling and simulation of ocean wave propagation from the deep sea to the shoreline. This requires high computational cost for simulation with large domain. We propose to couple a 1D shallow water equations (SWE) model with a 2D incompressible Navier-Stokes equations (NSE) model in order to reduce the computational cost. The coupled model is solved using the lattice Boltzmann method (LBM) with the lattice Bhatnagar-Gross-Krook (BGK) scheme. Additionally, a special method is implemented to treat the complex behavior of free surface close to the shoreline. The result shows the coupled model can reduce computational cost significantly compared to the full NSE model.

  15. Lattice Boltzmann algorithm for continuum multicomponent flow (United States)

    Halliday, I.; Hollis, A. P.; Care, C. M.


    We present a multicomponent lattice Boltzmann simulation for continuum fluid mechanics, paying particular attention to the component segregation part of the underlying algorithm. In the principal result of this paper, the dynamics of a component index, or phase field, is obtained for a segregation method after U. D’Ortona [Phys. Rev. E 51, 3718 (1995)], due to Latva-Kokko and Rothman [Phys. Rev. E 71 056702 (2005)]. The said dynamics accord with a simulation designed to address multicomponent flow in the continuum approximation and underwrite improved simulation performance in two main ways: (i) by reducing the interfacial microcurrent activity considerably and (ii) by facilitating simulational access to regimes of flow with a low capillary number and drop Reynolds number [I. Halliday, R. Law, C. M. Care, and A. Hollis, Phys. Rev. E 73, 056708 (2006)]. The component segregation method studied, used in conjunction with Lishchuk’s method [S. V. Lishchuk, C. M. Care, and I. Halliday, Phys. Rev. E 67, 036701 (2003)], produces an interface, which is distributed in terms of its component index; however, the hydrodynamic boundary conditions which emerge are shown to support the notion of a sharp, unstructured, continuum interface.

  16. Accurate lineshape spectroscopy and the Boltzmann constant. (United States)

    Truong, G-W; Anstie, J D; May, E F; Stace, T M; Luiten, A N


    Spectroscopy has an illustrious history delivering serendipitous discoveries and providing a stringent testbed for new physical predictions, including applications from trace materials detection, to understanding the atmospheres of stars and planets, and even constraining cosmological models. Reaching fundamental-noise limits permits optimal extraction of spectroscopic information from an absorption measurement. Here, we demonstrate a quantum-limited spectrometer that delivers high-precision measurements of the absorption lineshape. These measurements yield a very accurate measurement of the excited-state (6P1/2) hyperfine splitting in Cs, and reveals a breakdown in the well-known Voigt spectral profile. We develop a theoretical model that accounts for this breakdown, explaining the observations to within the shot-noise limit. Our model enables us to infer the thermal velocity dispersion of the Cs vapour with an uncertainty of 35 p.p.m. within an hour. This allows us to determine a value for Boltzmann's constant with a precision of 6 p.p.m., and an uncertainty of 71 p.p.m.

  17. A combined Lattice Boltzmann and Immersed Boundary approach for predicting the vascular transport of differently shaped particles

    CERN Document Server

    Coclite, Alessandro; Pascazio, Giuseppe; Decuzzi, Paolo


    Modelling the vascular transport and adhesion of man-made particles is crucial for optimizing their efficacy in the detection and treatment of diseases. Here, a Lattice Boltzmann and Immersed Boundary methods are combined together for predicting the near wall dynamics of particles with different shapes in a laminar flow. For the lattice Boltzmann modelling, a Gauss-Hermite projection is used to derive the lattice equation, wall boundary conditions are imposed through the Zou-He framework, and a moving least squares algorithm accurately reconstructs the forcing term accounting for the immersed boundary. First, the computational code is validated against two well-known test cases: the sedimentation of circular and elliptical cylinders in a quiescent fluid. A very good agreement is observed between the present results and those available in the literature. Then, the transport of circular, elliptical, rectangular, square and triangular particles is analyzed in a Couette flow, at Re=20. All particles drifted later...

  18. Radiative and rovibrational collisional relaxation of sodium dimer (United States)

    Bayram, Burcin; Horton, Tim; McFarland, Jacob


    Radiative and rovibrational collisional relaxation of sodium dimer of the A1Σu+ (8,30) state have been measured by direct observation of the decay fluorescence. Sodium molecular vapor is created in a heatpipe oven at 600 K and excited using a 6-ns pulsed dye laser pumped by a Nd:YAG, operating at 532 nm. The preliminary lifetime measurement was done by directly acquiring lifetime data through boxcar averager from the stored oscilloscope trace of the fluorescence. Analysis of the exponential decay of the fluorescence allows us to obtain the radiative lifetime. By introducing the argon buffer gas and varying the pressure of the heatpipe, a collisional cross section between excited sodium dimer and ground state argon atom collision can be extracted using Stern-Volmer relation.

  19. Collisional broadening of angular correlations in a multiphase transport model (United States)

    Edmonds, Terrence; Li, Qingfeng; Wang, Fuqiang


    Systematic comparisons of jetlike correlation data to radiative and collisional energy loss model calculations are essential to extract transport properties of the quark-gluon medium created in relativistic heavy ion collisions. This paper presents a transport study of collisional broadening of jetlike correlations, by following parton-parton collision history in a multiphase transport (AMPT) model. The correlation shape is studied as functions of the number of parton-parton collisions suffered by a high transverse momentum probe parton (Ncoll) and the azimuth of the probe relative to the reaction plane (ϕfin.probe). Correlation is found to broaden with increasing Ncoll and ϕfin.probe from in- to out-of-plane direction. This study provides a transport model reference for future jet-medium interaction studies.

  20. An integrated Boltzmann+hydrodynamics approach to heavy ion collisions

    Energy Technology Data Exchange (ETDEWEB)

    Petersen, Hannah


    In this thesis the first fully integrated Boltzmann+hydrodynamics approach to relativistic heavy ion reactions has been developed. After a short introduction that motivates the study of heavy ion reactions as the tool to get insights about the QCD phase diagram, the most important theoretical approaches to describe the system are reviewed. The hadron-string transport approach that this work is based on is the Ultra-relativistic Quantum Molecular Dynamics (UrQMD) approach. Predictions for the charged particle multiplicities at LHC energies are made. The next step is the development of a new framework to calculate the baryon number density in a transport approach. Time evolutions of the net baryon number and the quark density have been calculated at AGS, SPS and RHIC energies. Studies of phase diagram trajectories using hydrodynamics are performed. The hybrid approach that has been developed as the main part of this thesis is based on the UrQMD transport approach with an intermediate hydrodynamical evolution for the hot and dense stage of the collision. The full (3+1) dimensional ideal relativistic one fluid dynamics evolution is solved using the SHASTA algorithm. Three different equations of state have been used, namely a hadron gas equation of state without a QGP phase transition, a chiral EoS and a bag model EoS including a strong first order phase transition. For the freeze-out transition from hydrodynamics to the cascade calculation two different set-ups are employed. The parameter dependences of the model are investigated and the time evolution of different quantities is explored. The hybrid model calculation is able to reproduce the experimentally measured integrated as well as transverse momentum dependent v{sub 2} values for charged particles. The multiplicity and mean transverse mass excitation function is calculated for pions, protons and kaons in the energy range from E{sub lab}=2-160 A GeV. The HBT correlation of the negatively charged pion source

  1. A multi-component lattice Boltzmann scheme: towards the mesoscale simulation of blood flow. (United States)

    Dupin, M M; Halliday, I; Care, C M


    While blood at the macroscopic scale is frequently treated as a continuum by techniques such as computational fluid dynamics, its mesoscale behaviour is not so well investigated or understood. At this scale, the deformability of each cell within the plasma is important and cannot be ignored. However there is currently a lack of efficient computational techniques able to simulate a large number of deformable particles such as blood cells. This paper addresses this problem and demonstrates the applicability of the authors' recent multi-component lattice Boltzmann method for the simulation of a large number of mutually immiscible liquid species [Dupin MM, Halliday I, Care CM. Multi-component lattice boltzmann equation for mesoscale blood flow. J Phys A: Math Gen 2003;36:8517-34]. In here, biological cells are treated as immiscible, deformable, and relatively viscous drops (compared to the surrounding fluid). The validation of the model is based on the work of Goldsmith on the flow of solid particles, deformable particles and red blood cells [Goldsmith HL, Marlow JC. Flow behavior of erythrocytes. II. Particle motions in concentrated suspensions of ghost cells. J Colloid Interf Sci 1979;71:383-407]. We demonstrate, in particular, that the model recovers Goldsmith's observations on the flow properties of red blood cells and also the experimental observations of Frank on the flow of solid beads [Frank M, Anderson D, Weeks ER, Morris JF. Particle migration in pressure-driven flow of a brownian suspension. J Fluid Mech 2003;493:363-78]. The current article is the first validation of our new lattice Boltzmann model for a large number of deformable particles in this context and demonstrates that the method provides a new, and effective, approach for the modeling of mesoscale blood flow.

  2. Normal and Anomalous Diffusion: An Analytical Study Based on Quantum Collision Dynamics and Boltzmann Transport Theory. (United States)

    Mahakrishnan, Sathiya; Chakraborty, Subrata; Vijay, Amrendra


    Diffusion, an emergent nonequilibrium transport phenomenon, is a nontrivial manifestation of the correlation between the microscopic dynamics of individual molecules and their statistical behavior observed in experiments. We present a thorough investigation of this viewpoint using the mathematical tools of quantum scattering, within the framework of Boltzmann transport theory. In particular, we ask: (a) How and when does a normal diffusive transport become anomalous? (b) What physical attribute of the system is conceptually useful to faithfully rationalize large variations in the coefficient of normal diffusion, observed particularly within the dynamical environment of biological cells? To characterize the diffusive transport, we introduce, analogous to continuous phase transitions, the curvature of the mean square displacement as an order parameter and use the notion of quantum scattering length, which measures the effective interactions between the diffusing molecules and the surrounding, to define a tuning variable, η. We show that the curvature signature conveniently differentiates the normal diffusion regime from the superdiffusion and subdiffusion regimes and the critical point, η = ηc, unambiguously determines the coefficient of normal diffusion. To solve the Boltzmann equation analytically, we use a quantum mechanical expression for the scattering amplitude in the Boltzmann collision term and obtain a general expression for the effective linear collision operator, useful for a variety of transport studies. We also demonstrate that the scattering length is a useful dynamical characteristic to rationalize experimental observations on diffusive transport in complex systems. We assess the numerical accuracy of the present work with representative experimental results on diffusion processes in biological systems. Furthermore, we advance the idea of temperature-dependent effective voltage (of the order of 1 μV or less in a biological environment, for example

  3. Lattice Boltzmann Methods to Address Fundamental Boiling and Two-Phase Problems

    Energy Technology Data Exchange (ETDEWEB)

    Uddin, Rizwan


    This report presents the progress made during the fourth (no cost extension) year of this three-year grant aimed at the development of a consistent Lattice Boltzmann formulation for boiling and two-phase flows. During the first year, a consistent LBM formulation for the simulation of a two-phase water-steam system was developed. Results of initial model validation in a range of thermo-dynamic conditions typical for Boiling Water Reactors (BWRs) were shown. Progress was made on several fronts during the second year. Most important of these included the simulation of the coalescence of two bubbles including the surface tension effects. Work during the third year focused on the development of a new lattice Boltzmann model, called the artificial interface lattice Boltzmann model (AILB model) for the 3 simulation of two-phase dynamics. The model is based on the principle of free energy minimization and invokes the Gibbs-Duhem equation in the formulation of non-ideal forcing function. This was reported in detail in the last progress report. Part of the efforts during the last (no-cost extension) year were focused on developing a parallel capability for the 2D as well as for the 3D codes developed in this project. This will be reported in the final report. Here we report the work carried out on testing the AILB model for conditions including the thermal effects. A simplified thermal LB model, based on the thermal energy distribution approach, was developed. The simplifications are made after neglecting the viscous heat dissipation and the work done by pressure in the original thermal energy distribution model. Details of the model are presented here, followed by a discussion of the boundary conditions, and then results for some two-phase thermal problems.

  4. Dual Lattice Boltzmann method for electrokinetic coupling : behavior at high and low salinities in rough channels. (United States)

    Fiorentino, Eve-Agnès; Toussaint, Renaud; Jouniaux, Laurence


    We study the coupling between hydraulic and electric flows in a porous medium at small scale using the Lattice Boltzmann method. This method is a computational fluid dynamics technique that is used for advection and diffusion modeling. We implement a coupled Lattice Boltzmann algorithm that solves both the mass transport and the electric field arising from charges displacements. The streaming potential and electroosmosis phenomena occur in a variety of situations and derive from this coupling. We focus on the streaming potential which is described using the ratio between the created potential difference and the applied pressure gradient. The streaming potential is assumed to be a linear function of the fluid conductivity, but experimental results highlight anomalous behaviors at low and high salinity. We try to account for them by setting extreme conditions that are likely to generate non-linearities. Several pore radii are tested so as to determine what is the effect of a radius that is comparable to the Debye length, the screening length of the electric potential, due to the ions in the electrolyte. The volumetric integral of the electrical current is calculated for comparison with the 2D simulations. High values of zeta potential are tested to verify if the discrepancy regarding the theoretical result is concentration-dependent. We try to include a surface conductivity term in the coefficient formulation. Some tests including a rugosity on the channel walls are performed. All of these attempts show a normal behaviour of the streaming potential at high salinity. We observe a decrease of the ratio at low conductivity, showing that this ratio is modified when the pore radius becomes negligible compared with the Debye length, which is physically meaningful in little pores at low concentrations. References : S. Pride. Governing equations for the coupled electromagnetics and acoustics of porous media. Physical Review B, 50 : 15678-15696, 1994. D. A. Wolf

  5. Kinetic model for the collisionless sheath of a collisional plasma

    Energy Technology Data Exchange (ETDEWEB)

    Tang, Xian-Zhu, E-mail:; Guo, Zehua, E-mail: [Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545 (United States)


    Collisional plasmas typically have mean-free-path still much greater than the Debye length, so the sheath is mostly collisionless. Once the plasma density, temperature, and flow are specified at the sheath entrance, the profile variation of electron and ion density, temperature, flow speed, and conductive heat fluxes inside the sheath is set by collisionless dynamics, and can be predicted by an analytical kinetic model distribution. These predictions are contrasted here with direct kinetic simulations, showing good agreement.



    E. Berthier; Delpech, J.-F.; Vuillemin, M.


    Collisional -Radiative numerical models are commonly used to design or interpret experiments in atomic physics of laser-created plasmas, including X-Ray laser studies. We describe our new code containing several options : average ion, more or less detailed configurations. It consists of an atomic data base coupled to subroutines evaluating ionic populations and emission and absorption coefficients. Numerical results are given to illustrate the capabilities of the code and to compare different...

  7. The radiative transport equation in flatland with separation of variables

    Energy Technology Data Exchange (ETDEWEB)

    Machida, Manabu, E-mail: [Department of Mathematical Sciences, The University of Tokyo, Komaba, Meguro, Tokyo 153-8914 (Japan)


    The linear Boltzmann equation can be solved with separation of variables in one dimension, i.e., in three-dimensional space with planar symmetry. In this method, solutions are given by superpositions of eigenmodes which are sometimes called singular eigenfunctions. In this paper, we explore the singular-eigenfunction approach in flatland or two-dimensional space.

  8. The radiative transport equation in flatland with separation of variables (United States)

    Machida, Manabu


    The linear Boltzmann equation can be solved with separation of variables in one dimension, i.e., in three-dimensional space with planar symmetry. In this method, solutions are given by superpositions of eigenmodes which are sometimes called singular eigenfunctions. In this paper, we explore the singular-eigenfunction approach in flatland or two-dimensional space.

  9. Phase-field-based multiple-relaxation-time lattice Boltzmann model for incompressible multiphase flows. (United States)

    Liang, H; Shi, B C; Guo, Z L; Chai, Z H


    In this paper, a phase-field-based multiple-relaxation-time lattice Boltzmann (LB) model is proposed for incompressible multiphase flow systems. In this model, one distribution function is used to solve the Chan-Hilliard equation and the other is adopted to solve the Navier-Stokes equations. Unlike previous phase-field-based LB models, a proper source term is incorporated in the interfacial evolution equation such that the Chan-Hilliard equation can be derived exactly and also a pressure distribution is designed to recover the correct hydrodynamic equations. Furthermore, the pressure and velocity fields can be calculated explicitly. A series of numerical tests, including Zalesak's disk rotation, a single vortex, a deformation field, and a static droplet, have been performed to test the accuracy and stability of the present model. The results show that, compared with the previous models, the present model is more stable and achieves an overall improvement in the accuracy of the capturing interface. In addition, compared to the single-relaxation-time LB model, the present model can effectively reduce the spurious velocity and fluctuation of the kinetic energy. Finally, as an application, the Rayleigh-Taylor instability at high Reynolds numbers is investigated.

  10. Lattice Boltzmann modelling of streaming potentials: variations with salinity in monophasic conditions (United States)

    Fiorentino, Eve-Agnès; Toussaint, Renaud; Jouniaux, Laurence


    The streaming potential phenomenon is produced by the flow of an electrolyte in a porous medium and is used for geophysical prospecting. It is quantified through an electrokinetic (EK) coefficient. The dependence of the EK coefficient on the conductivity of the electrolyte is described by the Helmholtz-Smoluchowski (HS) equation. This equation provides successful forecasts of the EK coefficient in the standard range of concentration. However, experimental measurements show deviations to this equation at extreme low and extreme high salinities. The aim of this study is to model the EK coefficient using Lattice Boltzmann simulations in a 2-D capillary channel, with a view to understanding these deviations. The effect of the constitutive parameters of the HS equation such as the permittivity and the viscosity is discussed. The validity of the HS equation using strong ζ potentials is assessed. Finally, a model of bulk fluid conductivity is derived. This model allows to take into account the change of local ionic distribution in the vicinity of the mineral. It appears to have a significant impact on the derivation of ζ potentials at low salinities and in the presence of polyvalent counterions.

  11. Lattice Boltzmann Model for Electronic Structure Simulations

    CERN Document Server

    Mendoza, M; Succi, S


    Recently, a new connection between density functional theory and kinetic theory has been proposed. In particular, it was shown that the Kohn-Sham (KS) equations can be reformulated as a macroscopic limit of the steady-state solution of a suitable single-particle kinetic equation. By using a discrete version of this new formalism, the exchange and correlation energies of simple atoms and the geometrical configuration of the methane molecule were calculated accurately. Here, we discuss the main ideas behind the lattice kinetic approach to electronic structure computations, offer some considerations for prospective extensions, and also show additional numerical results, namely the geometrical configuration of the water molecule.

  12. Surface Tension of Acid Solutions: Fluctuations beyond the Nonlinear Poisson-Boltzmann Theory. (United States)

    Markovich, Tomer; Andelman, David; Podgornik, Rudi


    We extend our previous study of surface tension of ionic solutions and apply it to acids (and salts) with strong ion-surface interactions, as described by a single adhesivity parameter for the ionic species interacting with the interface. We derive the appropriate nonlinear boundary condition with an effective surface charge due to the adsorption of ions from the bulk onto the interface. The calculation is done using the loop-expansion technique, where the zero loop (mean field) corresponds of the full nonlinear Poisson-Boltzmann equation. The surface tension is obtained analytically to one-loop order, where the mean-field contribution is a modification of the Poisson-Boltzmann surface tension and the one-loop contribution gives a generalization of the Onsager-Samaras result. Adhesivity significantly affects both contributions to the surface tension, as can be seen from the dependence of surface tension on salt concentration for strongly absorbing ions. Comparison with available experimental data on a wide range of different acids and salts allows the fitting of the adhesivity parameter. In addition, it identifies the regime(s) where the hypotheses on which the theory is based are outside their range of validity.

  13. A multi-component discrete Boltzmann model for nonequilibrium reactive flows. (United States)

    Lin, Chuandong; Luo, Kai Hong; Fei, Linlin; Succi, Sauro


    We propose a multi-component discrete Boltzmann model (DBM) for premixed, nonpremixed, or partially premixed nonequilibrium reactive flows. This model is suitable for both subsonic and supersonic flows with or without chemical reaction and/or external force. A two-dimensional sixteen-velocity model is constructed for the DBM. In the hydrodynamic limit, the DBM recovers the modified Navier-Stokes equations for reacting species in a force field. Compared to standard lattice Boltzmann models, the DBM presents not only more accurate hydrodynamic quantities, but also detailed nonequilibrium effects that are essential yet long-neglected by traditional fluid dynamics. Apart from nonequilibrium terms (viscous stress and heat flux) in conventional models, specific hydrodynamic and thermodynamic nonequilibrium quantities (high order kinetic moments and their departure from equilibrium) are dynamically obtained from the DBM in a straightforward way. Due to its generality, the developed methodology is applicable to a wide range of phenomena across many energy technologies, emissions reduction, environmental protection, mining accident prevention, chemical and process industry.

  14. A Non-Isothermal Chemical Lattice Boltzmann Model Incorporating Thermal Reaction Kinetics and Enthalpy Changes

    Directory of Open Access Journals (Sweden)

    Stuart Bartlett


    Full Text Available The lattice Boltzmann method is an efficient computational fluid dynamics technique that can accurately model a broad range of complex systems. As well as single-phase fluids, it can simulate thermohydrodynamic systems and passive scalar advection. In recent years, it also gained attention as a means of simulating chemical phenomena, as interest in self-organization processes increased. This paper will present a widely-used and versatile lattice Boltzmann model that can simultaneously incorporate fluid dynamics, heat transfer, buoyancy-driven convection, passive scalar advection, chemical reactions and enthalpy changes. All of these effects interact in a physically accurate framework that is simple to code and readily parallelizable. As well as a complete description of the model equations, several example systems will be presented in order to demonstrate the accuracy and versatility of the method. New simulations, which analyzed the effect of a reversible reaction on the transport properties of a convecting fluid, will also be described in detail. This extra chemical degree of freedom was utilized by the system to augment its net heat flux. The numerical method outlined in this paper can be readily deployed for a vast range of complex flow problems, spanning a variety of scientific disciplines.

  15. Higher-order Galilean-invariant lattice Boltzmann model for microflows: single-component gas. (United States)

    Yudistiawan, Wahyu Perdana; Kwak, Sang Kyu; Patil, D V; Ansumali, Santosh


    We introduce a scheme which gives rise to additional degree of freedom for the same number of discrete velocities in the context of the lattice Boltzmann model. We show that an off-lattice D3Q27 model exists with correct equilibrium to recover Galilean-invariant form of Navier-Stokes equation (without any cubic error). In the first part of this work, we show that the present model can capture two important features of the microflow in a single component gas: Knudsen boundary layer and Knudsen Paradox. Finally, we present numerical results corresponding to Couette flow for two representative Knudsen numbers. We show that the off-lattice D3Q27 model exhibits better accuracy as compared to more widely used on-lattice D3Q19 or D3Q27 model. Finally, our construction of discrete velocity model shows that there is no contradiction between entropic construction and quadrature-based procedure for the construction of the lattice Boltzmann model.

  16. Thermal transport in isotopically disordered carbon nanotubes: a comparison between Green's functions and Boltzmann approaches. (United States)

    Stoltz, G; Lazzeri, M; Mauri, F


    We present a study of the phononic thermal conductivity of isotopically disordered carbon nanotubes. In particular, the behaviour of the thermal conductivity as a function of the system length is investigated, using Green's function techniques to compute the transmission across the system. The method is implemented using linear scaling algorithms, which allow us to reach systems of lengths up to L = 2.5 µm (with up to 200 000 atoms). As for 1D systems, it is observed that the conductivity diverges with the system size L. We also observe a dramatic decrease of the thermal conductance for systems of experimental sizes (roughly 80% at room temperature for L = 2.5 µm), when a large fraction of isotopic disorder is introduced. The results obtained with Green's function techniques are compared to results obtained with a Boltzmann description of thermal transport. There is a good agreement between both approaches for systems of experimental sizes, even in the presence of Anderson localization. This is particularly interesting since the computation of the transmission using Boltzmann's equation is much less computationally expensive, so that larger systems may be studied with this method.

  17. Effects of Nanoparticles on Melting Process with Phase-Change Using the Lattice Boltzmann Method

    KAUST Repository

    Ibrahem, Ahmed M.


    In this work, the problem of nanoparticles dispersion effects on coupled heat transfer and solid-liquid phase change has been studied. The lattice Boltzmann method (LBM) enthalpy-based is employed. The collision model of lattice Bhatangar-Gross-Krook (LBGK) is used to solve the problem of 1D melting by conduction. On the other hand, we use the model of multi-distribution functions (MDF) to calculate the density, the velocity and the temperature for the problem of 2D melting by free convection, associated with different boundary conditions. In these simulations, the volume fractions of copper nanoparticles (0-2%) added to water-base fluid and Rayleigh numbers of 103to105. We use the Chapman-Enskog expansion to derive the governing macroscopic quantities from the mesoscopic lattice Boltzmann equation. The results obtained by these models have been compared to an analytical solution or other numerical methods. The effects of nanoparticles on conduction and natural convection during the melting process have been investigated. Moreover, the influences of nanoparticles on moving of the phase change front, the thermal conductivity and the latent heat of fusion are also studied.

  18. Tomography and generative training with quantum Boltzmann machines (United States)

    Kieferová, Mária; Wiebe, Nathan


    The promise of quantum neural nets, which utilize quantum effects to model complex data sets, has made their development an aspirational goal for quantum machine learning and quantum computing in general. Here we provide methods of training quantum Boltzmann machines. Our work generalizes existing methods and provides additional approaches for training quantum neural networks that compare favorably to existing methods. We further demonstrate that quantum Boltzmann machines enable a form of partial quantum state tomography that further provides a generative model for the input quantum state. Classical Boltzmann machines are incapable of this. This verifies the long-conjectured connection between tomography and quantum machine learning. Finally, we prove that classical computers cannot simulate our training process in general unless BQP=BPP , provide lower bounds on the complexity of the training procedures and numerically investigate training for small nonstoquastic Hamiltonians.

  19. Finite-element lattice Boltzmann simulations of contact line dynamics (United States)

    Matin, Rastin; Krzysztof Misztal, Marek; Hernández-García, Anier; Mathiesen, Joachim


    The lattice Boltzmann method has become one of the standard techniques for simulating a wide range of fluid flows. However, the intrinsic coupling of momentum and space discretization restricts the traditional lattice Boltzmann method to regular lattices. Alternative off-lattice Boltzmann schemes exist for both single- and multiphase flows that decouple the velocity discretization from the underlying spatial grid. The current study extends the applicability of these off-lattice methods by introducing a finite element formulation that enables simulating contact line dynamics for partially wetting fluids. This work exemplifies the implementation of the scheme and furthermore presents benchmark experiments that show the scheme reduces spurious currents at the liquid-vapor interface by at least two orders of magnitude compared to a nodal implementation and allows for predicting the equilibrium states accurately in the range of moderate contact angles.

  20. AFMPB: An adaptive fast multipole Poisson-Boltzmann solver for calculating electrostatics in biomolecular systems (United States)

    Lu, Benzhuo; Cheng, Xiaolin; Huang, Jingfang; McCammon, J. Andrew


    A Fortran program package is introduced for rapid evaluation of the electrostatic potentials and forces in biomolecular systems modeled by the linearized Poisson-Boltzmann equation. The numerical solver utilizes a well-conditioned boundary integral equation (BIE) formulation, a node-patch discretization scheme, a Krylov subspace iterative solver package with reverse communication protocols, and an adaptive new version of fast multipole method in which the exponential expansions are used to diagonalize the multipole-to-local translations. The program and its full description, as well as several closely related libraries and utility tools are available at and a mirror site at This paper is a brief summary of the program: the algorithms, the implementation and the usage. Program summaryProgram title: AFMPB: Adaptive fast multipole Poisson-Boltzmann solver Catalogue identifier: AEGB_v1_0 Program summary URL: Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: GPL 2.0 No. of lines in distributed program, including test data, etc.: 453 649 No. of bytes in distributed program, including test data, etc.: 8 764 754 Distribution format: tar.gz Programming language: Fortran Computer: Any Operating system: Any RAM: Depends on the size of the discretized biomolecular system Classification: 3 External routines: Pre- and post-processing tools are required for generating the boundary elements and for visualization. Users can use MSMS ( for pre-processing, and VMD ( for visualization. Sub-programs included: An iterative Krylov subspace solvers package from SPARSKIT by Yousef Saad (, and the fast multipole methods subroutines from FMMSuite ( http

  1. On the dispute between Boltzmann and Gibbs entropy

    Energy Technology Data Exchange (ETDEWEB)

    Buonsante, Pierfrancesco; Franzosi, Roberto, E-mail:; Smerzi, Augusto


    The validity of the concept of negative temperature has been recently challenged by arguing that the Boltzmann entropy (that allows negative temperatures) is inconsistent from a mathematical and statistical point of view, whereas the Gibbs entropy (that does not admit negative temperatures) provides the correct definition for the microcanonical entropy. Here we prove that the Boltzmann entropy is thermodynamically and mathematically consistent. Analytical results on two systems supporting negative temperatures illustrate the scenario we propose. In addition we numerically study a lattice system to show that negative temperature equilibrium states are accessible and obey standard statistical mechanics prediction.

  2. Revisiting Boltzmann learning: parameter estimation in Markov random fields

    DEFF Research Database (Denmark)

    Hansen, Lars Kai; Andersen, Lars Nonboe; Kjems, Ulrik


    This article presents a generalization of the Boltzmann machine that allows us to use the learning rule for a much wider class of maximum likelihood and maximum a posteriori problems, including both supervised and unsupervised learning. Furthermore, the approach allows us to discuss regularization...... and generalization in the context of Boltzmann machines. We provide an illustrative example concerning parameter estimation in an inhomogeneous Markov field. The regularized adaptation produces a parameter set that closely resembles the “teacher” parameters, hence, will produce segmentations that closely reproduce...

  3. On a Boltzmann-type price formation model

    KAUST Repository

    Burger, Martin


    In this paper, we present a Boltzmann-type price formation model, which is motivated by a parabolic free boundary model for the evolution of price presented by Lasry and Lions in 2007. We discuss the mathematical analysis of the Boltzmann-type model and show that its solutions converge to solutions of the model by Lasry and Lions as the transaction rate tends to infinity. Furthermore, we analyse the behaviour of the initial layer on the fast time scale and illustrate the price dynamics with various numerical experiments. © 2013 The Author(s) Published by the Royal Society. All rights reserved.

  4. Diffusion equation and spin drag in spin-polarized transport

    DEFF Research Database (Denmark)

    Flensberg, Karsten; Jensen, Thomas Stibius; Mortensen, Asger


    We study the role of electron-electron interactions for spin-polarized transport using the Boltzmann equation, and derive a set of coupled transport equations. For spin-polarized transport the electron-electron interactions are important, because they tend to equilibrate the momentum of the two......: it equilibrates the spin density imbalance and, provided it has a non-s-wave component, also a current imbalance....

  5. Evaluation of permeability and non-Darcy flow in vuggy macroporous limestone aquifer samples with lattice Boltzmann methods (United States)

    Sukop, Michael C.; Huang, Haibo; Alvarez, Pedro F.; Variano, Evan A.; Cunningham, Kevin J.


    Lattice Boltzmann flow simulations provide a physics-based means of estimating intrinsic permeability from pore structure and accounting for inertial flow that leads to departures from Darcy's law. Simulations were used to compute intrinsic permeability where standard measurement methods may fail and to provide better understanding of departures from Darcy's law under field conditions. Simulations also investigated resolution issues. Computed tomography (CT) images were acquired at 0.8 mm interscan spacing for seven samples characterized by centimeter-scale biogenic vuggy macroporosity from the extremely transmissive sole-source carbonate karst Biscayne aquifer in southeastern Florida. Samples were as large as 0.3 m in length; 7–9 cm-scale-length subsamples were used for lattice Boltzmann computations. Macroporosity of the subsamples was as high as 81%. Matrix porosity was ignored in the simulations. Non-Darcy behavior led to a twofold reduction in apparent hydraulic conductivity as an applied hydraulic gradient increased to levels observed at regional scale within the Biscayne aquifer; larger reductions are expected under higher gradients near wells and canals. Thus, inertial flows and departures from Darcy's law may occur under field conditions. Changes in apparent hydraulic conductivity with changes in head gradient computed with the lattice Boltzmann model closely fit the Darcy-Forchheimer equation allowing estimation of the Forchheimer parameter. CT-scan resolution appeared adequate to capture intrinsic permeability; however, departures from Darcy behavior were less detectable as resolution coarsened.

  6. Path-integral Mayer-sampling calculations of the quantum Boltzmann contribution to virial coefficients of helium-4. (United States)

    Shaul, Katherine R S; Schultz, Andrew J; Kofke, David A


    We present Mayer-sampling Monte Carlo calculations of the quantum Boltzmann contribution to the virial coefficients B(n), as defined by path integrals, for n = 2 to 4 and for temperatures from 2.6 K to 1000 K, using state-of-the-art ab initio potentials for interactions within pairs and triplets of helium-4 atoms. Effects of exchange are not included. The vapor-liquid critical temperature of the resulting fourth-order virial equation of state is 5.033(16) K, a value only 3% less than the critical temperature of helium-4: 5.19 K. We describe an approach for parsing the Boltzmann contribution into components that reduce the number of Mayer-sampling Monte Carlo steps required for components with large per-step time requirements. We estimate that in this manner the calculation of the Boltzmann contribution to B(3) at 2.6 K is completed at least 100 times faster than the previously reported approach.

  7. Lattice Boltzmann simulations of pressure-driven flows in microchannels using Navier–Maxwell slip boundary conditions

    KAUST Repository

    Reis, Tim


    We present lattice Boltzmann simulations of rarefied flows driven by pressure drops along two-dimensional microchannels. Rarefied effects lead to non-zero cross-channel velocities, nonlinear variations in the pressure along the channel. Both effects are absent in flows driven by uniform body forces. We obtain second-order accuracy for the two components of velocity the pressure relative to asymptotic solutions of the compressible Navier-Stokes equations with slip boundary conditions. Since the common lattice Boltzmann formulations cannot capture Knudsen boundary layers, we replace the usual discrete analogs of the specular diffuse reflection conditions from continuous kinetic theory with a moment-based implementation of the first-order Navier-Maxwell slip boundary conditions that relate the tangential velocity to the strain rate at the boundary. We use these conditions to solve for the unknown distribution functions that propagate into the domain across the boundary. We achieve second-order accuracy by reformulating these conditions for the second set of distribution functions that arise in the derivation of the lattice Boltzmann method by an integration along characteristics. Our moment formalism is also valuable for analysing the existing boundary conditions. It reveals the origin of numerical slip in the bounce-back other common boundary conditions that impose conditions on the higher moments, not on the local tangential velocity itself. © 2012 American Institute of Physics.

  8. Poisson-Boltzmann model for protein-surface electrostatic interactions and grid-convergence study using the PyGBe code

    CERN Document Server

    Cooper, Christopher D


    Interactions between surfaces and proteins occur in many vital processes and are crucial in biotechnology: the ability to control specific interactions is essential in fields like biomaterials, biomedical implants and biosensors. In the latter case, biosensor sensitivity hinges on ligand proteins adsorbing on bioactive surfaces with a favorable orientation, exposing reaction sites to target molecules. Protein adsorption, being a free-energy-driven process, is difficult to study experimentally. This paper develops and evaluates a computational model to study electrostatic interactions of proteins and charged nanosurfaces, via the Poisson-Boltzmann equation. We extended the implicit-solvent model used in the open-source code PyGBe to include surfaces of imposed charge or potential. This code solves the boundary integral formulation of the Poisson-Boltzmann equation, discretized with surface elements. PyGBe has at its core a treecode-accelerated Krylov iterative solver, resulting in O(N log N) scaling, with furt...

  9. Collisional-Radiative Nonequilibrium and Precursor Effects in a Nitrogen Shock Wave (United States)

    Cambier, Jean-Luc; Edwards, Thomas A. (Technical Monitor)


    Improvements to a plasma code with a Collisional-Radiative (CR) non-equilibrium model are made, allowing for a more accurate description of the physical processes. The code allows for non-Boltzmann distributions of the electronic excited states by convecting separately each excited state, as a pseudo-specie. Each molecular state has also its own vibrational temperature, while a global rotational temperature is assumed. The free electron temperature is different from those of the excited states, and the electron heat conduction is also included. The CR model also uses a unique coupling between chemistry and vibrational energy (C-V coupling), which is fully coherent, and has the property of establishing thermal equilibrium as well as chemical equilibrium, on its own. We have also included a coupling between electronic excitations and vibrational energy (X-V coupling), which can have a strong influence on the vibrational temperature of some states. The recent improvements include the multi- temperature dependence of the chemical rates for associative ionization, as well as the estimation of the internal energies transferred during this process. Additionally, the distribution of energy into different translational modes (electron and heavy particles) is now correctly modeled. This provides a very rapid heating mechanism for the free electrons, since it is found that the electrons are generated with an average thermal energy of the same order as the heavy particle translational energy. This effect was observed by Gorelov et al in a recent paper, and lead to pronounced peaks in electron temperature immediately behind the shock. We will attempt ro reproduce this phenomenon. The last modification concerns the inclusion of the radiative terms into the calculations, thus enabling us to observe the effect of radiative losses and radiation transport. Preliminary tests have shown that the radiative losses are not negligible, i.e. the shock velocity drops when the radiative

  10. Entropic Lattice Boltzmann study of hydrodynamics in a microcavity - Part 1

    Energy Technology Data Exchange (ETDEWEB)

    Karlin, I. V.; Ansumali, S.; Frouzakis, Ch. E.; Boulouchos, K. [Eidgenoessische Technische Hochschule (ETH), Labor fuer Aerothermochemie und Verbrennungssysteme ETHZ, ETH-Zentrum, Zuerich (Switzerland)


    This yearly report for 2004 presents a review of work being done on behalf of the Swiss Federal Office of Energy (SFOE) at the Laboratory for Aero-thermochemistry and Combustion Systems at the Federal Institute of Technology ETH in Zurich, Switzerland, on the development of a new approximation method for use in micrometer-scale flow calculations. The method, based on recently-developed so-called minimal entropy-kinetic models of the Boltzmann-kinetic equation, is discussed. Two detailed studies of micro-flows in specific geometries are discussed. The potential of the new method as a replacement for costly microscopic simulation methods is examined. The development and testing of a new thermal model - the so-called Thermal D2Q9 model - is discussed. A second study examined flows in a micro-cavity. A detailed parametric study of the quantitative and qualitative properties of the flows for a comprehensive range of dilution is mentioned.

  11. Spherical Harmonic Expansion Method for Coupled Electron-Phonon Boltzmann Transport (United States)

    Santia, Marco; Albrecht, John


    Thermoelectric transport modeling often relies on independent Boltzmann transport equations (BTEs) for electrons and phonons which work best near equilibrium (linearized) and steady-state. Device design relies heavily on this baseline approximation. Monte Carlo methods can allow for complex physical interactions (e.g., anharmonicity) but their stochastic nature has practical limits. Distribution functions with wide disparities in population (e.g., ratios >108 between majority and minority carriers.[1]) are a computational challenge. We present a coupled BTE solver based on a k-space spherical harmonic expansion (SHE) of the distribution functions and eigenstates of electrons and phonons. The method is deterministic and allows for detailed treatments of scattering processes, yet ameliorates the issues with population disparity within phase space. We set the formalism and examine the accuracy of the SHE for phonon band structures, calculate scattering rates determined within that representation, and compare our preliminary results for distribution statistics in control examples such as thermal conductivity and drift velocity.

  12. Lattice Boltzmann Method for Two-phase Flows on Unstructured Mesh (United States)

    Lee, Taehun; Baroudi, Lina; Wardle, Kent


    A lattice Boltzmann method with Galerkin finite element discretization (FE-LBM) is proposed to simulate incompressible two-phase flows on unstructured mesh. Two-distribution functions are used to recover the transport equations for the order parameter, pressure, and momentum. Consistent treatment of streaming and intermolecular forcing terms in FE-LBM enables us to use small equilibrium interface thickness compared with the existing two-phase LBMs and thus to achieve numerical stability at higher Reynolds number and large material property contrast. Several benchmark test cases with non-trivial wall boundaries will be presented, which include turbulent free surface flow inside a concentric rotating cylinder, drop impact on patterned surfaces, and bubbly flows. This work is partially supported by the DOE's NEUP.

  13. Planet signatures in collisionally active debris discs: scattered light images (United States)

    Thebault, P.; Kral, Q.; Ertel, S.


    Context. Planet perturbations have been often invoked as a potential explanation for many spatial structures that have been imaged in debris discs. So far this issue has been mostly investigated with pure N-body numerical models, which neglect the crucial effect collisions within the disc can have on the disc's response to dynamical perturbations. Aims: We numerically investigate how the coupled effect of collisions and radiation pressure can affect the formation and survival of radial and azimutal structures in a disc perturbed by a planet. We consider two different set-ups: a planet embedded within an extended disc and a planet exterior to an inner debris ring. One important issue we want to address is under which conditions a planet's signature can be observable in a collisionally active disc. Methods: We use our DyCoSS code, which is designed to investigate the structure of perturbed debris discs at dynamical and collisional steady-state, and derive synthetic images of the system in scattered light. The planet's mass and orbit, as well as the disc's collisional activity (parameterized by its average vertical optical depth τ0) are explored as free parameters. Results: We find that collisions always significantly damp planet-induced spatial structures. For the case of an embedded planet, the planet's signature, mostly a density gap around its radial position, should remain detectable in head-on images if Mplanet ≥ MSaturn. If the system is seen edge-on, however, inferring the presence of the planet is much more difficult, as only weak asymmetries remain in a collisionally active disc, although some planet-induced signatures might be observable under very favourable conditions. For the case of an inner ring and an external planet, planetary perturbations cannot prevent collision-produced small fragments from populating the regions beyond the ring. The radial luminosity profile exterior to the ring is in most cases close to the one it should have in the absence

  14. Multi-length and time scale thermal transport using the lattice Boltzmann method with application to electronics cooling

    Energy Technology Data Exchange (ETDEWEB)

    Escobar, Rodrigo A.; Amon, Cristina H. [Department of Mechanical Engineering and Institute for Complex Engineered Systems, Carnegie Mellon University, Pittsburgh, PA 15213 (United States); Ghai, Sartaj S.; Jhon, Myung S. [Department of Chemical Engineering and Institute for Complex Engineered Systems, Carnegie Mellon University, Pittsburgh, PA 15213 (United States)


    The lattice Boltzmann method (LBM) is used to investigate one-dimensional, multi-length and -time scale transient heat conduction in crystalline semiconductor solids, in which sub-continuum effects are important. The implementation of this method and its application to electronic devices are described. A silicon-on-insulator transistor subject to Joule heating conditions is used as a case study to illustrate the essence of the LBM. We compare our LBM results, for the diffusive to the ballistic transport regimes, with various hierarchical methodologies of heat transport such as the Fourier, Cattaneo, and ballistic-diffusive transport equations. (author)

  15. Numerical simulation of fluid flow and heat transfer inside a rotating disk-cylinder configuration by a lattice Boltzmann model (United States)

    Chen, Sheng; Tölke, Jonas; Krafczyk, Manfred


    A simple lattice Boltzmann model for numerical simulation of fluid flow and heat transfer inside a rotating disk-cylinder configuration, which is of fundamental interest and practical importance in science as well as in engineering, is proposed in this paper. Unlike existing lattice Boltzmann models for such flows, which were based on “primitive-variable” Navier-Stokes equations, the target macroscopic equations of the present model for the flow field are vorticity-stream function equations, inspired by our recent work designed for nonrotating flows [S. Chen, J. Tölke, and M. Krafczyk, Phys. Rev. E 79, 016704 (2009); S. Chen, J. Tölke, S. Geller, and M. Krafczyk, Phys. Rev. E 78, 046703 (2008)]. The flow field and the temperature field both are solved by the D2Q5 model. Compared with the previous models, the present model is more efficient, more stable, and much simpler. It was found that, even though with a relatively low grid resolution, the present model can still work well when the Grashof number is very high. The advantages of the present model are validated by numerical experiments.

  16. Immiscible multicomponent lattice Boltzmann model for fluids with ...

    Indian Academy of Sciences (India)

    An immiscible multicomponent lattice Boltzmann model is developed for fluids with high relaxation time ratios, which is based on the model proposed by Shan and Chen (SC). In the SC model, ... Ruofan Qiu1 Anlin Wang1. College of Mechanical Engineering, Tongji University, 4800# Cao'an Road, Shanghai 201804, China ...

  17. Distribution Learning in Evolutionary Strategies and Restricted Boltzmann Machines

    DEFF Research Database (Denmark)

    Krause, Oswin

    The thesis is concerned with learning distributions in the two settings of Evolutionary Strategies (ESs) and Restricted Boltzmann Machines (RBMs). In both cases, the distributions are learned from samples, albeit with different goals. Evolutionary Strategies are concerned with finding an optimum...

  18. Lattice Boltzmann scheme for diffusion on triangular grids

    NARCIS (Netherlands)

    Sman, van der R.G.M.


    In this paper we present a Lattice Boltzmann scheme for diffusion on it unstructured triangular grids. In this formulation of a LB for irregular grids there is no need for interpolation, which is required in other LB schemes on irregular grids. At the end of the propagation step the lattice gas

  19. Convection-diffusion lattice Boltzmann scheme for irregular lattices

    NARCIS (Netherlands)

    Sman, van der R.G.M.; Ernst, M.H.


    In this paper, a lattice Boltzmann (LB) scheme for convection diffusion on irregular lattices is presented, which is free of any interpolation or coarse graining step. The scheme is derived using the axioma that the velocity moments of the equilibrium distribution equal those of the

  20. Entropy and Galilean invariance of lattice Boltzmann theories. (United States)

    Chikatamarla, Shyam S; Karlin, Iliya V


    A theory of lattice Boltzmann (LB) models for hydrodynamic simulation is developed upon a novel relation between entropy construction and roots of Hermite polynomials. A systematic procedure is described for constructing numerically stable and complete Galilean invariant LB models. The stability of the new LB models is illustrated with a shock tube simulation.

  1. Diffusion on unstructured triangular grids using Lattice Boltzmann

    NARCIS (Netherlands)

    Sman, van der R.G.M.


    In this paper, we present a Lattice Boltzmann scheme for diffusion on unstructured triangular grids. In this formulation there is no need for interpolation, as is required in other LB schemes on irregular grids. At the end of the propagation step, the lattice gas particles arrive exactly at

  2. Measuring Boltzmann's Constant with Carbon Dioxide (United States)

    Ivanov, Dragia; Nikolov, Stefan


    In this paper we present two experiments to measure Boltzmann's constant--one of the fundamental constants of modern-day physics, which lies at the base of statistical mechanics and thermodynamics. The experiments use very basic theory, simple equipment and cheap and safe materials yet provide very precise results. They are very easy and…

  3. Lattice Boltzmann simulations of droplet formation during microchannel emulsification

    NARCIS (Netherlands)

    Zwan, van der E.A.; Sman, van der R.G.M.; Schroën, C.G.P.H.; Boom, R.M.


    In this study, we compared microchannel droplet formation in a microfluidics device with a two phase lattice Boltzmann simulation. The droplet formation was found to be qualitatively described, with a slightly smaller droplet in the simulation. This was due to the finite thickness of the interface

  4. Thermal, Microchannel, and Immersed Boundary Extension Validation for the Lattice-Boltzmann Method: Report 2 in Discrete Nano Scale Mechanics and Simulations Series (United States)


    Boundary Conditions D2Q9 2-dimentional lattice (D2) with nine discrete velocities (Q9) model IMB Immersed Moving Boundary Kn Knudsen number LBE...Lattice Boltzmann Equation LBM Lattice Boltzzmann Method NS Navier-Stokes method Nu Nusselt number Pr Prandtl number Ra Reynolds number ...kinetic theory was introduced. The LBM includes no continuum assumption with regard to the flow; rather the fluid is described by individual

  5. Progress in lattice Boltzmann methods for magnetohydrodynamic flows relevant to fusion applications

    Energy Technology Data Exchange (ETDEWEB)

    Pattison, M.J. [MetaHeuristics LLC, 3944 State St., Ste. 350, Santa Barbara, CA 93105 (United States)], E-mail:; Premnath, K.N. [MetaHeuristics LLC, 3944 State St., Ste. 350, Santa Barbara, CA 93105 (United States); UCSB, Chemical Engineering Department, Santa Barbara, CA 93106 (United States); Morley, N.B.; Abdou, M.A. [UCLA, MAE Department, 44-114 Engineering IV, 420 Westwood Pza, Los Angeles, CA 90095-1597 (United States)


    In this paper, an approach to simulating magnetohydrodynamic (MHD) flows based on the lattice Boltzmann method (LBM) is presented. The dynamics of the flow are simulated using a so-called multiple relaxation time (MRT) lattice Boltzmann equation (LBE), in which a source term is included for the Lorentz force. The evolution of the magnetic induction is represented by introducing a vector distribution function and then solving an appropriate lattice kinetic equation for this function. The solution of both distribution functions are obtained through a simple, explicit, and computationally efficient stream-and-collide procedure. The use of the MRT collision term enhances the numerical stability over that of a single relaxation time approach. To apply the methodology to solving practical problems, a new extrapolation-based method for imposing magnetic boundary conditions is introduced and a technique for simulating steady-state flows with low magnetic Prandtl number is developed. In order to resolve thin layers near the walls arising in the presence of high magnetic fields, a non-uniform gridding strategy is introduced through an interpolated-streaming step applied to both distribution functions. These advances are particularly important for applications in fusion engineering where liquid metal flows with low magnetic Prandtl numbers and high Hartmann numbers are introduced. A number of MHD benchmark problems, under various physical and geometrical conditions are presented, including 3-D MHD lid driven cavity flow, high Hartmann number flows and turbulent MHD flows, with good agreement with prior data. Due to the local nature of the method, the LBM also demonstrated excellent performance on parallel machines, with almost linear scaling up to 128 processors for a MHD flow problem.

  6. Effect of collisional energy loss on particle correlations in AMPT (United States)

    Wang, Fuqiang; Edmonds, Terrence; Li, Qingfeng


    Jet quenching is a powerful tool to study medium properties of relativistic heavy ion collisions via jet-medium interactions. Jet quenching studies have so far focused on high transverse momentum (pT) particle suppression. Jet shapes at low to intermediate pT, containing rich information on jet-medium interactions, have been less explored. In this talk, I will present a recent study, using a multiphase transport (AMPT) model, of effects on particle correlations from collisional energy loss of partons traversing the heavy ion medium. We follow the parton cascading history so that medium partons (associated particles) which have interacted with a high-pT probe parton (hard probe trigger particle) can be uniquely identified and hence no subtraction of combinatorial background is needed. Results on particle correlation shapes will be presented as a function of pT, the number of parton-parton collisions suffered by the probe parton, and the azimuthal angle of the probe parton relative to the reaction plane. These results reveal pathlength dependence of collisional energy loss.


    Energy Technology Data Exchange (ETDEWEB)

    Nesvold, Erika R. [Department of Physics, University of Maryland Baltimore County 1000 Hilltop Circle Baltimore, MD 21250 (United States); Kuchner, Marc J., E-mail:, E-mail: [NASA Goddard Space Flight Center Exoplanets and Stellar Astrophysics Laboratory, Code 667 Greenbelt, MD 21230 (United States)


    We apply our 3D debris disk model, SMACK, to simulate a planet on a circular orbit near a ring of planetesimals that are experiencing destructive collisions. Previous simulations of a planet opening a gap in a collisionless debris disk have found that the width of the gap scales as the planet mass to the 2/7th power (α = 2/7). We find that gap sizes in a collisional disk still obey a power law scaling with planet mass, but that the index α of the power law depends on the age of the system t relative to the collisional timescale t {sub coll} of the disk by α = 0.32(t/t {sub coll}){sup –0.04}, with inferred planet masses up to five times smaller than those predicted by the classical gap law. The increased gap sizes likely stem from the interaction between collisions and the mean motion resonances near the chaotic zone. We investigate the effects of the initial eccentricity distribution of the disk particles and find a negligible effect on the gap size at Jovian planet masses, since collisions tend to erase memory of the initial particle eccentricity distributions. Finally, we find that the presence of Trojan analogs is a potentially powerful diagnostic of planets in the mass range ∼1-10 M {sub Jup}. We apply our model to place new upper limits on planets around Fomalhaut, HR 4796 A, HD 202628, HD 181327, and β Pictoris.

  8. Dynamical and collisional evolution of Kuiper belt binaries (United States)

    Brunini, Adrián; Zanardi, Macarena


    We present numerical simulations of the evolution of synthetic transneptunian binaries (TNBs) under the influence of the solar perturbation, tidal friction, and collisions with the population of classical Kuiper belt objects (KBOs). We show that these effects, acting together, have strongly sculpted the primordial population of TNBs. If the population of classical KBOs have a power-law size distribution as the ones that are inferred from recent observational surveys, the fraction of surviving binaries at present would be ˜70 per cent of the primordial population. The orbits of the surviving synthetic systems match reasonably well the observed sample. The collisional process excites the mutual orbital eccentricity of the binaries, acting against the effect of tides. Therefore only ˜10 per cent of the objects reach total orbital circularization (e ≤ 10-4). In addition, our results predict that the population of contact binaries in the transneptunian region should be small. Ultrawide binaries are naturally obtained by the combined action of Kozai cycles and tidal friction and collisional evolution, being the number and orbital distribution of them very similar to the ones of the observed population.

  9. Forcing scheme analysis for the axisymmetric lattice Boltzmann method under incompressible limit. (United States)

    Zhang, Liangqi; Yang, Shiliang; Zeng, Zhong; Chen, Jie; Yin, Linmao; Chew, Jia Wei


    Because the standard lattice Boltzmann (LB) method is proposed for Cartesian Navier-Stokes (NS) equations, additional source terms are necessary in the axisymmetric LB method for representing the axisymmetric effects. Therefore, the accuracy and applicability of the axisymmetric LB models depend on the forcing schemes adopted for discretization of the source terms. In this study, three forcing schemes, namely, the trapezium rule based scheme, the direct forcing scheme, and the semi-implicit centered scheme, are analyzed theoretically by investigating their derived macroscopic equations in the diffusive scale. Particularly, the finite difference interpretation of the standard LB method is extended to the LB equations with source terms, and then the accuracy of different forcing schemes is evaluated for the axisymmetric LB method. Theoretical analysis indicates that the discrete lattice effects arising from the direct forcing scheme are part of the truncation error terms and thus would not affect the overall accuracy of the standard LB method with general force term (i.e., only the source terms in the momentum equation are considered), but lead to incorrect macroscopic equations for the axisymmetric LB models. On the other hand, the trapezium rule based scheme and the semi-implicit centered scheme both have the advantage of avoiding the discrete lattice effects and recovering the correct macroscopic equations. Numerical tests applied for validating the theoretical analysis show that both the numerical stability and the accuracy of the axisymmetric LB simulations are affected by the direct forcing scheme, which indicate that forcing schemes free of the discrete lattice effects are necessary for the axisymmetric LB method.

  10. Hypocoercivity for linear kinetic equations conserving mass

    KAUST Repository

    Dolbeault, Jean


    We develop a new method for proving hypocoercivity for a large class of linear kinetic equations with only one conservation law. Local mass conservation is assumed at the level of the collision kernel, while transport involves a confining potential, so that the solution relaxes towards a unique equilibrium state. Our goal is to evaluate in an appropriately weighted $ L^2$ norm the exponential rate of convergence to the equilibrium. The method covers various models, ranging from diffusive kinetic equations like Vlasov-Fokker-Planck equations, to scattering models or models with time relaxation collision kernels corresponding to polytropic Gibbs equilibria, including the case of the linear Boltzmann model. In this last case and in the case of Vlasov-Fokker-Planck equations, any linear or superlinear growth of the potential is allowed. - See more at:

  11. Study of the Dynamics of a Condensing Bubble Using Lattice Boltzmann Method

    Directory of Open Access Journals (Sweden)

    Shahnawaz Ahmed


    Full Text Available Mesoscopic lattice Boltzmann method (LBM is used to discretize the governing equations for a steam bubble inside a tube filled with water. The bubbles are kept at higher temperature compared to its boiling point while the liquid is kept subcooled. Heat transfer is allowed to take place between the two phases by virtue of which the bubble will condense. Three separate probability distribution functions are used in LBM to handle continuity, momentum and energy equations separately. The interface is considered to be diffused within a narrow zone and it has been modeled using convective Cahn-Hillard equation. Combined diffused interface-LBM framework is adapted accordingly to handle complex interface separating two phases having high density ratio. Developed model is validated with respect to established correlations for instantaneous equivalent radius of a spherical condensing bubble. Numerical snapshots of the simulation depict that the bubble volume decreases faster for higher degree of superheat. The degrees of superheat are varied over a wide range to note its effect on bubble shape and size. Effect of initial volume of the bubble on the condensation rate is also studied. It has been observed that for a fixed degree of superheat, the condensation rate is not exactly proportional to its volume. Due to the variation in interfacial configuration for different sized bubbles, condensation rate changes drastically. Influence of gravity on the rate of condensation is also studied using the developed methodology.

  12. Free-energy functionals of the electrostatic potential for Poisson-Boltzmann theory. (United States)

    Jadhao, Vikram; Solis, Francisco J; de la Cruz, Monica Olvera


    In simulating charged systems, it is often useful to treat some ionic components of the system at the mean-field level and solve the Poisson-Boltzmann (PB) equation to get their respective density profiles. The numerically intensive task of solving the PB equation at each step of the simulation can be bypassed using variational methods that treat the electrostatic potential as a dynamic variable. But such approaches require the access to a true free-energy functional: a functional that not only provides the correct solution of the PB equation upon extremization, but also evaluates to the true free energy of the system at its minimum. Moreover, the numerical efficiency of such procedures is further enhanced if the free-energy functional is local and is expressed in terms of the electrostatic potential. Existing PB functionals of the electrostatic potential, while possessing the local structure, are not free-energy functionals. We present a variational formulation with a local free-energy functional of the potential. In addition, we also construct a nonlocal free-energy functional of the electrostatic potential. These functionals are suited for employment in simulation schemes based on the ideas of dynamical optimization.

  13. Plasma-statistical models of the atom in the theory of some collisional and radiative processes

    NARCIS (Netherlands)

    Astapenko, VA


    A plasma-statistical model was used to describe collisional and radiative processes involving target ionization, namely, collisional ionization of atoms and incoherent polarization bremsstrahlung. The cross sections of these processes were expressed through the Compton profile of X-ray scattering,

  14. Hill's equation

    CERN Document Server

    Magnus, Wilhelm


    The hundreds of applications of Hill's equation in engineering and physics range from mechanics and astronomy to electric circuits, electric conductivity of metals, and the theory of the cyclotron. New applications are continually being discovered and theoretical advances made since Liapounoff established the equation's fundamental importance for stability problems in 1907. Brief but thorough, this volume offers engineers and mathematicians a complete orientation to the subject.""Hill's equation"" connotes the class of homogeneous, linear, second order differential equations with real, period

  15. Implementation of a high-order compact finite-difference lattice Boltzmann method in generalized curvilinear coordinates (United States)

    Hejranfar, Kazem; Ezzatneshan, Eslam


    In this work, the implementation of a high-order compact finite-difference lattice Boltzmann method (CFDLBM) is performed in the generalized curvilinear coordinates to improve the computational efficiency of the solution algorithm to handle curved geometries with non-uniform grids. The incompressible form of the discrete Boltzmann equation with the Bhatnagar-Gross-Krook (BGK) approximation with the pressure as the independent dynamic variable is transformed into the generalized curvilinear coordinates. Herein, the spatial derivatives in the resulting lattice Boltzmann (LB) equation in the computational plane are discretized by using the fourth-order compact finite-difference scheme and the temporal term is discretized with the fourth-order Runge-Kutta scheme to provide an accurate and efficient incompressible flow solver. A high-order spectral-type low-pass compact filter is used to regularize the numerical solution and remove spurious waves generated by boundary conditions, flow non-linearities and grid non-uniformity. All boundary conditions are implemented based on the solution of governing equations in the generalized curvilinear coordinates. The accuracy and efficiency of the solution methodology presented are demonstrated by computing different benchmark steady and unsteady incompressible flow problems. A sensitivity study is also conducted to evaluate the effects of grid size and filtering on the accuracy and convergence rate of the solution. Four test cases considered herein for validating the present computations and demonstrating the accuracy and robustness of the solution algorithm are: unsteady Couette flow and steady flow in a 2-D cavity with non-uniform grid and steady and unsteady flows over a circular cylinder and the NACA0012 hydrofoil at different flow conditions. Results obtained for the above test cases are in good agreement with the existing numerical and experimental results. The study shows the present solution methodology based on the

  16. Application of Lattice Boltzmann Methods in Complex Mass Transfer Systems (United States)

    Sun, Ning

    Lattice Boltzmann Method (LBM) is a novel computational fluid dynamics method that can easily handle complex and dynamic boundaries, couple local or interfacial interactions/reactions, and be easily parallelized allowing for simulation of large systems. While most of the current studies in LBM mainly focus on fluid dynamics, however, the inherent power of this method makes it an ideal candidate for the study of mass transfer systems involving complex/dynamic microstructures and local reactions. In this thesis, LBM is introduced to be an alternative computational method for the study of electrochemical energy storage systems (Li-ion batteries (LIBs) and electric double layer capacitors (EDLCs)) and transdermal drug design on mesoscopic scale. Based on traditional LBM, the following in-depth studies have been carried out: (1) For EDLCs, the simulation of diffuse charge dynamics is carried out for both the charge and the discharge processes on 2D systems of complex random electrode geometries (pure random, random spheres and random fibers). Steric effect of concentrated solutions is considered by using modified Poisson-Nernst-Plank (MPNP) equations and compared with regular Poisson-Nernst-Plank (PNP) systems. The effects of electrode microstructures (electrode density, electrode filler morphology, filler size, etc.) on the net charge distribution and charge/discharge time are studied in detail. The influence of applied potential during discharging process is also discussed. (2) For the study of dendrite formation on the anode of LIBs, it is shown that the Lattice Boltzmann model can capture all the experimentally observed features of microstructure evolution at the anode, from smooth to mossy to dendritic. The mechanism of dendrite formation process in mesoscopic scale is discussed in detail and compared with the traditional Sand's time theories. It shows that dendrite formation is closely related to the inhomogeneous reactively at the electrode-electrolyte interface

  17. Numerical investigation of the pseudopotential lattice Boltzmann modeling of liquid-vapor for multi-phase flows (United States)

    Nemati, Maedeh; Shateri Najaf Abady, Ali Reza; Toghraie, Davood; Karimipour, Arash


    The incorporation of different equations of state into single-component multiphase lattice Boltzmann model is considered in this paper. The original pseudopotential model is first detailed, and several cubic equations of state, the Redlich-Kwong, Redlich-Kwong-Soave, and Peng-Robinson are then incorporated into the lattice Boltzmann model. A comparison of the numerical simulation achievements on the basis of density ratios and spurious currents is used for presentation of the details of phase separation in these non-ideal single-component systems. The paper demonstrates that the scheme for the inter-particle interaction force term as well as the force term incorporation method matters to achieve more accurate and stable results. The velocity shifting method is demonstrated as the force term incorporation method, among many, with accuracy and stability results. Kupershtokh scheme also makes it possible to achieve large density ratio (up to 104) and to reproduce the coexistence curve with high accuracy. Significant reduction of the spurious currents at vapor-liquid interface is another observation. High-density ratio and spurious current reduction resulted from the Redlich-Kwong-Soave and Peng-Robinson EOSs, in higher accordance with the Maxwell construction results.

  18. Riemann-Theta Boltzmann Machine arXiv

    CERN Document Server

    Krefl, Daniel; Haghighat, Babak; Kahlen, Jens

    A general Boltzmann machine with continuous visible and discrete integer valued hidden states is introduced. Under mild assumptions about the connection matrices, the probability density function of the visible units can be solved for analytically, yielding a novel parametric density function involving a ratio of Riemann-Theta functions. The conditional expectation of a hidden state for given visible states can also be calculated analytically, yielding a derivative of the logarithmic Riemann-Theta function. The conditional expectation can be used as activation function in a feedforward neural network, thereby increasing the modelling capacity of the network. Both the Boltzmann machine and the derived feedforward neural network can be successfully trained via standard gradient- and non-gradient-based optimization techniques.

  19. Accelerated Monte Carlo simulations with restricted Boltzmann machines (United States)

    Huang, Li; Wang, Lei


    Despite their exceptional flexibility and popularity, Monte Carlo methods often suffer from slow mixing times for challenging statistical physics problems. We present a general strategy to overcome this difficulty by adopting ideas and techniques from the machine learning community. We fit the unnormalized probability of the physical model to a feed-forward neural network and reinterpret the architecture as a restricted Boltzmann machine. Then, exploiting its feature detection ability, we utilize the restricted Boltzmann machine to propose efficient Monte Carlo updates to speed up the simulation of the original physical system. We implement these ideas for the Falicov-Kimball model and demonstrate an improved acceptance ratio and autocorrelation time near the phase transition point.

  20. Comparison of Einstein-Boltzmann solvers for testing general relativity (United States)

    Bellini, E.; Barreira, A.; Frusciante, N.; Hu, B.; Peirone, S.; Raveri, M.; Zumalacárregui, M.; Avilez-Lopez, A.; Ballardini, M.; Battye, R. A.; Bolliet, B.; Calabrese, E.; Dirian, Y.; Ferreira, P. G.; Finelli, F.; Huang, Z.; Ivanov, M. M.; Lesgourgues, J.; Li, B.; Lima, N. A.; Pace, F.; Paoletti, D.; Sawicki, I.; Silvestri, A.; Skordis, C.; Umiltà, C.; Vernizzi, F.


    We compare Einstein-Boltzmann solvers that include modifications to general relativity and find that, for a wide range of models and parameters, they agree to a high level of precision. We look at three general purpose codes that primarily model general scalar-tensor theories, three codes that model Jordan-Brans-Dicke (JBD) gravity, a code that models f (R ) gravity, a code that models covariant Galileons, a code that models Hořava-Lifschitz gravity, and two codes that model nonlocal models of gravity. Comparing predictions of the angular power spectrum of the cosmic microwave background and the power spectrum of dark matter for a suite of different models, we find agreement at the subpercent level. This means that this suite of Einstein-Boltzmann solvers is now sufficiently accurate for precision constraints on cosmological and gravitational parameters.

  1. LAD Dissertation Prize Talk: Molecular Collisional Excitation in Astrophysical Environments (United States)

    Walker, Kyle M.


    While molecular excitation calculations are vital in determining particle velocity distributions, internal state distributions, abundances, and ionization balance in gaseous environments, both theoretical calculations and experimental data for these processes are lacking. Reliable molecular collisional data with the most abundant species - H2, H, He, and electrons - are needed to probe material in astrophysical environments such as nebulae, molecular clouds, comets, and planetary atmospheres. However, excitation calculations with the main collider, H2, are computationally expensive and therefore various approximations are used to obtain unknown rate coefficients. The widely-accepted collider-mass scaling approach is flawed, and alternate scaling techniques based on physical and mathematical principles are presented here. The most up-to-date excitation data are used to model the chemical evolution of primordial species in the Recombination Era and produce accurate non-thermal spectra of the molecules H2+, HD, and H2 in a primordial cloud as it collapses into a first generation star.

  2. The violent collisional history of asteroid 4 Vesta. (United States)

    Marchi, S; McSween, H Y; O'Brien, D P; Schenk, P; De Sanctis, M C; Gaskell, R; Jaumann, R; Mottola, S; Preusker, F; Raymond, C A; Roatsch, T; Russell, C T


    Vesta is a large differentiated rocky body in the main asteroid belt that accreted within the first few million years after the formation of the earliest solar system solids. The Dawn spacecraft extensively imaged Vesta's surface, revealing a collision-dominated history. Results show that Vesta's cratering record has a strong north-south dichotomy. Vesta's northern heavily cratered terrains retain much of their earliest history. The southern hemisphere was reset, however, by two major collisions in more recent times. We estimate that the youngest of these impact structures, about 500 kilometers across, formed about 1 billion years ago, in agreement with estimates of Vesta asteroid family age based on dynamical and collisional constraints, supporting the notion that the Vesta asteroid family was formed during this event.

  3. Signal Propagation in Collisional Plasma with Negative Ions

    Energy Technology Data Exchange (ETDEWEB)

    I. Kaganovich; S.V. Berezhnoi; C.B. Shin


    The transport of charged species in collisional currentless plasmas is traditionally thought of as a diffusion-like process. In this paper, it is demonstrated that, in contrast to two-component plasma, containing electrons and positive ions, the transport of additional ions in multi-species plasmas is not governed by diffusion, rather described by nonlinear convection. As a particular example, plasmas with the presence of negative ions have been studied. The velocity of a small perturbation of negative ions was found analytically and validated by numerical simulation. As a result of nonlinear convection, initially smooth ion density profiles break and form strongly inhomogeneous shock-like fronts. These fronts are different from collisionless shocks and shocks in fully ionized plasma. The structure of the fronts has been found analytically and numerically.

  4. Collisional damping rates for electron plasma waves reassessed (United States)

    Banks, J. W.; Brunner, S.; Berger, R. L.; Arrighi, W. J.; Tran, T. M.


    Collisional damping of electron plasma waves, the primary damping for high phase velocity waves, is proportional to the electron-ion collision rate, νei ,th. Here, it is shown that the damping rate normalized to νei ,th depends on the charge state, Z , on the magnitude of νei ,th and the wave number k in contrast with the commonly used damping rate in plasma wave research. Only for weak collision rates in low-Z plasmas for which the electron self-collision rate is comparable to the electron-ion collision rate is the damping rate given by the commonly accepted value. The result presented here corrects the result presented in textbooks at least as early as 1973. The complete linear theory requires the inclusion of both electron-ion pitch-angle and electron-electron scattering, which itself contains contributions to both pitch-angle scattering and thermalization.

  5. Modern methods in collisional-radiative modeling of plasmas

    CERN Document Server


    This book provides a compact yet comprehensive overview of recent developments in collisional-radiative (CR) modeling of laboratory and astrophysical plasmas. It describes advances across the entire field, from basic considerations of model completeness to validation and verification of CR models to calculation of plasma kinetic characteristics and spectra in diverse plasmas. Various approaches to CR modeling are presented, together with numerous examples of applications. A number of important topics, such as atomic models for CR modeling, atomic data and its availability and quality, radiation transport, non-Maxwellian effects on plasma emission, ionization potential lowering, and verification and validation of CR models, are thoroughly addressed. Strong emphasis is placed on the most recent developments in the field, such as XFEL spectroscopy. Written by leading international research scientists from a number of key laboratories, the book offers a timely summary of the most recent progress in this area. It ...

  6. Non-resonant Particle Heating due to Collisional Separatrix Crossings (United States)

    Driscoll, C. Fred; Anderegg, F.; Affolter, M.; Dubin, D. H. E.


    We observe plasma heating when a pure ion column is ``sloshed'' back and forth across a trapping separatrix, with heating rate larger than expected from simple collisional viscosity. Here, an externally applied theta-symmetric ``squeeze'' potential creates a velocity separatrix between trapped and passing particles, and weak collisions at rate νc cause separatrix crossings. The trapped particles are repeatedly compressed and expanded (by δL at rate fsl) whereas the passing particles counter-stream and Debye shield the resultant potential variations. LIF diagnostics clearly show the separatrix energy Esep (r) , in close agreement with (r , z) Boltmann-Poisson equilibrium calculations. With νc Science Foundation Grant PHY-1414570, Department of Energy Grants DE-SC0002451.

  7. Boltzmann learning of parameters in cellular neural networks

    DEFF Research Database (Denmark)

    Hansen, Lars Kai


    The use of Bayesian methods to design cellular neural networks for signal processing tasks and the Boltzmann machine learning rule for parameter estimation is discussed. The learning rule can be used for models with hidden units, or for completely unsupervised learning. The latter is exemplified...... by unsupervised adaptation of an image segmentation cellular network. The learning rule is applied to adaptive segmentation of satellite imagery...

  8. The high-order Boltzmann machine: learned distribution and topology. (United States)

    Albizuri, F X; Danjou, A; Grana, M; Torrealdea, J; Hernandez, M C


    In this paper we give a formal definition of the high-order Boltzmann machine (BM), and extend the well-known results on the convergence of the learning algorithm of the two-order BM. From the Bahadur-Lazarsfeld expansion we characterize the probability distribution learned by the high order BM. Likewise a criterion is given to establish the topology of the BM depending on the significant correlations of the particular probability distribution to be learned.

  9. Planar Approximation as Two-Field Boltzmann Theory


    Aref'eva, I. Ya.


    A modified interaction representation for the master field describing connected $SU(N)$-invariant Wightman's functions in the large $N$ limit of matrix fields is constructed. This construction is based on the representation of the master field in terms of Boltzmannian field theory found before. In the modified interaction representation we deal with two scalar Boltzmann fields ({\\it up} and {\\it down} fields). For up and down fields only half-planar diagrams contribute and this could help to ...

  10. Evolution of intense laser pulse spot size propagating in collisional plasma embedded in magnetic field with variable direction (United States)

    Malekshahi, Moslem


    In this study, propagation of an intense laser pulse through collisional, homogenous, magnetized plasma has been investigated. The plasma is embedded in an external magnetic field with the amplitude and variable direction being constant. The complex dispersion relation of the plasma medium has been obtained that predicates the Faraday rotation effect. The paraxial wave equation has been used for the study of propagation of laser pulse in plasma. The nonlinear current density vector as a source of wave equation is obtained by motion equation and continuity equation of plasma free electrons. Using the source dependent expansion method, the evolution of laser pulse spot size has been investigated. It is shown that the spot size of the laser pulse is dependent on the strength and direction of the external magnetic field significantly. The effect of collision frequency on the evolution of spot size has been studied. The space damping rate of laser pulse power along the propagation length due to collision is obtained. Results show that the increase in the external magnetic field strength increases the rate of laser energy loss.

  11. Quasi-linear landau kinetic equations for magnetized plasmas: compact propagator formalism, rotation matrices and interaction

    Energy Technology Data Exchange (ETDEWEB)

    Misguich, J.H


    As a first step toward a nonlinear renormalized description of turbulence phenomena in magnetized plasmas, the lowest order quasi-linear description is presented here from a unified point of view for collisionless as well as for collisional plasmas in a constant magnetic field. The quasi-linear approximation is applied to a general kinetic equation obtained previously from the Klimontovich exact equation, by means of a generalised Dupree-Weinstock method. The so-obtained quasi-linear description of electromagnetic turbulence in a magnetoplasma is applied to three separate physical cases: -) weak electrostatic turbulence, -) purely magnetic field fluctuations (the classical quasi-linear results are obtained for cosmic ray diffusion in the 'slab model' of magnetostatic turbulence in the solar wind), and -) collisional kinetic equations of magnetized plasmas. This mathematical technique has allowed us to derive basic kinetic equations for turbulent plasmas and collisional plasmas, respectively in the quasi-linear and Landau approximation. In presence of a magnetic field we have shown that the systematic use of rotation matrices describing the helical particle motion allows for a much more compact derivation than usually performed. Moreover, from the formal analogy between turbulent and collisional plasmas, the results derived here in detail for the turbulent plasmas, can be immediately translated to obtain explicit results for the Landau kinetic equation.

  12. Entropic multirelaxation lattice Boltzmann models for turbulent flows. (United States)

    Bösch, Fabian; Chikatamarla, Shyam S; Karlin, Ilya V


    We present three-dimensional realizations of a class of lattice Boltzmann models introduced recently by the authors [I. V. Karlin, F. Bösch, and S. S. Chikatamarla, Phys. Rev. E 90, 031302(R) (2014)] and review the role of the entropic stabilizer. Both coarse- and fine-grid simulations are addressed for the Kida vortex flow benchmark. We show that the outstanding numerical stability and performance is independent of a particular choice of the moment representation for high-Reynolds-number flows. We report accurate results for low-order moments for homogeneous isotropic decaying turbulence and second-order grid convergence for most assessed statistical quantities. It is demonstrated that all the three-dimensional lattice Boltzmann realizations considered herein converge to the familiar lattice Bhatnagar-Gross-Krook model when the resolution is increased. Moreover, thanks to the dynamic nature of the entropic stabilizer, the present model features less compressibility effects and maintains correct energy and enstrophy dissipation. The explicit and efficient nature of the present lattice Boltzmann method renders it a promising candidate for both engineering and scientific purposes for highly turbulent flows.

  13. Self-Focusing of Elliptical Laser Beam in Collisional Plasma and Its Effect on Stimulated Brillouin Scattering Process (United States)

    Singh, Arvinder; Walia, Keshav


    This paper presents an investigation of self-focusing of elliptical laser beam in a collisional plasma and its effect on stimulated Brillouin scattering (SBS) process. The non-linearity arising through non-uniform heating leads to redistribution of carriers, which modifies the background plasma density profile in a direction transverse to pump beam axis. This modification affects the incident laser beam, ion-acoustic wave and back scattered beam. Non-linear differential equations for the beam width parameters of the pump laser beam, ion-acoustic wave and back scattered beam are set up and solved numerically. It is observed from the analysis that the focusing of waves greatly enhances the SBS back-reflectivity.

  14. Proton collisional excitation in the lowest lying 3P terms of ions in the Be and Mg isoelectronic sequences (United States)

    Landman, D. A.; Brown, T.


    Proton collisional excitation cross sections and rate constants are presented for transitions between the 3P(J) fine-structure levels of the lowest-lying sp configurations in a number of astrophysically important ions belonging to the Be and Mg isoelectronic sequences. The calculations were made by direct integration of the Schroedinger equation resulting from semiclassical Coulomb excitation theory. The cross sections and rate constants for the 3P(J) transitions in the lowest-lying P(2) configurations are expected to be similar to those for the corresponding sp configuration transitions, and this is illustrated for C III. For the high-temperature ion Ca XVII alpha particle excitation is shown to be unimportant for situations involving ordinary values of the He/H abundance ratio. A simple, but apparently accurate method for determining certain radial integrals for low-lying excited configurations is proposed.

  15. Development of a Prototype Lattice Boltzmann Code for CFD of Fusion Systems.

    Energy Technology Data Exchange (ETDEWEB)

    Pattison, Martin J; Premnath, Kannan N; Banerjee, Sanjoy; Dwivedi, Vinay


    Designs of proposed fusion reactors, such as the ITER project, typically involve the use of liquid metals as coolants in components such as heat exchangers, which are generally subjected to strong magnetic fields. These fields induce electric currents in the fluids, resulting in magnetohydrodynamic (MHD) forces which have important effects on the flow. The objective of this SBIR project was to develop computational techniques based on recently developed lattice Boltzmann techniques for the simulation of these MHD flows and implement them in a computational fluid dynamics (CFD) code for the study of fluid flow systems encountered in fusion engineering. The code developed during this project, solves the lattice Boltzmann equation, which is a kinetic equation whose behaviour represents fluid motion. This is in contrast to most CFD codes which are based on finite difference/finite volume based solvers. The lattice Boltzmann method (LBM) is a relatively new approach which has a number of advantages compared with more conventional methods such as the SIMPLE or projection method algorithms that involve direct solution of the Navier-Stokes equations. These are that the LBM is very well suited to parallel processing, with almost linear scaling even for very large numbers of processors. Unlike other methods, the LBM does not require solution of a Poisson pressure equation leading to a relatively fast execution time. A particularly attractive property of the LBM is that it can handle flows in complex geometries very easily. It can use simple rectangular grids throughout the computational domain -- generation of a body-fitted grid is not required. A recent advance in the LBM is the introduction of the multiple relaxation time (MRT) model; the implementation of this model greatly enhanced the numerical stability when used in lieu of the single relaxation time model, with only a small increase in computer time. Parallel processing was implemented using MPI and demonstrated the

  16. Two-Dimensional Lattice Boltzmann for Reactive Rayleigh–Bénard and Bénard–Poiseuille Regimes

    Directory of Open Access Journals (Sweden)

    Suemi Rodríguez-Romo


    Full Text Available We perform a computer simulation of the reaction-diffusion and convection that takes place in Rayleigh–Bénard and Bénard–Poiseuille regimes. The lattice Boltzmann equation (LBE is used along with the Boussinesq approximation to solve the non-linear coupled differential equations that govern the systems’ thermo-hydrodynamics. Another LBE, is introduced to calculate the evolution concentration of the chemical species involved in the chemical reactions. The simulations are conducted at low Reynolds numbers and in terms of steady state between the first and second thermo-hydrodynamics instability. The results presented here (with no chemical reactions are in good agreement with those reported in the scientific literature which gives us high expectations about the reliability of the chemical kinetics simulation. Some examples are provided.

  17. Nonlinear wave structures in collisional plasma of auroral E-region ionosphere

    Directory of Open Access Journals (Sweden)

    A. V. Volosevich


    Full Text Available Studies of the auroral plasma with small-scale inhomogenieties producing the VHF-radar reflections (radar aurora when observed in conditions of the saturated Farley-Buneman instability within the auroral E region, show strong nonlinear interactions and density fluctuations of 5–15%. Such nonlinearity and high fluctation amplitudes are inconsistent with the limitations of the weak turbulence theory, and thus a theory for arbitrary amplitudes is needed. To this end, a nonlinear theory is described for electrostatic MHD moving plasma structures of arbitrary amplitude for conditions throughout the altitude range of the collisional auroral E region. The equations are derived, from electron and ion motion self-consistent with the electric field, for the general case of the one-dimensional problem. They take into account nonlinearity, electron and ion inertia, diffusion, deviation from quasi-neutrality, and dynamical ion viscosity. The importance of the ion viscosity for dispersion is stressed, while deviation from the quasi-neutrality can be important only at rather low plasma densities, not typical for the auroral E region. In a small amplitude limit these equations have classical nonlinear solutions of the type of "electrostatic shock wave" or of knoidal waves. In a particular case these knoidal waves degrade to a dissipative soliton. A two-dimensional case of a quasi-neutral plasma is considered in the plane perpendicular to the magnetic field by way of the Poisson brackets, but neglecting the nonlinearity and ion inertia. It is shown that in these conditions an effective saturation can be achieved at the stationary turbulence level of order of 10%.

  18. Nonlinear wave structures in collisional plasma of auroral E-region ionosphere

    Directory of Open Access Journals (Sweden)

    A. V. Volosevich

    Full Text Available Studies of the auroral plasma with small-scale inhomogenieties producing the VHF-radar reflections (radar aurora when observed in conditions of the saturated Farley-Buneman instability within the auroral E region, show strong nonlinear interactions and density fluctuations of 5–15%. Such nonlinearity and high fluctation amplitudes are inconsistent with the limitations of the weak turbulence theory, and thus a theory for arbitrary amplitudes is needed. To this end, a nonlinear theory is described for electrostatic MHD moving plasma structures of arbitrary amplitude for conditions throughout the altitude range of the collisional auroral E region. The equations are derived, from electron and ion motion self-consistent with the electric field, for the general case of the one-dimensional problem. They take into account nonlinearity, electron and ion inertia, diffusion, deviation from quasi-neutrality, and dynamical ion viscosity. The importance of the ion viscosity for dispersion is stressed, while deviation from the quasi-neutrality can be important only at rather low plasma densities, not typical for the auroral E region. In a small amplitude limit these equations have classical nonlinear solutions of the type of "electrostatic shock wave" or of knoidal waves. In a particular case these knoidal waves degrade to a dissipative soliton. A two-dimensional case of a quasi-neutral plasma is considered in the plane perpendicular to the magnetic field by way of the Poisson brackets, but neglecting the nonlinearity and ion inertia. It is shown that in these conditions an effective saturation can be achieved at the stationary turbulence level of order of 10%.

  19. Integral equations

    CERN Document Server

    Moiseiwitsch, B L


    Two distinct but related approaches hold the solutions to many mathematical problems--the forms of expression known as differential and integral equations. The method employed by the integral equation approach specifically includes the boundary conditions, which confers a valuable advantage. In addition, the integral equation approach leads naturally to the solution of the problem--under suitable conditions--in the form of an infinite series.Geared toward upper-level undergraduate students, this text focuses chiefly upon linear integral equations. It begins with a straightforward account, acco

  20. Lattice Boltzmann simulation of thermofluidic transport phenomena in a DC magnetohydrodynamic (MHD) micropump. (United States)

    Chatterjee, Dipankar; Amiroudine, Sakir


    A comprehensive non-isothermal Lattice Boltzmann (LB) algorithm is proposed in this article to simulate the thermofluidic transport phenomena encountered in a direct-current (DC) magnetohydrodynamic (MHD) micropump. Inside the pump, an electrically conducting fluid is transported through the microchannel by the action of an electromagnetic Lorentz force evolved out as a consequence of the interaction between applied electric and magnetic fields. The fluid flow and thermal characteristics of the MHD micropump depend on several factors such as the channel geometry, electromagnetic field strength and electrical property of the conducting fluid. An involved analysis is carried out following the LB technique to understand the significant influences of the aforementioned controlling parameters on the overall transport phenomena. In the LB framework, the hydrodynamics is simulated by a distribution function, which obeys a single scalar kinetic equation associated with an externally imposed electromagnetic force field. The thermal history is monitored by a separate temperature distribution function through another scalar kinetic equation incorporating the Joule heating effect. Agreement with analytical, experimental and other available numerical results is found to be quantitative.

  1. Modeling fluid flow and heat transfer in PEM fuel cell using lattice Boltzmann approach (United States)

    Afsharpoya, Behnam


    The fluid flow and species transport in fuel cells are affected by diffusion, advection, thermal gradients, material properties, electrochemical effects, and interfacial forces. A consistent approach capable of modeling these processes has not yet been developed. There have been studies addressing transport of reactants and products in the gas phase, however, water management and convective / thermal effects are still poorly understood. While most modeling efforts in fuel-cell research adopt the traditional CFD approach based on the continuum governing equations, we are developing lattice- Boltzmann (LB) methods to model fluid and thermal transport inside flow channels and gas diffusion layers in proton exchange membrane fuel cells. Specifically, we have developed and tested a new method for implementing structured non-uniform mesh using Lagrangian interpolations. A three-dimensional LB code has been developed for thermal flows through a section of serpentine channel with a gas diffusion layer. The gas diffusion layer is modelled as a porous medium using a modified LB equation and a forcing term. A separate distribution is used to model thermal effects. Methods of validating the approach and preliminary results will be presented.

  2. Improved thermal lattice Boltzmann model for simulation of liquid-vapor phase change (United States)

    Li, Qing; Zhou, P.; Yan, H. J.


    In this paper, an improved thermal lattice Boltzmann (LB) model is proposed for simulating liquid-vapor phase change, which is aimed at improving an existing thermal LB model for liquid-vapor phase change [S. Gong and P. Cheng, Int. J. Heat Mass Transfer 55, 4923 (2012), 10.1016/j.ijheatmasstransfer.2012.04.037]. First, we emphasize that the replacement of ∇ .(λ ∇ T ) /∇.(λ ∇ T ) ρ cV ρ cV with ∇ .(χ ∇ T ) is an inappropriate treatment for diffuse interface modeling of liquid-vapor phase change. Furthermore, the error terms ∂t 0(T v ) +∇ .(T vv ) , which exist in the macroscopic temperature equation recovered from the previous model, are eliminated in the present model through a way that is consistent with the philosophy of the LB method. Moreover, the discrete effect of the source term is also eliminated in the present model. Numerical simulations are performed for droplet evaporation and bubble nucleation to validate the capability of the model for simulating liquid-vapor phase change. It is shown that the numerical results of the improved model agree well with those of a finite-difference scheme. Meanwhile, it is found that the replacement of ∇ .(λ ∇ T ) /∇ .(λ ∇ T ) ρ cV ρ cV with ∇ .(χ ∇ T ) leads to significant numerical errors and the error terms in the recovered macroscopic temperature equation also result in considerable errors.

  3. Lattice Boltzmann Explicit Schemes for 3D MHD on Non-Uniform Grids (United States)

    Schleif, C.; Vahala, G.; Vahala, L.; Macnab, A.; Soe, M.; Carter, J.


    Lattice-Boltzmann Model (LBM) is a very promising alternative computational approach to MHD and to other nonlinear macroscopic systems because of its simplicity, ease of imposition of geometric boundary conditions and ideal parallelization on multi-PE (and especially vector) platforms. For example, on the Earth Simulator our 2D explicit LBM-MHD code has achieved over 3.6 TFlops/sec. The disparate length and time scales that appear in the solutions of dissipative MHD require careful treatment of ill-conditioned matrices in direct solvers. In LBM-MHD one introduces a scalar distribution function for the velocity field and a vector distribution function for the magnetic field. Since the magnetic evolution equation is obtained at the 1st moment closures, less speeds are needed than to recover the momentum equation. We are also investigating the least square LBM for non-uniform spatial grids. In one approach, the standard LBM is applied to the fine scales while the least square LBM is applied to the large scales. Since the least square algorithm involves matrices that are only grid-dependent, these matrices need only be calculated once leading to an efficient algorithm. Our algorithm will be applied to the 3D Orszag-Tang vortex and compare our results to the 3D pseudo-spectral results of Poquet et. al.

  4. A hydrodynamically-consistent MRT lattice Boltzmann model on a 2D rectangular grid (United States)

    Peng, Cheng; Min, Haoda; Guo, Zhaoli; Wang, Lian-Ping


    A multiple-relaxation time (MRT) lattice Boltzmann (LB) model on a D2Q9 rectangular grid is designed theoretically and validated numerically in the present work. By introducing stress components into the equilibrium moments, this MRT-LB model restores the isotropy of diffusive momentum transport at the macroscopic level (or in the continuum limit), leading to moment equations that are fully consistent with the Navier-Stokes equations. The model is derived by an inverse design process which is described in detail. Except one moment associated with the energy square, all other eight equilibrium moments can be theoretically and uniquely determined. The model is then carefully validated using both the two-dimensional decaying Taylor-Green vortex flow and lid-driven cavity flow, with different grid aspect ratios. The corresponding results from an earlier model (Bouzidi et al. (2001) [28]) are also presented for comparison. The results of Bouzidi et al.'s model show problems associated with anisotropy of viscosity coefficients, while the present model exhibits full isotropy and is accurate and stable.

  5. Multiple-relaxation-time lattice Boltzmann simulation for flow, mass transfer, and adsorption in porous media (United States)

    Ma, Qiang; Chen, Zhenqian; Liu, Hao


    In this paper, to predict the dynamics behaviors of flow and mass transfer with adsorption phenomena in porous media at the representative elementary volume (REV) scale, a multiple-relaxation-time (MRT) lattice Boltzmann (LB) model for the convection-diffusion equation is developed to solve the transfer problem with an unsteady source term in porous media. Utilizing the Chapman-Enskog analysis, the modified MRT-LB model can recover the macroscopic governing equations at the REV scale. The coupled MRT-LB model for momentum and mass transfer is validated by comparing with the finite-difference method and the analytical solution. Moreover, using the MRT-LB method coupled with the linear driving force model, the fluid transfer and adsorption behaviors of the carbon dioxide in a porous fixed bed are explored. The breakthrough curve of adsorption from MRT-LB simulation is compared with the experimental data and the finite-element solution, and the transient concentration distributions of the carbon dioxide along the porous fixed bed are elaborated upon in detail. In addition, the MRT-LB simulation results show that the appearance time of the breakthrough point in the breakthrough curve is advanced as the mass transfer resistance in the linear driving force model increases; however, the saturation point is prolonged inversely.

  6. Effect of polarization force on the Jeans instability in collisional dusty plasmas (United States)



    The Jeans instability in collisional dusty plasmas has been analytically investigated by considering the polarization force effect. Instabilities due to dust-neutral and ion-neutral drags can occur in electrostatic waves of collisional dusty plasmas with self-gravitating particles. In this study, the effect of gravitational force on heavy dust particles is considered in tandem with both the polarization and electrostatic forces. The theoretical framework has been developed and the dispersion relation and instability growth rate have been derived, assuming the plane wave approximation. The derived instability growth rate shows that, in collisional dusty plasmas, the Jeans instability strongly depends on the magnitude of the polarization force.

  7. Derivation of transient relativistic fluid dynamics from the Boltzmann equation for a multi-component system

    Energy Technology Data Exchange (ETDEWEB)

    Denicol, G.S. [Department of Physics, McGill University, 3600 University Street, Montreal, Quebec, H3A2T8 (Canada); Institut für Theoretische Physik, Johann Wolfgang Goethe-Universität, Max-von-Laue-Str. 1, D-60438 Frankfurt am Main (Germany); Niemi, H. [Department of Physics, P.O. Box 35, FI-40014 University of Jyväskylä (Finland)


    We derive the non-equilibrium single-particle momentum distribution function of a hadron resonance gas. We then study the effects that this newly derived expression can have in the freeze-out description of fluid-dynamical models of heavy ion collisions and compare it with the method traditionally employed, the 14-moment approximation.

  8. Transport and fluctuations in granular fluids from Boltzmann equation to hydrodynamics, diffusion and motor effects

    CERN Document Server

    Puglisi, Andrea


    This brief offers a concise presentation of granular fluids from the  point of view of non-equilibrium statistical physics. The emphasis is on fluctuations, which can be large in granular fluids due to the small system size (the number of grains is many orders of magnitude smaller than in molecular fluids). Firstly, readers will be introduced to the most intriguing experiments on fluidized granular fluids. Then granular fluid theory, which goes through increasing levels of coarse-graining and emerging collective phenomena, is described. Problems and questions are initially posed at the level of kinetic theory, which describes particle densities in full or reduced phase-space. Some answers become clear through hydrodynamics, which describes the evolution of slowly evolving fields. Granular fluctuating hydrodynamics, which builds a bridge to the most recent results in non-equilibrium statistical mechanics, is also introduced. Further and more interesting answers come when the dynamics of a massive intruder are...

  9. Electrokinetics of diffuse soft interfaces. 2. Analysis based on the nonlinear Poisson-Boltzmann equation

    NARCIS (Netherlands)

    Duval, J.F.L.


    In a previous study (Langmuir 2004, 20, 10324), the electrokinetic properties of diffuse soft layers were theoretically investigated within the framework of the Debye-H¿ckel approximation valid in the limit of sufficiently low values for the Donnan potential. In the current paper, the

  10. Numerical Simulations of Particulate Suspensions via a Discretized Boltzmann Equation; 2, Numerical Results

    CERN Document Server

    Ladd, A J C


    A new and very general technique for simulating solid-fluid suspensions has been described in a previous paper (Part I); the most important feature of the new method is that the computational cost scales with the number of particles. In this paper (Part II), extensive numerical tests of the method are described; for creeping flows, both with and without Brownian motion, and at finite Reynolds numbers. Hydrodynamic interactions, transport coefficients, and the short-time dynamics of random dispersions of up to 1024 colloidal particles have been simulated.

  11. A Photoplethysmography Melanin Evaluation System by Modified Boltzmann Transport Equation (BTE

    Directory of Open Access Journals (Sweden)

    Sheng-Chieh Huang


    Full Text Available With the advance in cosmetic medical technology in recent years, more and more people get cosmetic medical treatments, especially skin whitening treatments. Nevertheless, people usually assess the effect of skin whitening products by vision, which is subjective and will be different from each person. To acquire the value of melanin concentration objectively, people need to go to cosmetic medical clinics. This will cause inconvenience to people. This paper develops a novel evaluation platform based on optical assessment methods, which employ different absorption and scattering properties to different wavelengths of light in human tissue to obtain melanin concentration. Moreover, this paper proposes a new method that compensates the interaction between epidermis and dermis to acquire the melanin concentration more accurately. The novel platform designed in this paper is smaller and consumes lower-power and smaller when comparing to other conventional devices in market.

  12. Penetration equations

    Energy Technology Data Exchange (ETDEWEB)

    Young, C.W. [Applied Research Associates, Inc., Albuquerque, NM (United States)


    In 1967, Sandia National Laboratories published empirical equations to predict penetration into natural earth materials and concrete. Since that time there have been several small changes to the basic equations, and several more additions to the overall technique for predicting penetration into soil, rock, concrete, ice, and frozen soil. The most recent update to the equations was published in 1988, and since that time there have been changes in the equations to better match the expanding data base, especially in concrete penetration. This is a standalone report documenting the latest version of the Young/Sandia penetration equations and related analytical techniques to predict penetration into natural earth materials and concrete. 11 refs., 6 tabs.

  13. Dust production by collisional grinding during Planetesimal-Driven Migration (United States)

    Salmon, Julien; Walsh, Kevin J.; Levison, Harold F.


    Many main-sequence stars are surrounded by optically thin disks of dust in the absence of any detectable gas (e.g. Su et al. 2006, Meyer et al. 2008). IR and sub-millimeter observations suggest that most of the observed emission comes from grains with sizes between 1-100 microns. Since radiation forces are expected to remove these grains on timescales much shorter than the age of the parent stars (Backman & Parsce 1993, Wyatt 2008), it implies that some process is replenishing the dust, such as collisional grinding. The latter requires large impact velocities between planetesimals, which can be achieved if large objects are dynamically exciting a disk of 1-10km planetesimals. Such debris disks could be hosting ongoing planet formation, and present a powerful tool to test planet formation theories.If a planet is embedded in a gas-free planetesimal disk, the mutual gravitational interactions will force the planet to migrate (e.g. Fernandez & Ip 1984). Planetesimals situated along the direction of migration can be trapped in mean motion resonances (MMRs) with the planet (Malhotra 1993, 1995, Hahn & Malholtra 1999). Planetesimals trapped in such resonances will have their eccentricities pumped to large values as the planet continues to migrate, thereby leading to energetic collisions and dust production (Wyatt 2003, Reche et al. 2008, Mustill & Wyatt 2011).We have performed an extensive suite of simulations in which we explore the likelihood that a given set of disk parameters (mass, surface density slope, number of planetesimals) can sustain planetesimal-driven migration (PDM). We confirm the strong dependence on resolution found in previous works (e.g. Kirsch et al 2009), and find that an embryo to planetesimal mass ratio of 400 is necessary to mitigate the effects of stochasticity, which may cause migration to stall and/or reverse. After having identified disks suitable for sustained PDM, we model their evolution using LIPAD (Levison et al. 2012) taking into account

  14. Impurity transport and bulk ion flow in a mixed collisionality stellarator plasma (United States)

    Newton, S. L.; Helander, P.; Mollén, A.; Smith, H. M.


    The accumulation of impurities in the core of magnetically confined plasmas, resulting from standard collisional transport mechanisms, is a known threat to their performance as fusion energy sources. Whilst the axisymmetric tokamak systems have been shown to benefit from the effect of temperature screening, that is an outward flux of impurities driven by the temperature gradient, impurity accumulation in stellarators was thought to be inevitable, driven robustly by the inward pointing electric field characteristic of hot fusion plasmas. We have shown in Helander et al. (Phys. Rev. Lett., vol. 118, 2017a, 155002) that such screening can in principle also appear in stellarators, in the experimentally relevant mixed collisionality regime, where a highly collisional impurity species is present in a low collisionality bulk plasma. Details of the analytic calculation are presented here, along with the effect of the impurity on the bulk ion flow, which will ultimately affect the bulk contribution to the bootstrap current.

  15. Differential equations

    CERN Document Server

    Tricomi, FG


    Based on his extensive experience as an educator, F. G. Tricomi wrote this practical and concise teaching text to offer a clear idea of the problems and methods of the theory of differential equations. The treatment is geared toward advanced undergraduates and graduate students and addresses only questions that can be resolved with rigor and simplicity.Starting with a consideration of the existence and uniqueness theorem, the text advances to the behavior of the characteristics of a first-order equation, boundary problems for second-order linear equations, asymptotic methods, and diff

  16. On the equivalence of Hopfield networks and Boltzmann Machines. (United States)

    Barra, Adriano; Bernacchia, Alberto; Santucci, Enrica; Contucci, Pierluigi


    A specific type of neural networks, the Restricted Boltzmann Machines (RBM), are implemented for classification and feature detection in machine learning. They are characterized by separate layers of visible and hidden units, which are able to learn efficiently a generative model of the observed data. We study a "hybrid" version of RBMs, in which hidden units are analog and visible units are binary, and we show that thermodynamics of visible units are equivalent to those of a Hopfield network, in which the N visible units are the neurons and the P hidden units are the learned patterns. We apply the method of stochastic stability to derive the thermodynamics of the model, by considering a formal extension of this technique to the case of multiple sets of stored patterns, which may act as a benchmark for the study of correlated sets. Our results imply that simulating the dynamics of a Hopfield network, requiring the update of N neurons and the storage of N(N-1)/2 synapses, can be accomplished by a hybrid Boltzmann Machine, requiring the update of N+P neurons but the storage of only NP synapses. In addition, the well known glass transition of the Hopfield network has a counterpart in the Boltzmann Machine: it corresponds to an optimum criterion for selecting the relative sizes of the hidden and visible layers, resolving the trade-off between flexibility and generality of the model. The low storage phase of the Hopfield model corresponds to few hidden units and hence a overly constrained RBM, while the spin-glass phase (too many hidden units) corresponds to unconstrained RBM prone to overfitting of the observed data. Copyright © 2012 Elsevier Ltd. All rights reserved.

  17. Lattice Boltzmann modeling and simulation of liquid jet breakup (United States)

    Saito, Shimpei; Abe, Yutaka; Koyama, Kazuya


    A three-dimensional color-fluid lattice Boltzmann model for immiscible two-phase flows is developed in the framework of a three-dimensional 27-velocity (D3Q27) lattice. The collision operator comprises the D3Q27 versions of three suboperators: a multiple-relaxation-time (MRT) collision operator, a generalized Liu-Valocchi-Kang perturbation operator, and a Latva-Kokko-Rothman recoloring operator. A D3Q27 version of an enhanced equilibrium distribution function is also incorporated into this model to improve the Galilean invariance. Three types of numerical tests, namely, a static droplet, an oscillating droplet, and the Rayleigh-Taylor instability, show a good agreement with analytical solutions and numerical simulations. Following these numerical tests, this model is applied to liquid-jet-breakup simulations. The simulation conditions are matched to the conditions of the previous experiments. In this case, numerical stability is maintained throughout the simulation, although the kinematic viscosity for the continuous phase is set as low as 1.8 ×10-4 , in which case the corresponding Reynolds number is 3.4 ×103 ; the developed lattice Boltzmann model based on the D3Q27 lattice enables us to perform the simulation with parameters directly matched to the experiments. The jet's liquid column transitions from an asymmetrical to an axisymmetrical shape, and entrainment occurs from the side of the jet. The measured time history of the jet's leading-edge position shows a good agreement with the experiments. Finally, the reproducibility of the regime map for liquid-liquid systems is assessed. The present lattice Boltzmann simulations well reproduce the characteristics of predicted regimes, including varicose breakup, sinuous breakup, and atomization.


    Energy Technology Data Exchange (ETDEWEB)

    Dobinson, Jack; Leinhardt, Zoë M.; Lines, Stefan; Carter, Philip J. [University of Bristol, School of Physics, H. H. Wills Physics Laboratory, University of Bristol, Bristol BS8 1TL (United Kingdom); Dodson-Robinson, Sarah E. [University of Delaware, Department of Physics and Astronomy, 217 Sharp Lab, Newark, DE 19716 (United States); Teanby, Nick A. [University of Bristol, School of Earth Sciences, H. H. Wills Physics Laboratory, University of Bristol, Bristol BS8 1TL (United Kingdom)


    Observations of the youngest planets (∼1–10 Myr for a transitional disk) will increase the accuracy of our planet formation models. Unfortunately, observations of such planets are challenging and time-consuming to undertake, even in ideal circumstances. Therefore, we propose the determination of a set of markers that can preselect promising exoplanet-hosting candidate disks. To this end, N-body simulations were conducted to investigate the effect of an embedded Jupiter-mass planet on the dynamics of the surrounding planetesimal disk and the resulting creation of second-generation collisional dust. We use a new collision model that allows fragmentation and erosion of planetesimals, and dust-sized fragments are simulated in a post-process step including non-gravitational forces due to stellar radiation and a gaseous protoplanetary disk. Synthetic images from our numerical simulations show a bright double ring at 850 μm for a low-eccentricity planet, whereas a high-eccentricity planet would produce a characteristic inner ring with asymmetries in the disk. In the presence of first-generation primordial dust these markers would be difficult to detect far from the orbit of the embedded planet, but would be detectable inside a gap of planetary origin in a transitional disk.

  19. Modeling Non-Equilibrium Collisional Plasmas with AtomDB (United States)

    Foster, Adam; Yamaguchi, H.; Smith, R. K.; Brickhouse, N. S.; Ji, L.; Kallman, T.; Wilms, J.


    Collisionally ionized plasmas that are in non-equilibrium ionization (NEI) show distinctly different emission from those in equilibrium. Recombining, or overionized, plasmas show significant recombination-driven continuum features, while ionizing plasmas show strong inner-shell emission lines, such as the Iron Kα line at 6.4-6.7keV. Existing models in analysis tools such as XSPEC treat only the equilibrium case and part of the ionizing plasma case due to a significant lack of atomic data. We present major updates to the AtomDB database, and new models for use XSPEC, which allow all types of these non-equilibrium plasmas to be modeled in a simple yet accurate fashion. This model has been created using a large amount of data obtained from published sources, supplemented by data we have calculated using the Flexible Atomic Code where required. We identify the spectral features that have been seen and can now be modeled using this data for existing missions as well as Astro-H. We also revisit archival data where recombining plasma emission has previously been identified.

  20. Electrostatic thermal noise in a weakly ionized collisional plasma (United States)

    Martinović, M. M.; Zaslavsky, A.; Maksimović, M.; Å egan, S.


    Quasi-thermal noise (QTN) spectroscopy is a plasma diagnostic technique which enables precise measurements of local electron velocity distribution function moments. This technique is based on measurements and analysis of voltage fluctuations at the antenna terminals, induced by thermal motion of charged particles. In this work, we accommodate, for the first time, this technique to weakly ionized collisional plasmas. It turns out that the QTN spectrum is modified both at low frequencies, increasing the level of power spectrum, and around the plasma frequency, where collisions damp the plasma oscillations and therefore broaden and reduce the amplitude of so called "plasma peak," while the spectrum at high frequencies is nearly unmodified compared to the collisionless case. Based on these results, we show that QTN spectroscopy enables independent measurements of the collision frequency, electron density, and temperature, provided the ratio of collision frequency to plasma frequency is ν/ωp˜0.1. The method presented here can be used for precise estimation of plasma parameters in laboratory devices and unmagnetized ionospheres, while application in the ionosphere of Earth is possible but limited to small, low-frequency range due to magnetic field influence.

  1. Collisional disruption of gravitational aggregates in the tidal environment

    Energy Technology Data Exchange (ETDEWEB)

    Hyodo, Ryuki; Ohtsuki, Keiji [Department of Earth and Planetary Sciences, Kobe University, Kobe 657-8501 (Japan)


    The degree of disruption in collisions in free space is determined by specific impact energy, and the mass fraction of the largest remnant is a monotonically decreasing function of impact energy. However, it has not been shown whether such a relationship is applicable to collisions under the influence of a planet's tidal force, which is important in ring dynamics and satellite accretion. Here we examine the collisional disruption of gravitational aggregates in the tidal environment by using local N-body simulations. We find that outcomes of such a collision largely depend on the impact velocity, the direction of impact, and the radial distance from the planet. In the case of a strong tidal field corresponding to Saturn's F ring, collisions in the azimuthal direction are much more destructive than those in the radial direction. Numerical results of collisions sensitively depend on the impact velocity, and a complete disruption of aggregates can occur even in impacts with velocity much lower than their escape velocity. In such low-velocity collisions, the deformation of colliding aggregates plays an essential role in determining collision outcomes, because the physical size of the aggregate is comparable to its Hill radius. On the other hand, the dependence of collision outcomes on impact velocity becomes similar to the case in free space when the distance from the planet is sufficiently large. Our results are consistent with Cassini observations of the F ring, which suggest ongoing creation and disruption of aggregates within the ring.

  2. Cancellation of Collisional Frequency Shifts in Optical Lattice Clocks with Rabi Spectroscopy


    Lee, Sangkyung; Park, Chang Yong; Lee, Won-Kyu; Yu, Dai-Hyuk


    We analyze both the s- and p-wave collision induced frequency shifts and propose a over-$\\pi$ pulse scheme to cancel the shifts in optical lattice clocks interrogated by a Rabi pulse. The collisional frequency shifts are analytically solved as a function of the pulse area and the inhomogeneity of the Rabi frequencies. Experimentally measured collisional frequency shifts in an Yb optical lattice clock are in good agreement with the analytical calculations. Based on our analysis, the over-$\\pi$...

  3. A Hybrid Model for Multiscale Laser Plasma Simulations with Detailed Collisional Physics (United States)


    important physical process as possible with as little computational cost as possible. • To that end, we are in the early processes of characterizing...Detailed Collisional Physics David Bilyeu, Carl Lederman, Richard Abrantes Air Force Research Laboratory (AFMC) AFRL/RQRS 1 Ara Drive Edwards AFB, CA...for Public Release; Distribution is Unlimited. PA# 17383 A Hybrid Model for Multiscale Laser Plasma Simulations with Detailed Collisional Physics

  4. Modeling of collisional excited x-ray lasers using short pulse laser pumping

    Energy Technology Data Exchange (ETDEWEB)

    Sasaki, Akira; Moribayashi, Kengo; Utsumi, Takayuki; Tajima, Toshiki [Japan Atomic Energy Research Inst., Neyagawa, Osaka (Japan). Kansai Research Establishment


    A simple atomic kinetics model of electron collisional excited x-ray lasers has been developed. The model consists of a collisional radiative model using the average ion model (AIM) and a detailed term accounting (DTA) model of Ni-like Ta. An estimate of plasma condition to produce gain in Ni-like Ta ({lambda}=44A) is given. Use of the plasma confined in a cylinder is proposed to preform a uniform high density plasma from 1-D hydrodynamics calculations. (author)

  5. Classical Langevin equations for the free electron gas and blackbody radiation (United States)

    Frank, T. D.


    Among others, Uhling and Uhlenbeck, Kaniadakis and Quarati and Kadanoff have suggested to describe the evolution of quantum systems exhibiting Fermi-Dirac and Bose-Einstein statistics by means of classical but nonlinear evolution equations for density measures such as generalized Boltzmann equations and nonlinear Fokker-Planck equations. We use this approach in order to derive classical Langevin equations for quantum systems and apply the Langevin equations thus obtained to two fundamental quantum systems, namely, the free electron gas and blackbody radiation.

  6. Classical Langevin equations for the free electron gas and blackbody radiation

    Energy Technology Data Exchange (ETDEWEB)

    Frank, T D [Institute for Theoretical Physics, University of Muenster, Wilhelm-Klemm-Str. 9, 48149 Muenster (Germany)


    Among others, Uhling and Uhlenbeck, Kaniadakis and Quarati and Kadanoff have suggested to describe the evolution of quantum systems exhibiting Fermi-Dirac and Bose-Einstein statistics by means of classical but nonlinear evolution equations for density measures such as generalized Boltzmann equations and nonlinear Fokker-Planck equations. We use this approach in order to derive classical Langevin equations for quantum systems and apply the Langevin equations thus obtained to two fundamental quantum systems, namely, the free electron gas and blackbody radiation.

  7. A numerical theory of lattice gas and lattice Boltzmann methods in the computation of solutions to nonlinear advective-diffusive systems

    Energy Technology Data Exchange (ETDEWEB)

    Elton, A.B.H.


    A numerical theory for the massively parallel lattice gas and lattice Boltzmann methods for computing solutions to nonlinear advective-diffusive systems is introduced. The convergence theory is based on consistency and stability arguments that are supported by the discrete Chapman-Enskog expansion (for consistency) and conditions of monotonicity (in establishing stability). The theory is applied to four lattice methods: Two of the methods are for some two-dimensional nonlinear diffusion equations. One of the methods is for the one-dimensional lattice method for the one-dimensional viscous Burgers equation. And one of the methods is for a two-dimensional nonlinear advection-diffusion equation. Convergence is formally proven in the L{sub 1}-norm for the first three methods, revealing that they are second-order, conservative, conditionally monotone finite difference methods. Computational results which support the theory for lattice methods are presented. In addition, a domain decomposition strategy using mesh refinement techniques is presented for lattice gas and lattice Boltzmann methods. The strategy allows concentration of computational resources on regions of high activity. Computational evidence is reported for the strategy applied to the lattice gas method for the one-dimensional viscous Burgers equation. 72 refs., 19 figs., 28 tabs.

  8. Phase transitions in restricted Boltzmann machines with generic priors (United States)

    Barra, Adriano; Genovese, Giuseppe; Sollich, Peter; Tantari, Daniele


    We study generalized restricted Boltzmann machines with generic priors for units and weights, interpolating between Boolean and Gaussian variables. We present a complete analysis of the replica symmetric phase diagram of these systems, which can be regarded as generalized Hopfield models. We underline the role of the retrieval phase for both inference and learning processes and we show that retrieval is robust for a large class of weight and unit priors, beyond the standard Hopfield scenario. Furthermore, we show how the paramagnetic phase boundary is directly related to the optimal size of the training set necessary for good generalization in a teacher-student scenario of unsupervised learning.

  9. Thrombosis modeling in intracranial aneurysms: a lattice Boltzmann numerical algorithm (United States)

    Ouared, R.; Chopard, B.; Stahl, B.; Rüfenacht, D. A.; Yilmaz, H.; Courbebaisse, G.


    The lattice Boltzmann numerical method is applied to model blood flow (plasma and platelets) and clotting in intracranial aneurysms at a mesoscopic level. The dynamics of blood clotting (thrombosis) is governed by mechanical variations of shear stress near wall that influence platelets-wall interactions. Thrombosis starts and grows below a shear rate threshold, and stops above it. Within this assumption, it is possible to account qualitatively well for partial, full or no occlusion of the aneurysm, and to explain why spontaneous thrombosis is more likely to occur in giant aneurysms than in small or medium sized aneurysms.

  10. Lattice Boltzmann Modeling of Thrombosis in Giant Aneurysms (United States)

    Chopard, B.; Ouared, R.; Ruefenacht, D. A.; Yilmaz, H.

    We propose a numerical model of blood flow and blood clotting whose purpose is to describe thrombus formation in cerebral aneurysms. We identify possible mechanisms that can cause occurence of spontaneous thrombosis in unruptured giant intracranial aneurysms. Our main claim is that, under normal conditions, there is a low shear rate threshold below which thrombosis starts and growths. This assumption is supported by several evidences from literature. The proposed mechanisms are incorporated into a Lattice Boltzmann (LB) model for blood flow and platelets adhesion and aggregation. Numerical simulations show that the low shear rate threshold assumption together with aneurysm geometry account well for the observations.

  11. Restricted Boltzmann machine learning for solving strongly correlated quantum systems (United States)

    Nomura, Yusuke; Darmawan, Andrew S.; Yamaji, Youhei; Imada, Masatoshi


    We develop a machine learning method to construct accurate ground-state wave functions of strongly interacting and entangled quantum spin as well as fermionic models on lattices. A restricted Boltzmann machine algorithm in the form of an artificial neural network is combined with a conventional variational Monte Carlo method with pair product (geminal) wave functions and quantum number projections. The combination allows an application of the machine learning scheme to interacting fermionic systems. The combined method substantially improves the accuracy beyond that ever achieved by each method separately, in the Heisenberg as well as Hubbard models on square lattices, thus proving its power as a highly accurate quantum many-body solver.

  12. Jet propagation within a Linearized Boltzmann Transport model

    Energy Technology Data Exchange (ETDEWEB)

    Luo, Tan; He, Yayun [Key Laboratory of Quark and Lepton Physics (MOE) and Institute of Particle Physics, Central China Normal University, Wuhan 430079 (China); Wang, Xin-Nian [Key Laboratory of Quark and Lepton Physics (MOE) and Institute of Particle Physics, Central China Normal University, Wuhan 430079 (China); Nuclear Science Division, Mailstop 70R0319, Lawrence Berkeley National Laboratory, Berkeley, CA 94740 (United States); Zhu, Yan [Departamento de Física de Partículas and IGFAE, Universidade de Santiago de Compostela, E-15706 Santiago de Compostela, Galicia (Spain)


    A Linearized Boltzmann Transport (LBT) model has been developed for the study of parton propagation inside quark–gluon plasma. Both leading and thermal recoiled partons are tracked in order to include the effect of jet-induced medium excitation. In this talk, we present a study within the LBT model in which we implement the complete set of elastic parton scattering processes. We investigate elastic parton energy loss and their energy and length dependence. We further investigate energy loss and transverse shape of reconstructed jets. Contributions from the recoiled thermal partons and jet-induced medium excitations are found to have significant influences on the jet energy loss and transverse profile.

  13. Differential equations

    CERN Document Server

    Barbu, Viorel


    This textbook is a comprehensive treatment of ordinary differential equations, concisely presenting basic and essential results in a rigorous manner. Including various examples from physics, mechanics, natural sciences, engineering and automatic theory, Differential Equations is a bridge between the abstract theory of differential equations and applied systems theory. Particular attention is given to the existence and uniqueness of the Cauchy problem, linear differential systems, stability theory and applications to first-order partial differential equations. Upper undergraduate students and researchers in applied mathematics and systems theory with a background in advanced calculus will find this book particularly useful. Supplementary topics are covered in an appendix enabling the book to be completely self-contained.

  14. Integral Equations

    CERN Document Server

    Hochstadt, Harry


    This classic work is now available in an unabridged paperback edition. Hochstatdt's concise treatment of integral equations represents the best compromise between the detailed classical approach and the faster functional analytic approach, while developing the most desirable features of each. The seven chapters present an introduction to integral equations, elementary techniques, the theory of compact operators, applications to boundary value problems in more than dimension, a complete treatment of numerous transform techniques, a development of the classical Fredholm technique, and applicatio

  15. Two-phase Lattice Boltzmann modelling of streaming potentials: influence of the air-water interface on the electrokinetic coupling (United States)

    Fiorentino, Eve-Agnès; Toussaint, Renaud; Jouniaux, Laurence


    The streaming potential phenomenon is an electrokinetic effect that occurs in porous media. It is characterized by an electrokinetic (EK) coefficient. The aim of this paper is to simulate the EK coefficient in unsaturated conditions using the Lattice Boltzmann method in a 2-D capillary channel. The multiphase flow is simulated with the model of Shan & Chen. The Poisson-Boltzmann equation is solved by implementing the model of Chai & Shi. The streaming potential response shows a non-monotonous behaviour due to the combination of the increase of charge density and decrease of flow velocity with decreasing water saturation. Using a ζ potential of -20 mV at the air-water interface, an enhancement of a factor 5-30 of the EK coefficient, compared to the saturated state, can be observed due to the positive charge excess at this interface which is magnified by the fluid velocity away from the rock surface. This enhancement is correlated to the fractioning of the bubbles, and to the dynamic state of these bubbles, moving or entrapped in the crevices of the channel.

  16. Thermal transport in isotopically disordered carbon nanotubes: a comparison between Green's functions and Boltzmann approaches

    Energy Technology Data Exchange (ETDEWEB)

    Stoltz, G [Universite Paris Est, CERMICS, Project-team MICMAC, INRIA-Ecole des Ponts, 6 and 8 Avenue Pascal, F-77455 Marne-la-Vallee Cedex 2 (France); Lazzeri, M; Mauri, F [IMPMC, Universites Paris 6 et 7, CNRS, IGPG, 140 rue de Lourmel, F-75015 Paris (France)], E-mail:


    We present a study of the phononic thermal conductivity of isotopically disordered carbon nanotubes. In particular, the behaviour of the thermal conductivity as a function of the system length is investigated, using Green's function techniques to compute the transmission across the system. The method is implemented using linear scaling algorithms, which allow us to reach systems of lengths up to L = 2.5 {mu}m (with up to 200 000 atoms). As for 1D systems, it is observed that the conductivity diverges with the system size L. We also observe a dramatic decrease of the thermal conductance for systems of experimental sizes (roughly 80% at room temperature for L = 2.5 {mu}m), when a large fraction of isotopic disorder is introduced. The results obtained with Green's function techniques are compared to results obtained with a Boltzmann description of thermal transport. There is a good agreement between both approaches for systems of experimental sizes, even in the presence of Anderson localization. This is particularly interesting since the computation of the transmission using Boltzmann's equation is much less computationally expensive, so that larger systems may be studied with this method.

  17. Adjoint Parameter Sensitivity Analysis for the Hydrodynamic Lattice Boltzmann Method with Applications to Design Optimization

    DEFF Research Database (Denmark)

    Pingen, Georg; Evgrafov, Anton; Maute, Kurt


    We present an adjoint parameter sensitivity analysis formulation and solution strategy for the lattice Boltzmann method (LBM). The focus is on design optimization applications, in particular topology optimization. The lattice Boltzmann method is briefly described with an in-depth discussion of so...

  18. The role of coherent excitation and collisional energy transfer in atomic vapor filters and photon detectors (United States)

    Correll, Tiffany Lee

    Many optical techniques, including laser Doppler velocimetry, free space optical communications, and chemical imaging, require-or can be enhanced by-high spectral resolution photon detection. Such detection is characterized by spectral discrimination on the order of GHz or MHz i.e., approximately 10-4 nm in the near-infrared region. This spectral resolution has recently been achieved by exploiting the narrow absorption features of gas phase atoms. Absorption of light by alkali vapors is intrinsically selective and can be monitored by detecting the fluorescence resulting from laser excitation coupled to selectively excited atomic states. Imaging can be accomplished by spatially expanding the excitation lasers into two dimensions. Fluorescence photons are only created and detected when the interrogated object is forced to scatter radiation of an energy precisely matching one of the transitions of a pre-determined optimal excitation/fluorescence scheme. Devices based on resonance fluorescence photon detection have recently been described using cesium atoms. In this work, the sensitivity and spectral resolution of cesium-based photon detectors were evaluated and improved. To this end, initial experiments focused on laser induced fluorescence in room temperature cesium vapor. The fluorescence response of the detector was augmented by the use of cesium-induced collisional excitation energy transfer between states involved in the chosen excitation scheme. Additional studies focused on helium and argon-induced collisions in the vapor to increase the signal output while maintaining adequate spatial resolution in imaging mode. The probability or cross section of helium-cesium collisions at the operating temperature of the detector was determined by use of a simplified rate equation model. The spectral response of the detector was improved by the use of coherent optical effects resulting from the interaction of a multi-level atomic system with narrowband radiation. Superior

  19. Avoiding Boltzmann Brain domination in holographic dark energy models (United States)

    Horvat, R.


    In a spatially infinite and eternal universe approaching ultimately a de Sitter (or quasi-de Sitter) regime, structure can form by thermal fluctuations as such a space is thermal. The models of Dark Energy invoking holographic principle fit naturally into such a category, and spontaneous formation of isolated brains in otherwise empty space seems the most perplexing, creating the paradox of Boltzmann Brains (BB). It is thus appropriate to ask if such models can be made free from domination by Boltzmann Brains. Here we consider only the simplest model, but adopt both the local and the global viewpoint in the description of the Universe. In the former case, we find that if a dimensionless model parameter c, which modulates the Dark Energy density, lies outside the exponentially narrow strip around the most natural c = 1 line, the theory is rendered BB-safe. In the latter case, the bound on c is exponentially stronger, and seemingly at odds with those bounds on c obtained from various observational tests.

  20. Lattice Boltzmann simulations of genuinely multidimensional rarefied flows in microchannels (United States)

    Dellar, Paul; Reis, Tim


    We present lattice Boltzmann simulations of rarefied slip flows driven by applied pressure differences across microchannels of finite length. We correctly capture the nonlinear streamwise pressure variation and the cross-channel velocity component, as well as the streamwise velocity and volume flux. The former effects are both absent from almost all previous work that approximated the pressure difference using a uniform body force. We demonstrate second-order convergence of both velocity components towards the asymptotic solution for long microchannels, and slower convergence of the pressure. We use the standard lattice Boltzmann formulation that reduces to a second-order recurrence relation for the streamwise velocity in uniform shear, and whose analytical solution gives a parabolic profile from wall to wall. We therefore cannot capture Knudsen boundary layers, but instead implement Maxwell-Navier slip boundary conditions directly on the hydrodynamic moments of our discrete velocity model. Our only parameter is the tangential momentum accommodation coefficient, so we require no fitting to known solutions. Our moment-based approach shows that existing boundary conditions impose conditions on higher non-hydrodynamic moments rather than on the tangential fluid velocity itself. Supported by Award No. KUK-C1-013-04 made by King Abdullah University of Science and Technology (KAUST) and by EPSRC grant EP/E054625/1.

  1. Lattice Boltzmann simulation of flow around a confined circular cyclinder

    Energy Technology Data Exchange (ETDEWEB)

    Ashrafizaadeh, M.; Zadehgol, A. [Isfahan Univ. of Technology , Mechanical Engineering, Isfahan (Iran, Islamic Republic of)]. E-mail:


    A two dimensional lattice Boltzmann model (LBM) based on a single time relaxation BGK model has been developed. Several benchmark problems including the Poiseuille flow, the lid driven cavity flow and the flow around a circular cylinder have been performed employing a d2q9 lattice. The laminar flow around a circular cylinder within a channel has been extensively investigated using the present lattice Boltzmann model. Both symmetric and asymmetric placement configurations of the circular cylinder within the channel have been considered. A new treatment for the outlet velocity and pressure (density) boundary conditions has been proposed and validated. The present LBM results are in excellent agreement with those of the other existing CFD results. Careful examination of the LBM results and an appropriate calculation of the lift coefficient based on the rectangular lattice representation of the circular cylinder reveals that the periodic oscillation of the lift coefficient has a second harmonic when the cylinder is placed asymmetrically within the channel. The second harmonic could be associated with an asymmetrical shedding pattern of the vortices behind the cylinder from the upper and lower sides of the cylinder. (author)

  2. A Study of the Boltzmann Sequence-Structure Channel. (United States)

    Magner, Abram; Kihara, Daisuke; Szpankowski, Wojciech


    We rigorously study a channel that maps sequences from a finite alphabet to self-avoiding walks in the two-dimensional grid, inspired by a model of protein folding from statistical physics and studied empirically by biophysicists. This channel, which we call the Boltzmann sequence-structure channel, is characterized by a Boltzmann/Gibbs distribution with a free parameter corresponding to temperature. In our previous work, we verified empirically that the channel capacity appears to have a phase transition for small temperature and decays to zero for high temperature. In this paper, we make some progress toward theoretically explaining these phenomena. We first estimate the conditional entropy between the input sequence and the output fold, giving an upper bound which exhibits a phase transition with respect to temperature. Next, we formulate a class of parameter settings under which the dependence between walk energies is governed by their number of shared contacts. In this setting, we derive a lower bound on the conditional entropy. This lower bound allows us to conclude that the mutual information tends to zero in a nontrivial regime of high temperature, giving some support to the empirical fact regarding capacity. Finally, we construct an example setting of the parameters of the model for which the conditional entropy is exactly calculable and which does not exhibit a phase transition.

  3. Detection of Hypertension Retinopathy Using Deep Learning and Boltzmann Machines (United States)

    Triwijoyo, B. K.; Pradipto, Y. D.


    hypertensive retinopathy (HR) in the retina of the eye is disturbance caused by high blood pressure disease, where there is a systemic change of arterial in the blood vessels of the retina. Most heart attacks occur in patients caused by high blood pressure symptoms of undiagnosed. Hypertensive retinopathy Symptoms such as arteriolar narrowing, retinal haemorrhage and cotton wool spots. Based on this reasons, the early diagnosis of the symptoms of hypertensive retinopathy is very urgent to aim the prevention and treatment more accurate. This research aims to develop a system for early detection of hypertension retinopathy stage. The proposed method is to determine the combined features artery and vein diameter ratio (AVR) as well as changes position with Optic Disk (OD) in retinal images to review the classification of hypertensive retinopathy using Deep Neural Networks (DNN) and Boltzmann Machines approach. We choose this approach of because based on previous research DNN models were more accurate in the image pattern recognition, whereas Boltzmann machines selected because It requires speedy iteration in the process of learning neural network. The expected results from this research are designed a prototype system early detection of hypertensive retinopathy stage and analysed the effectiveness and accuracy of the proposed methods.

  4. Avoiding Boltzmann Brain domination in holographic dark energy models

    Directory of Open Access Journals (Sweden)

    R. Horvat


    Full Text Available In a spatially infinite and eternal universe approaching ultimately a de Sitter (or quasi-de Sitter regime, structure can form by thermal fluctuations as such a space is thermal. The models of Dark Energy invoking holographic principle fit naturally into such a category, and spontaneous formation of isolated brains in otherwise empty space seems the most perplexing, creating the paradox of Boltzmann Brains (BB. It is thus appropriate to ask if such models can be made free from domination by Boltzmann Brains. Here we consider only the simplest model, but adopt both the local and the global viewpoint in the description of the Universe. In the former case, we find that if a dimensionless model parameter c, which modulates the Dark Energy density, lies outside the exponentially narrow strip around the most natural c=1 line, the theory is rendered BB-safe. In the latter case, the bound on c is exponentially stronger, and seemingly at odds with those bounds on c obtained from various observational tests.

  5. Effects of ionization and ion loss on dust ion-acoustic solitary waves in a collisional dusty plasma with suprathermal electrons

    Energy Technology Data Exchange (ETDEWEB)

    Mayout, Saliha; Gougam, Leila Ait [Faculty of Physics, Theoretical Physics Laboratory, Plasma Physics Group, University of Bab-Ezzouar, USTHB, B.P. 32, El Alia, Algiers 16111 (Algeria); Tribeche, Mouloud, E-mail:, E-mail: [Faculty of Physics, Theoretical Physics Laboratory, Plasma Physics Group, University of Bab-Ezzouar, USTHB, B.P. 32, El Alia, Algiers 16111 (Algeria); Algerian Academy of Sciences and Technologies, Algiers (Algeria)


    The combined effects of ionization, ion loss, and electron suprathermality on dust ion-acoustic solitary waves in a collisional dusty plasma are examined. Carrying out a small but finite amplitude analysis, a damped Korteweg-de Vries (dK–dV) equation is derived. The damping term decreases with the increase of the spectral index and saturates for Maxwellian electrons. Choosing typical plasma parameters, the analytical approximate solution of the dK-dV equation is numerically analyzed. We first neglect the ionization and ion loss effects and account only for collisions to estimate the relative importance between these damping terms which can act concurrently. Interestingly, we found that as the suprathermal character of the electrons becomes important, the strength of the collisions related dissipation becomes more important and causes the dust ion-acoustic solitary wave amplitude to decay more rapidly. Moreover, the collisional damping may largely prevail over the ionization and ion loss related damping. The latter becomes more effective as the electrons evolve far away from their thermal equilibrium. Our results complement and provide new insights into previously published work on this problem.

  6. Non-Newtonian unconfined flow and heat transfer over a heated cylinder using the direct-forcing immersed boundary-thermal lattice Boltzmann method (United States)

    Amiri Delouei, A.; Nazari, M.; Kayhani, M. H.; Succi, S.


    In this study, the immersed boundary-thermal lattice Boltzmann method has been used to simulate non-Newtonian fluid flow over a heated circular cylinder. The direct-forcing algorithm has been employed to couple the off-lattice obstacles and on-lattice fluid nodes. To investigate the effect of boundary sharpness, two different diffuse interface schemes are considered to interpolate the velocity and temperature between the boundary and computational grid points. The lattice Boltzmann equation with split-forcing term is applied to consider the effects of the discrete lattice and the body force to the momentum flux, simultaneously. A method for calculating the Nusselt number based on diffuse interface schemes is developed. The rheological and thermal properties of non-Newtonian fluids are investigated under the different power-law indices and Reynolds numbers. The effect of numerical parameters on the accuracy of the proposed method has been investigated in detail. Results show that the rheological and thermal properties of non-Newtonian fluids in the presence of a heated immersed body can be suitably captured using the immersed boundary thermal lattice Boltzmann method.

  7. Spin states of asteroids in the Eos collisional family (United States)

    Hanuš, J.; Delbo', M.; Alí-Lagoa, V.; Bolin, B.; Jedicke, R.; Ďurech, J.; Cibulková, H.; Pravec, P.; Kušnirák, P.; Behrend, R.; Marchis, F.; Antonini, P.; Arnold, L.; Audejean, M.; Bachschmidt, M.; Bernasconi, L.; Brunetto, L.; Casulli, S.; Dymock, R.; Esseiva, N.; Esteban, M.; Gerteis, O.; de Groot, H.; Gully, H.; Hamanowa, Hiroko; Hamanowa, Hiromi; Krafft, P.; Lehký, M.; Manzini, F.; Michelet, J.; Morelle, E.; Oey, J.; Pilcher, F.; Reignier, F.; Roy, R.; Salom, P. A.; Warner, B. D.


    Eos family was created during a catastrophic impact about 1.3 Gyr ago. Rotation states of individual family members contain information about the history of the whole population. We aim to increase the number of asteroid shape models and rotation states within the Eos collision family, as well as to revise previously published shape models from the literature. Such results can be used to constrain theoretical collisional and evolution models of the family, or to estimate other physical parameters by a thermophysical modeling of the thermal infrared data. We use all available disk-integrated optical data (i.e., classical dense-in-time photometry obtained from public databases and through a large collaboration network as well as sparse-in-time individual measurements from a few sky surveys) as input for the convex inversion method, and derive 3D shape models of asteroids together with their rotation periods and orientations of rotation axes. We present updated shape models for 15 asteroids and new shape model determinations for 16 asteroids. Together with the already published models from the publicly available DAMIT database, we compiled a sample of 56 Eos family members with known shape models that we used in our analysis of physical properties within the family. Rotation states of asteroids smaller than ∼ 20 km are heavily influenced by the YORP effect, whilst the large objects more or less retained their rotation state properties since the family creation. Moreover, we also present a shape model and bulk density of asteroid (423) Diotima, an interloper in the Eos family, based on the disk-resolved data obtained by the Near InfraRed Camera (Nirc2) mounted on the W.M. Keck II telescope.

  8. On Quantum Fokker-Planck Equation (United States)

    Yano, Ryosuke


    The quantum Fokker-Planck equation (QFPE) is revisited. Provided that the molecule is the Maxwellian molecule, the quantum Landau-Fokker-Planck equation is divided into characteristic four terms. The characteristics of three terms among four terms are investigated on the basis of Grad's method, whereas the characteristics of the remained term, which is attributed to the collisional term of the QFPE proposed by Kaniadakis-Quarati, when the distribution function of the colliding partner is under the equilibrium state, are numerically investigated. The numerical result indicates that the time evolution of the distribution function obtained using such a remained term is instable, when the equilibrium or nonequilibrium state is given as initial data of the distribution function. Such an instability of the distribution function can be described by analyzing the propagation of the plane harmonic wave in one dimensional velocity space.

  9. Modeling mass transfer and reaction of dilute solutes in a ternary phase system by the lattice Boltzmann method (United States)

    Fu, Yu-Hang; Bai, Lin; Luo, Kai-Hong; Jin, Yong; Cheng, Yi


    In this work, we propose a general approach for modeling mass transfer and reaction of dilute solute(s) in incompressible three-phase flows by introducing a collision operator in lattice Boltzmann (LB) method. An LB equation was used to simulate the solute dynamics among three different fluids, in which the newly expanded collision operator was used to depict the interface behavior of dilute solute(s). The multiscale analysis showed that the presented model can recover the macroscopic transport equations derived from the Maxwell-Stefan equation for dilute solutes in three-phase systems. Compared with the analytical equation of state of solute and dynamic behavior, these results are proven to constitute a generalized framework to simulate solute distributions in three-phase flows, including compound soluble in one phase, compound adsorbed on single-interface, compound in two phases, and solute soluble in three phases. Moreover, numerical simulations of benchmark cases, such as phase decomposition, multilayered planar interfaces, and liquid lens, were performed to test the stability and efficiency of the model. Finally, the multiphase mass transfer and reaction in Janus droplet transport in a straight microchannel were well reproduced.

  10. Hydrodynamic equations for electrons in graphene obtained from the maximum entropy principle

    Energy Technology Data Exchange (ETDEWEB)

    Barletti, Luigi, E-mail: [Dipartimento di Matematica e Informatica “Ulisse Dini”, Università degli Studi di Firenze, Viale Morgagni 67/A, 50134 Firenze (Italy)


    The maximum entropy principle is applied to the formal derivation of isothermal, Euler-like equations for semiclassical fermions (electrons and holes) in graphene. After proving general mathematical properties of the equations so obtained, their asymptotic form corresponding to significant physical regimes is investigated. In particular, the diffusive regime, the Maxwell-Boltzmann regime (high temperature), the collimation regime and the degenerate gas limit (vanishing temperature) are considered.

  11. Fractional diffusion equation for heterogeneous medium

    Energy Technology Data Exchange (ETDEWEB)

    Polo L, M. A.; Espinosa M, E. G.; Espinosa P, G. [Universidad Autonoma Metropolitana, Unidad Iztapalapa, Area de Ingenieria en Recursos Energeticos, Av, San Rafael Atlixco 186, Col. Vicentina, 09340 Mexico D. F. (Mexico); Del Valle G, E., E-mail: [Instituto Politecnico Nacional, Escuela Superior de Fisica y Matematicas, Av. IPN s/n, Col. San Pedro Zacatenco, 07738 Mexico D. F. (Mexico)


    The asymptotic diffusion approximation for the Boltzmann (transport) equation was developed in 1950 decade in order to describe the diffusion of a particle in an isotropic medium, considers that the particles have a diffusion infinite velocity. In this work is developed a new approximation where is considered that the particles have a finite velocity, with this model is possible to describe the behavior in an anomalous medium. According with these ideas the model was obtained from the Fick law, where is considered that the temporal term of the current vector is not negligible. As a result the diffusion equation of fractional order which describes the dispersion of particles in a highly heterogeneous or disturbed medium is obtained, i.e., in a general medium. (Author)

  12. A lattice Boltzmann model for substrates with regularly structured surface roughness (United States)

    Yagub, A.; Farhat, H.; Kondaraju, S.; Singh, T.


    Superhydrophobic surface characteristics are important in many industrial applications, ranging from the textile to the military. It was observed that surfaces fabricated with nano/micro roughness can manipulate the droplet contact angle, thus providing an opportunity to control the droplet wetting characteristics. The Shan and Chen (SC) lattice Boltzmann model (LBM) is a good numerical tool, which holds strong potentials to qualify for simulating droplets wettability. This is due to its realistic nature of droplet contact angle (CA) prediction on flat smooth surfaces. But SC-LBM was not able to replicate the CA on rough surfaces because it lacks a real representation of the physics at work under these conditions. By using a correction factor to influence the interfacial tension within the asperities, the physical forces acting on the droplet at its contact lines were mimicked. This approach allowed the model to replicate some experimentally confirmed Wenzel and Cassie wetting cases. Regular roughness structures with different spacing were used to validate the study using the classical Wenzel and Cassie equations. The present work highlights the strength and weakness of the SC model and attempts to qualitatively conform it to the fundamental physics, which causes a change in the droplet apparent contact angle, when placed on nano/micro structured surfaces.

  13. Achieving tunable surface tension in the pseudopotential lattice Boltzmann modeling of multiphase flows. (United States)

    Li, Qing; Luo, K H


    In this paper, we aim to address an important issue about the pseudopotential lattice Boltzmann (LB) model, which has attracted much attention as a mesoscopic model for simulating interfacial dynamics of complex fluids, but suffers from the problem that the surface tension cannot be tuned independently of the density ratio. In the literature, a multirange potential was devised to adjust the surface tension [Sbragaglia et al., Phys. Rev. E 75, 026702 (2007)]. However, it was recently found that the density ratio of the system will be changed when the multirange potential is employed to adjust the surface tension. An alternative approach is therefore proposed in the present work. The basic strategy is to add a source term to the LB equation so as to tune the surface tension of the pseudopotential LB model. The proposed approach can guarantee that the adjustment of the surface tension does not affect the mechanical stability condition of the pseudopotential LB model, and thus provides a separate control of the surface tension and the density ratio. Meanwhile, it still retains the mesoscopic feature and the computational simplicity of the pseudopotential LB model. Numerical simulations are carried out for stationary droplets, capillary waves, and droplet splashing on a thin liquid film. The numerical results demonstrate that the proposed approach is capable of achieving a tunable surface tension over a very wide range and can keep the density ratio unchanged when adjusting the surface tension.

  14. Simulation by using the lattice Boltzmann method of microscopic particle motion induced by artificial cilia (United States)

    Alapati, Suresh; Che, Woo Seong; Mannoor, Madhusoodanan; Suh, Yong Kweon


    In this paper, we present the results obtained from the simulation of particle motion induced by the fluid flow driven by an array of beating artificial cilia inside a micro-channel. A worm-like-chain model is used to simulate the elastic cilia, and the lattice Boltzmann equation is used to compute the fluid flow. We employ a harmonic force at the extreme tip of each cilium to actuate it. Our simulation methods are first validated by applying them to the motion of a single cilium and a freely falling sphere. After validation, we simulate the fluid flow generated by an array of beating cilia and find that a maximum flow rate is achieved at an optimum sperm number. Next, we simulate the motion of a neutrally buoyant spherical particle at this optimum sperm number by tracking the particle motion with a smoothed profile method. We address the effect of the following parameters on the particle velocity: the gap between cilia and particle, the particle size, the cilia density, and the presence of an array of intermediate particles.

  15. An immersed boundary-lattice Boltzmann model for biofilm growth in porous media (United States)

    Benioug, M.; Golfier, F.; Oltéan, C.; Buès, M. A.; Bahar, T.; Cuny, J.


    In this paper, we present a two-dimensional pore-scale numerical model to investigate the main mechanisms governing biofilm growth in porous media. The fluid flow and solute transport equations are coupled with a biofilm evolution model. Fluid flow is simulated with an immersed boundary-lattice Boltzmann model while solute transport is described with a volume-of-fluid-type approach. A cellular automaton algorithm combined with immersed boundary methods was developed to describe the spreading and distribution of biomass. Bacterial attachment and detachment mechanisms are also taken into account. The capability of this model to describe correctly the couplings involved between fluid circulation, nutrient transport and bacterial growth is tested under different hydrostatic and hydrodynamic conditions (i) on a flat medium and (ii) for a complex porous medium. For the second case, different regimes of biofilm growth are identified and are found to be related to the dimensionless parameters of the model, Damköhler and Péclet numbers and dimensionless shear stress. Finally, the impact of biofilm growth on the macroscopic properties of the porous medium is investigated and we discuss the unicity of the relationships between hydraulic conductivity and biofilm volume fraction.

  16. Consistent simulation of droplet evaporation based on the phase-field multiphase lattice Boltzmann method. (United States)

    Safari, Hesameddin; Rahimian, Mohammad Hassan; Krafczyk, Manfred


    In the present article, we extend and generalize our previous article [H. Safari, M. H. Rahimian, and M. Krafczyk, Phys. Rev. E 88, 013304 (2013)] to include the gradient of the vapor concentration at the liquid-vapor interface as the driving force for vaporization allowing the evaporation from the phase interface to work for arbitrary temperatures. The lattice Boltzmann phase-field multiphase modeling approach with a suitable source term, accounting for the effect of the phase change on the velocity field, is used to solve the two-phase flow field. The modified convective Cahn-Hilliard equation is employed to reconstruct the dynamics of the interface topology. The coupling between the vapor concentration and temperature field at the interface is modeled by the well-known Clausius-Clapeyron correlation. Numerous validation tests including one-dimensional and two-dimensional cases are carried out to demonstrate the consistency of the presented model. Results show that the model is able to predict the flow features around and inside an evaporating droplet quantitatively in quiescent as well as convective environments.

  17. A lattice Boltzmann method for axisymmetric multicomponent flows with high viscosity ratio (United States)

    Liu, Haihu; Wu, Lei; Ba, Yan; Xi, Guang; Zhang, Yonghao


    A color-gradient lattice Boltzmann method (LBM) is proposed to simulate axisymmetric multicomponent flows. This method uses a collision operator that is a combination of three separate parts, namely single-component collision operator, perturbation operator, and recoloring operator. A source term is added into the single-component collision operator such that in each single-component region the axisymmetric continuity and momentum equations can be exactly recovered. The interfacial tension effect is realized by the perturbation operator, in which an interfacial force of axisymmetric form is derived using the concept of continuum surface force. A recoloring operator proposed by Latva-Kokko and Rothman is extended to the axisymmetric case for phase segregation and maintenance of the interface. To enhance the method's numerical stability for handling binary fluids with high viscosity ratio, a multiple-relaxation-time model is used for the collision operator. Several numerical examples, including static droplet test, oscillation of a viscous droplet, and breakup of a liquid thread, are presented to test the capability and accuracy of the proposed color-gradient LBM. It is found that the present method is able to accurately capture the phase interface and produce low spurious velocities. Also, the LBM results are all in good agreement with the analytical solutions and/or available experimental data for a very broad range of viscosity ratios.

  18. Entropic lattice Boltzmann method for multiphase flows: Fluid-solid interfaces. (United States)

    Mazloomi M, Ali; Chikatamarla, Shyam S; Karlin, Iliya V


    The recently introduced entropic lattice Boltzmann model (ELBM) for multiphase flows [A. Mazloomi M., S. S. Chikatamarla, and I. V. Karlin, Phys. Rev. Lett. 114, 174502 (2015)] is extended to the simulation of dynamic fluid-solid interface problems. The thermodynamically consistent, nonlinearly stable ELBM together with a polynomial representation of the equation of state enables us to investigate the dynamics of the contact line in a wide range of applications, from capillary filling to liquid drop impact onto a flat surfaces with different wettability. The static interface behavior is tested by means of the liquid column in a channel to verify the Young-Laplace law. The numerical results of a capillary filling problem in a channel with wettability gradient show an excellent match with the existing analytical solution. Simulations of drop impact onto both wettable and nonwettable surfaces show that the ELBM reproduces the experimentally observed drop behavior in a quantitative manner. Results reported herein demonstrate that the present model is a promising alternative for studying the vapor-liquid-solid interface dynamics.

  19. The ionic atmosphere around A-RNA: Poisson-Boltzmann and molecular dynamics simulations. (United States)

    Kirmizialtin, Serdal; Silalahi, Alexander R J; Elber, Ron; Fenley, Marcia O


    The distributions of different cations around A-RNA are computed by Poisson-Boltzmann (PB) equation and replica exchange molecular dynamics (MD). Both the nonlinear PB and size-modified PB theories are considered. The number of ions bound to A-RNA, which can be measured experimentally, is well reproduced in all methods. On the other hand, the radial ion distribution profiles show differences between MD and PB. We showed that PB results are sensitive to ion size and functional form of the solvent dielectric region but not the solvent dielectric boundary definition. Size-modified PB agrees with replica exchange molecular dynamics much better than nonlinear PB when the ion sizes are chosen from atomistic simulations. The distribution of ions 14 Å away from the RNA central axis are reasonably well reproduced by size-modified PB for all ion types with a uniform solvent dielectric model and a sharp dielectric boundary between solvent and RNA. However, this model does not agree with MD for shorter distances from the A-RNA. A distance-dependent solvent dielectric function proposed by another research group improves the agreement for sodium and strontium ions, even for shorter distances from the A-RNA. However, Mg(2+) distributions are still at significant variances for shorter distances. Copyright © 2012 Biophysical Society. Published by Elsevier Inc. All rights reserved.

  20. Poisson-Boltzmann model of electrolytes containing uniformly charged spherical nanoparticles. (United States)

    Bohinc, Klemen; Volpe Bossa, Guilherme; Gavryushov, Sergei; May, Sylvio


    Like-charged macromolecules typically repel each other in aqueous solutions that contain small mobile ions. The interaction tends to turn attractive if mobile ions with spatially extended charge distributions are added. Such systems can be modeled within the mean-field Poisson-Boltzmann formalism by explicitly accounting for charge-charge correlations within the spatially extended ions. We consider an aqueous solution that contains a mixture of spherical nanoparticles with uniform surface charge density and small mobile salt ions, sandwiched between two like-charged planar surfaces. We perform the minimization of an appropriate free energy functional, which leads to a non-linear integral-differential equation for the electrostatic potential that we solve numerically and compare with predictions from Monte Carlo simulations. Nanoparticles with uniform surface charge density are contrasted with nanoparticles that have all their charges relocated at the center. Our mean-field model predicts that only the former (especially when large and highly charged particles) but not the latter are able to mediate attractive interactions between like-charged planar surfaces. We also demonstrate that at high salt concentration attractive interactions between like-charged planar surfaces turn into repulsion.

  1. Acceleration of Linear Finite-Difference Poisson-Boltzmann Methods on Graphics Processing Units. (United States)

    Qi, Ruxi; Botello-Smith, Wesley M; Luo, Ray


    Electrostatic interactions play crucial roles in biophysical processes such as protein folding and molecular recognition. Poisson-Boltzmann equation (PBE)-based models have emerged as widely used in modeling these important processes. Though great efforts have been put into developing efficient PBE numerical models, challenges still remain due to the high dimensionality of typical biomolecular systems. In this study, we implemented and analyzed commonly used linear PBE solvers for the ever-improving graphics processing units (GPU) for biomolecular simulations, including both standard and preconditioned conjugate gradient (CG) solvers with several alternative preconditioners. Our implementation utilizes the standard Nvidia CUDA libraries cuSPARSE, cuBLAS, and CUSP. Extensive tests show that good numerical accuracy can be achieved given that the single precision is often used for numerical applications on GPU platforms. The optimal GPU performance was observed with the Jacobi-preconditioned CG solver, with a significant speedup over standard CG solver on CPU in our diversified test cases. Our analysis further shows that different matrix storage formats also considerably affect the efficiency of different linear PBE solvers on GPU, with the diagonal format best suited for our standard finite-difference linear systems. Further efficiency may be possible with matrix-free operations and integrated grid stencil setup specifically tailored for the banded matrices in PBE-specific linear systems.

  2. Simulation of Thermal Flow Problems via a Hybrid Immersed Boundary-Lattice Boltzmann Method

    Directory of Open Access Journals (Sweden)

    J. Wu


    Full Text Available A hybrid immersed boundary-lattice Boltzmann method (IB-LBM is presented in this work to simulate the thermal flow problems. In current approach, the flow field is resolved by using our recently developed boundary condition-enforced IB-LBM (Wu and Shu, (2009. The nonslip boundary condition on the solid boundary is enforced in simulation. At the same time, to capture the temperature development, the conventional energy equation is resolved. To model the effect of immersed boundary on temperature field, the heat source term is introduced. Different from previous studies, the heat source term is set as unknown rather than predetermined. Inspired by the idea in (Wu and Shu, (2009, the unknown is calculated in such a way that the temperature at the boundary interpolated from the corrected temperature field accurately satisfies the thermal boundary condition. In addition, based on the resolved temperature correction, an efficient way to compute the local and average Nusselt numbers is also proposed in this work. As compared with traditional implementation, no approximation for temperature gradients is required. To validate the present method, the numerical simulations of forced convection are carried out. The obtained results show good agreement with data in the literature.

  3. Dielectric Boundary Force in Molecular Solvation with the Poisson-Boltzmann Free Energy: A Shape Derivative Approach. (United States)

    Li, Bo; Cheng, Xiaoliang; Zhang, Zhengfang


    In an implicit-solvent description of molecular solvation, the electrostatic free energy is given through the electrostatic potential. This potential solves a boundary-value problem of the Poisson-Boltzmann equation in which the dielectric coefficient changes across the solute-solvent interface-the dielectric boundary. The dielectric boundary force acting on such a boundary is the negative first variation of the electrostatic free energy with respect to the location change of the boundary. In this work, the concept of shape derivative is used to define such variations and formulas of the dielectric boundary force are derived. It is shown that such a force is always in the direction toward the charged solute molecules.

  4. Dielectric Boundary Force in Molecular Solvation with the Poisson–Boltzmann Free Energy: A Shape Derivative Approach (United States)

    Li, Bo; Cheng, Xiaoliang; Zhang, Zhengfang


    In an implicit-solvent description of molecular solvation, the electrostatic free energy is given through the electrostatic potential. This potential solves a boundary-value problem of the Poisson–Boltzmann equation in which the dielectric coefficient changes across the solute-solvent interface—the dielectric boundary. The dielectric boundary force acting on such a boundary is the negative first variation of the electrostatic free energy with respect to the location change of the boundary. In this work, the concept of shape derivative is used to define such variations and formulas of the dielectric boundary force are derived. It is shown that such a force is always in the direction toward the charged solute molecules. PMID:24058212

  5. Self-focusing and defocusing of Gaussian laser beams in collisional inhomogeneous plasmas with linear density and temperature ramps (United States)

    Hashemzadeh, M.


    Self-focusing and defocusing of Gaussian laser beams in collisional inhomogeneous plasmas are investigated in the presence of various laser intensities and linear density and temperature ramps. Considering the ponderomotive force and using the momentum transfer and energy equations, the nonlinear electron density is derived. Taking into account the paraxial approximation and nonlinear electron density, a nonlinear differential equation, governing the focusing and defocusing of the laser beam, is obtained. Results show that in the absence of ramps the laser beam is focused between a minimum and a maximum value of laser intensity. For a certain value of laser intensity and initial electron density, the self-focusing process occurs in a temperature range which reaches its maximum at turning point temperature. However, the laser beam is converged in a narrow range for various amounts of initial electron density. It is indicated that the σ2 parameter and its sign can affect the self-focusing process for different values of laser intensity, initial temperature, and initial density. Finally, it is found that although the electron density ramp-down diverges the laser beam, electron density ramp-up improves the self-focusing process.

  6. Differential equations

    CERN Document Server

    Hochstadt, Harry


    Modern approach to differential equations presents subject in terms of ideas and concepts rather than special cases and tricks which traditional courses emphasized. No prerequisites needed other than a good calculus course. Certain concepts from linear algebra used throughout. Problem section at end of each chapter.

  7. Viscous dissipation effects on MHD slip flow and heat transfer in porous micro duct with LTNE assumptions using modified lattice Boltzmann method (United States)

    Rabhi, R.; Amami, B.; Dhahri, H.; Mhimid, A.


    This paper deals with heat transfer and fluid flow in a porous micro duct under local thermal non equilibrium conditions subjected to an external oriented magnetic field. The considered sample is a micro duct filled with porous media assumed to be homogenous, isotropic and saturated. The slip velocity and the temperature jump were uniformly imposed to the wall. In modeling the flow, the Brinkmann-Forchheimer extended Darcy model was incorporated into the momentum equations. In the energy equation, the local thermal non equilibrium between the two phases was adopted. A modified axisymmetric lattice Boltzmann method was used to solve the obtained governing equation system. Attention was focused on the influence of the emerging parameters such as Knudsen number, Kn, Hartmann number, Ha, Eckert number, Ec, Biot number, Bi and the magnetic field inclination γ on flow and heat transfer throughout this paper.

  8. Great moments in kinetic theory: 150 years of Maxwell’s (other) equations (United States)

    Robson, Robert E.; Mehrling, Timon J.; Osterhoff, Jens


    In 1867, just two years after laying the foundations of electromagnetism, J. Clerk Maxwell presented a fundamental paper on kinetic gas theory, in which he described the evolution of the gas in terms of certain ‘moments’ of its velocity distribution function. This inspired Ludwig Boltzmann to formulate his famous kinetic equation, from which followed the H-theorem and the connection with entropy. On the occasion of the 150th anniversary of publication of Maxwell's paper, we review the Maxwell–Boltzmann formalism and discuss how its generality and adaptability enable it to play a key role in describing the behaviour of a variety of systems of current interest, in both gaseous and condensed matter, and in modern-day physics and technologies which Maxwell and Boltzmann could not possibly have foreseen. In particular, we illustrate the relevance and applicability of Maxwell's formalism to the dynamic field of plasma-wakefield acceleration.

  9. Application of Lattice Boltzmann Techniques to Modeling Millimeter Scale Water-Rock Interaction in Porous Media (United States)

    Fritz, D. E.; Farmer, G. L.; Janecky, D. R.


    Lattice Boltzmann (LB) techniques are being applied to models of reactive transport processes in groundwaters with the purpose of modeling small scale (mm) interactions between water and rock in complex solid media. This method was used to create a simple porous media structure in which chemical reactions including sorption, dissolution and precipitation occur. A pre-existing lattice Boltzmann code developed at LANL for physical fluid flow was used as the basis for overlaying chemical exchange controls at each lattice node. Initially, Langmuir sorption isotherms were applied on open channel models varying from 10x4000 to 30x1000 nodes. The resulting breakthrough curves from these chemical slug models are compatible with analytical solutions that apply the isotherm equations on larger scales. The model was then further developed to include multiple solid compositions and dissolution and precipitation reactions to study the small scale spatial and temporal distributions of materials in porous media. Mineral nodes of varying composition which undergo specific chemical reactions can be created in the open channels or among evenly spaced blocks of solid nodes which simulate homogeneous porous media. This model structure is being used to investigate millimeter scale Sr isotopic variations observed in the K-metasomatized Lemitar Tuff from Southern New Mexico. Multiple samples of the adularia will provide an isochron age that may represent the time of K-metasomatism. Physical microsampling of this tuff has demonstrated the secondary adularia has a unique Sr isotopic characteristic (87}Sr/{86Sr near 0.727) relative to the fresh plagioclase and sanidine phenocrysts (0.709). The model is being used to discern between the possibilities of the Sr now present in the altered material originating from the surround matrix or alternatively from an outside source in the altering fluids. An outside source of Sr in the K-metasomatic groundwater would be likely with the failure of the

  10. The role of collisional compaction in primitive asteroids and comets (United States)

    Trigo-Rodríguez, J. M.; Blum, J.


    During the early stages of solar system formation the consolidation of asteroids and comets took place. We have just learnt from recent space missions that some of these minor bodies have been preserved in a pristine way in several regions of our Solar System. From our experience on primitive meteorites we know that these bodies should contain valuable clues on the origin of the Solar System. Studies of the physical, chemical, and isotopic properties of the components of these minor bodies will provide important clues on their origin. We expect very different collisional histories undergone by these bodies depending on their particular formation, migration, and storage regions [1]. In Fig. 1 appears a schematic representation of the protoplanetary disk in the region of consolidation of the terrestrial planets about 4565 million years ago. Bodies located in the outer part of the main belt would have incorporated significant amounts of ice in their volume, but their migration to and residence times in other regions would have defined their physico-chemical properties. Recent laboratory studies and observational data compiled from comets, meteorites and meteoroids [2] suggest that the porosity of these bodies should have decreased with time depending on the degree of collisions, aqueous alteration and heating. For typical stony targets, the tensile strength and gravity are the main properties that are defining the formation of impact craters and subsequently the degree of impact metamorphism and mineralogy of the shocked materials. However, little is known about the influence of porosity on the impact process although the crushing of pore space is an efficient mechanism for absorbing shock waves, also increasing the postshock temperatures [2]. In this context, a Near-Earth Object (NEO) sample return mission called Marco Polo is being studied within the Cosmic Vision programme. Such kind of mission would be returning to the Earth unaltered material from a NEO, just

  11. Exploring the collisional evolution of the asteroid belt (United States)

    Bottke, W.; Broz, M.; O'Brien, D.; Campo Bagatin, A.; Morbidelli, A.


    The asteroid belt is a remnant of planet-formation processes. By modeling its collisional and dynamical history, and linking the results to constraints, we can probe how the planets and small bodies formed and evolved. Some key model constraints are: (i) The wavy shape of the main-belt size distribution (SFD), with inflection points near 100-km, 10--20-km, 1 to a few km, and ˜0.1-km diameter; (ii) The number of asteroid families created by the catastrophic breakup of large asteroid bodies over the last ˜ 4 Gy, with the number of disrupted D > 100 km bodies as small as ˜20 or as large as 60; (iii) the flux of small asteroids derived from the main belt that have struck the Moon over the last 3.5 Ga --- crater SFDs on lunar terrains with known ages suggest the D value may seem strange, considering the solar system is only 4.56 Gy old. One way to interpret it is that the main belt once had more mass that was eliminated by early dynamical processes between 4--4.56 Ga. This would allow for more early grinding, and it would suggest the main belt's wavy-shaped SFD is a ''fossil'' from a more violent early epoch. Simulations suggest that most D > 100 km bodies have been significantly battered, but only a fraction have been catastrophically disrupted. Conversely, most small asteroids today are byproducts of fragmentation events. These results are consistent with growing evidence that most of the prominent meteorite classes were produced by young asteroid families. The big question is how to use what we know to determine the main belt's original size and state. This work is ongoing, but dynamical models hint at many possibilities, including both the late arrival and late removal of material from the main belt. In addition, no model has yet properly accounted for the bombardment of the primordial main belt by leftover planetesimals in the terrestrial planet region. It is also possible to use additional constraints, such as the apparent paucity of Vesta-like or V

  12. Influence of collisional rate coefficients on water vapour excitation (United States)

    Daniel, F.; Goicoechea, J. R.; Cernicharo, J.; Dubernet, M.-L.; Faure, A.


    Context. Water is a key molecule in many astrophysical studies that deal with star or planet forming regions, evolved stars, and galaxies. Its high dipole moment makes this molecule subthermally populated under the typical conditions of most astrophysical objects. This motivated calculation of various sets of collisional rate coefficients (CRC) for H2O (with He or H2), which are needed to model its rotational excitation and line emission. Aims: The most accurate set of CRC are the quantum rates that involve H2. However, they have been published only recently, and less accurate CRC (quantum with He or quantum classical trajectory (QCT) with H2) were used in many studies before that. This work aims to underline the impact that the new available set of CRC have on interpretations of water vapour observations. Methods: We performed accurate non-local, non-LTE radiative transfer calculations using different sets of CRC to predict the line intensities from transitions that involve the lowest energy levels of H2O (E find that the intensities based on the quantum and QCT CRC are in good agreement. However, at relatively low H2 volume density (n(H2) find differences in the predicted line intensities of up to a factor of ~3 for the bulk of the lines. Most of the recent studies interpreting early Herschel Space Observatory spectra have used the QCT CRC. Our results show that, although the global conclusions from those studies will not be drastically changed, each case has to be considered individually, since depending on the physical conditions, the use of the QCT CRC may lead to a mis-estimate of the water vapour abundance of up to a factor of ~3. Additionally, the comparison of the quantum state-to-state and thermalised CRC, including the description of the population of the H2 rotational levels, show that above TK ~ 100 K, large differences are expected from those two sets for the p-H2 symmetry. Finally, we find that at low temperature (i.e. TK < 100 K) modelled line

  13. Implementation of a Combined Elastic-Viscous-Plastic and Collisional Sea Ice Rheology (United States)

    Rynders, Stefanie; Aksenov, Yevgeny; Feltham, Daniel


    The Marginal Ice Zone (MIZ) is a transitional area between the open ocean and pack ice. The MIZ is present in the Arctic and Southern Ocean and measures up to several hundred kilometers across. It is characterized by high surface ocean waves and consists of severely fragmented sea ice with ice floes less than 100m in diameter. With declining summer Arctic sea ice cover and increased wave heights in the Arctic Ocean, in the Arctic the MIZ widened by about 40 percent during the last three decades. The changes in sea ice and growing economic activity in the Polar Oceans necessitate new climate and forecasting models that can simulate the MIZ. Current models are not fit for the purpose since they do not model the surface ocean waves, which determine the MIZ width, or the sea ice rheology that represents MIZ ice dynamics. This study presents an implementation of collisional ice rheology that takes into account jostling of ice floes and also includes the effects of the ice floe distribution on internal ice stresses. The collisional contribution is derived from the magnitude of velocity fluctuations of ice floes. These are calculated from a kinetic energy evolution equation for the ice floes. Properties taken from a coupled wave-in-ice module determine the maximum floe size. This information is taken form a coupled wave-in-ice module. The rheology is derived in the framework of the Elastic-Viscous-Plastic rheology. This allows combination with the Elastic-Viscous-Plastic rheology and thus formulation of a unified sea ice rheology suitable for both the central pack ice and MIZ. The combined ice rheology is implemented in the Los Alamos CICE model and tested in the 2-degree resolution global NEMO Ocean General Circulation model. The 10-year run is forced by CORE2 climatological forcing. Prelimary results show that in the Arctic the new rheology decreases ice thicknesses near the coasts where ice is stationary. Overall, the change in the basin-scale Arctic ice thickness is

  14. The peeling process of infinite Boltzmann planar maps

    DEFF Research Database (Denmark)

    Budd, Timothy George


    We start by studying a peeling process on finite random planar maps with faces of arbitrary degrees determined by a general weight sequence, which satisfies an admissibility criterion. The corresponding perimeter process is identified as a biased random walk, in terms of which the admissibility...... criterion has a very simple interpretation. The finite random planar maps under consideration were recently proved to possess a well-defined local limit known as the infinite Boltzmann planar map (IBPM). Inspired by recent work of Curien and Le Gall, we show that the peeling process on the IBPM can...... be obtained from the peeling process of finite random maps by conditioning the perimeter process to stay positive. The simplicity of the resulting description of the peeling process allows us to obtain the scaling limit of the associated perimeter and volume process for arbitrary regular critical weight...

  15. Approximate Message Passing with Restricted Boltzmann Machine Priors

    CERN Document Server

    Tramel, Eric W; Krzakala, Florent


    Approximate Message Passing (AMP) has been shown to be an excellent statistical approach to signal inference and compressed sensing problem. The AMP framework provides modularity in the choice of signal prior; here we propose a hierarchical form of the Gauss-Bernouilli prior which utilizes a Restricted Boltzmann Machine (RBM) trained on the signal support to push reconstruction performance beyond that of simple iid priors for signals whose support can be well represented by a trained binary RBM. We present and analyze two methods of RBM factorization and demonstrate how these affect signal reconstruction performance within our proposed algorithm. Finally, using the MNIST handwritten digit dataset, we show experimentally that using an RBM allows AMP to approach oracle-support performance.

  16. Lattice Boltzmann Simulation of Multiple Bubbles Motion under Gravity

    Directory of Open Access Journals (Sweden)

    Deming Nie


    Full Text Available The motion of multiple bubbles under gravity in two dimensions is numerically studied through the lattice Boltzmann method for the Eotvos number ranging from 1 to 12. Two kinds of initial arrangement are taken into account: vertical and horizontal arrangement. In both cases the effects of Eotvos number on the bubble coalescence and rising velocity are investigated. For the vertical arrangement, it has been found that the coalescence pattern is similar. The first coalescence always takes place between the two uppermost bubbles. And the last coalescence always takes place between the coalesced bubble and the bottommost bubble. For four bubbles in a horizontal arrangement, the outermost bubbles travel into the wake of the middle bubbles in all cases, which allows the bubbles to coalesce. The coalescence pattern is more complex for the case of eight bubbles, which strongly depends on the Eotvos number.

  17. Lattice Boltzmann modeling an introduction for geoscientists and engineers

    CERN Document Server

    Sukop, Michael C


    Lattice Boltzmann models have a remarkable ability to simulate single- and multi-phase fluids and transport processes within them. A rich variety of behaviors, including higher Reynolds numbers flows, phase separation, evaporation, condensation, cavitation, buoyancy, and interactions with surfaces can readily be simulated. This book provides a basic introduction that emphasizes intuition and simplistic conceptualization of processes. It avoids the more difficult mathematics that underlies LB models. The model is viewed from a particle perspective where collisions, streaming, and particle-particle/particle-surface interactions constitute the entire conceptual framework. Beginners and those with more interest in model application than detailed mathematical foundations will find this a powerful "quick start" guide. Example simulations, exercises, and computer codes are included. Working code is provided on the Internet.

  18. Lattice Boltzmann model for melting with natural convection

    Energy Technology Data Exchange (ETDEWEB)

    Huber, Christian [Department of Earth and Planetary Science, University of California - Berkeley, 307 McCone Hall 4767, Berkeley, CA 94720-4767 (United States)], E-mail:; Parmigiani, Andrea [Computer Science Department, University of Geneva, 24, Rue du General Dufour, 1211 Geneva 4 (Switzerland)], E-mail:; Chopard, Bastien [Computer Science Department, University of Geneva, 24, Rue du General Dufour, 1211 Geneva 4 (Switzerland)], E-mail:; Manga, Michael [Department of Earth and Planetary Science, University of California - Berkeley, 177 McCone Hall 4767, Berkeley, CA 94720-4767 (United States)], E-mail:; Bachmann, Olivier [Department of Earth and Space Science, University of Washington, Johnson Hall 070, Seattle WA 98195-1310 (United States)], E-mail:


    We develop a lattice Boltzmann method to couple thermal convection and pure-substance melting. The transition from conduction-dominated heat transfer to fully-developed convection is analyzed and scaling laws and previous numerical results are reproduced by our numerical method. We also investigate the limit in which thermal inertia (high Stefan number) cannot be neglected. We use our results to extend the scaling relations obtained at low Stefan number and establish the correlation between the melting front propagation and the Stefan number for fully-developed convection. We conclude by showing that the model presented here is particularly well-suited to study convection melting in geometrically complex media with many applications in geosciences.

  19. Emergence of Compositional Representations in Restricted Boltzmann Machines (United States)

    Tubiana, J.; Monasson, R.


    Extracting automatically the complex set of features composing real high-dimensional data is crucial for achieving high performance in machine-learning tasks. Restricted Boltzmann machines (RBM) are empirically known to be efficient for this purpose, and to be able to generate distributed and graded representations of the data. We characterize the structural conditions (sparsity of the weights, low effective temperature, nonlinearities in the activation functions of hidden units, and adaptation of fields maintaining the activity in the visible layer) allowing RBM to operate in such a compositional phase. Evidence is provided by the replica analysis of an adequate statistical ensemble of random RBMs and by RBM trained on the handwritten digits data set MNIST.

  20. Boundary slip from the immersed boundary lattice Boltzmann models. (United States)

    Le, Guigao; Zhang, Junfeng


    We report an interesting and important observation of the velocity fields from immersed boundary lattice Boltzmann methods (IB-LBM). The computed velocity profiles can deviate from theoretical predictions greatly even for very simple flow situations, both in the immersed boundary layer and the bulk region. A rigorous analysis of the IB-LBM simulated velocity for a symmetric shear flow is carried out, and the analytical solutions indicate a strong dependence of velocity on the relaxation parameter (kinetic viscosity). Also our simulations demonstrate that simply increasing the immersed boundary layer thickness is not an efficient approach to reduce such velocity discrepancy. We hope this work will bring the awareness of this essential issue to people using IB-LBM for various flow situations.

  1. Using an Interactive Lattice Boltzmann Solver in Fluid Mechanics Instruction

    Directory of Open Access Journals (Sweden)

    Mirjam S. Glessmer


    Full Text Available This article gives an overview of the diverse range of teaching applications that can be realized using an interactive lattice Boltzmann simulation tool in fluid mechanics instruction and outreach. In an inquiry-based learning framework, examples are given of learning scenarios that address instruction on scientific results, scientific methods or the scientific process at varying levels of student activity, from consuming to applying to researching. Interactive live demonstrations on portable hardware enable new and innovative teaching concepts for fluid mechanics, also for large audiences and in the early stages of the university education. Moreover, selected examples successfully demonstrate that the integration of high-fidelity CFD methods into fluid mechanics teaching facilitates high-quality student research work within reach of the current state of the art in the respective field of research.

  2. Emergence of Compositional Representations in Restricted Boltzmann Machines. (United States)

    Tubiana, J; Monasson, R


    Extracting automatically the complex set of features composing real high-dimensional data is crucial for achieving high performance in machine-learning tasks. Restricted Boltzmann machines (RBM) are empirically known to be efficient for this purpose, and to be able to generate distributed and graded representations of the data. We characterize the structural conditions (sparsity of the weights, low effective temperature, nonlinearities in the activation functions of hidden units, and adaptation of fields maintaining the activity in the visible layer) allowing RBM to operate in such a compositional phase. Evidence is provided by the replica analysis of an adequate statistical ensemble of random RBMs and by RBM trained on the handwritten digits data set MNIST.

  3. Determination of the Boltzmann Constant Using the Differential - Cylindrical Procedure

    CERN Document Server

    Feng, X J; Lin, H; Gillis, K A; Moldover, M R


    We report in this paper the progresses on the determination of the Boltzmann constant using the acoustic gas thermometer (AGT) of fixed-length cylindrical cavities. First, we present the comparison of the molar masses of pure argon gases through comparing speeds of sound of gases. The procedure is independent from the methodology by Gas Chromatography-Mass Spectrometry (GC-MS). The experimental results show good agreement between both methods. The comparison offers an independent inspection of the analytical results by GC-MS. Second, we present the principle of the novel differential-cylindrical procedure based on the AGT of two fixed-length cavities. The deletion mechanism for some major perturbations is analyzed for the new procedure. The experimental results of the differential-cylindrical procedure demonstrate some major improvements on the first, second acoustic and third virial coefficients, and the excess half-widths. The three acoustic virial coefficients agree well with the stated-of-the-art experime...

  4. Lattice Boltzmann simulations of convection heat transfer in porous media (United States)

    Liu, Qing; He, Ya-Ling


    A non-orthogonal multiple-relaxation-time (MRT) lattice Boltzmann (LB) method is developed to study convection heat transfer in porous media at the representative elementary volume scale based on the generalized non-Darcy model. In the method, two different LB models are constructed: one is constructed in the framework of the double-distribution-function approach, and the other is constructed in the framework of the hybrid approach. In particular, the transformation matrices used in the MRT-LB models are non-orthogonal matrices. The present method is applied to study mixed convection flow in a porous channel and natural convection flow in a porous cavity. It is found that the numerical results are in good agreement with the analytical solutions and/or other results reported in previous studies. Furthermore, the non-orthogonal MRT-LB method shows better numerical stability in comparison with the BGK-LB method.

  5. Consistent forcing scheme in the cascaded lattice Boltzmann method (United States)

    Fei, Linlin; Luo, Kai Hong


    In this paper, we give an alternative derivation for the cascaded lattice Boltzmann method (CLBM) within a general multiple-relaxation-time (MRT) framework by introducing a shift matrix. When the shift matrix is a unit matrix, the CLBM degrades into an MRT LBM. Based on this, a consistent forcing scheme is developed for the CLBM. The consistency of the nonslip rule, the second-order convergence rate in space, and the property of isotropy for the consistent forcing scheme is demonstrated through numerical simulations of several canonical problems. Several existing forcing schemes previously used in the CLBM are also examined. The study clarifies the relation between MRT LBM and CLBM under a general framework.

  6. Training Scalable Restricted Boltzmann Machines Using a Quantum Annealer (United States)

    Kumar, V.; Bass, G.; Dulny, J., III


    Machine learning and the optimization involved therein is of critical importance for commercial and military applications. Due to the computational complexity of many-variable optimization, the conventional approach is to employ meta-heuristic techniques to find suboptimal solutions. Quantum Annealing (QA) hardware offers a completely novel approach with the potential to obtain significantly better solutions with large speed-ups compared to traditional computing. In this presentation, we describe our development of new machine learning algorithms tailored for QA hardware. We are training restricted Boltzmann machines (RBMs) using QA hardware on large, high-dimensional commercial datasets. Traditional optimization heuristics such as contrastive divergence and other closely related techniques are slow to converge, especially on large datasets. Recent studies have indicated that QA hardware when used as a sampler provides better training performance compared to conventional approaches. Most of these studies have been limited to moderately-sized datasets due to the hardware restrictions imposed by exisitng QA devices, which make it difficult to solve real-world problems at scale. In this work we develop novel strategies to circumvent this issue. We discuss scale-up techniques such as enhanced embedding and partitioned RBMs which allow large commercial datasets to be learned using QA hardware. We present our initial results obtained by training an RBM as an autoencoder on an image dataset. The results obtained so far indicate that the convergence rates can be improved significantly by increasing RBM network connectivity. These ideas can be readily applied to generalized Boltzmann machines and we are currently investigating this in an ongoing project.

  7. Partitioned learning of deep Boltzmann machines for SNP data. (United States)

    Hess, Moritz; Lenz, Stefan; Blätte, Tamara J; Bullinger, Lars; Binder, Harald


    Learning the joint distributions of measurements, and in particular identification of an appropriate low-dimensional manifold, has been found to be a powerful ingredient of deep leaning approaches. Yet, such approaches have hardly been applied to single nucleotide polymorphism (SNP) data, probably due to the high number of features typically exceeding the number of studied individuals. After a brief overview of how deep Boltzmann machines (DBMs), a deep learning approach, can be adapted to SNP data in principle, we specifically present a way to alleviate the dimensionality problem by partitioned learning. We propose a sparse regression approach to coarsely screen the joint distribution of SNPs, followed by training several DBMs on SNP partitions that were identified by the screening. Aggregate features representing SNP patterns and the corresponding SNPs are extracted from the DBMs by a combination of statistical tests and sparse regression. In simulated case-control data, we show how this can uncover complex SNP patterns and augment results from univariate approaches, while maintaining type 1 error control. Time-to-event endpoints are considered in an application with acute myeloid leukemia patients, where SNP patterns are modeled after a pre-screening based on gene expression data. The proposed approach identified three SNPs that seem to jointly influence survival in a validation dataset. This indicates the added value of jointly investigating SNPs compared to standard univariate analyses and makes partitioned learning of DBMs an interesting complementary approach when analyzing SNP data. A Julia package is provided at ''. Supplementary data are available at Bioinformatics online.

  8. Derivation of stable Burnett equations for rarefied gas flows. (United States)

    Singh, Narendra; Jadhav, Ravi Sudam; Agrawal, Amit


    A set of constitutive relations for the stress tensor and heat flux vector for the hydrodynamic description of rarefied gas flows is derived in this work. A phase density function consistent with Onsager's reciprocity principle and H theorem is utilized to capture nonequilibrium thermodynamics effects. The phase density function satisfies the linearized Boltzmann equation and the collision invariance property. Our formulation provides the correct value of the Prandtl number as it involves two different relaxation times for momentum and energy transport by diffusion. Generalized three-dimensional constitutive equations for different kinds of molecules are derived using the phase density function. The derived constitutive equations involve cross single derivatives of field variables such as temperature and velocity, with no higher-order derivative in higher-order terms. This is remarkable feature of the equations as the number of boundary conditions required is the same as needed for conventional Navier-Stokes equations. Linear stability analysis of the equations is performed, which shows that the derived equations are unconditionally stable. A comparison of the derived equations with existing Burnett-type equations is presented and salient features of our equations are outlined. The classic internal flow problem, force-driven compressible plane Poiseuille flow, is chosen to verify the stable Burnett equations and the results for equilibrium variables are presented.

  9. Collisional electron broadening of Stark sublevels of an atom of hydrogen in a plasma

    Energy Technology Data Exchange (ETDEWEB)

    Sholin, G.V.; Demura, A.V.; Lisitsa, V.S.


    A theoretical examination was made of the contours of the hydrogen lines in plasma, excited as a result of the static interaction of ions and collisional electrons with consideration of the special characteristics connected with the presence of supplementary degeneration in the Coulomb field. General expressions were obtained for the half-width of the Stark component of every line of the hydrogen spectrum in parabolic coordinates. A subsequent analysis was carried out of the transition from the overlapping of the Stark component to the isolated lines. It was found that such a transition is accompanied by annulment of the nondiagornl matrix elements of the operator of the collisional electron broadening. The form of the interference member in the operator of the collisional electron broadening is corrected with consideration of the perturbation of the upper and lower levels. The calculations carried out made it possible to improve the agreement between theory and experiment for lines with strong central components. (tr-auth)

  10. Cancellation of Collisional Frequency Shifts in Optical Lattice Clocks with Rabi Spectroscopy

    CERN Document Server

    Lee, Sangkyung; Lee, Won-Kyu; Yu, Dai-Hyuk


    We analyze both the s- and p-wave collision induced frequency shifts and propose a over-$\\pi$ pulse scheme to cancel the shifts in optical lattice clocks interrogated by a Rabi pulse. The collisional frequency shifts are analytically solved as a function of the pulse area and the inhomogeneity of the Rabi frequencies. Experimentally measured collisional frequency shifts in an Yb optical lattice clock are in good agreement with the analytical calculations. Based on our analysis, the over-$\\pi$ pulse combined with a small inhomogeneity below 0.1 allows a fractional uncertainty on a level of $10^{-18}$ in both Sr and Yb optical lattice clocks by canceling the collisional frequency shift. \\end{abstract}

  11. Cancellation of collisional frequency shifts in optical lattice clocks with Rabi spectroscopy (United States)

    Lee, Sangkyung; Park, Chang Yong; Lee, Won-Kyu; Yu, Dai-Hyuk


    We analyze both the s- and p-wave collision induced frequency shifts and propose an over-π pulse scheme to cancel the shifts in optical lattice clocks interrogated by a Rabi pulse. The collisional frequency shifts are analytically solved as a function of the pulse area and the inhomogeneity of the Rabi frequencies. Experimentally measured collisional frequency shifts in an Yb optical lattice clock are in good agreement with the analytical calculations. Based on our analysis, the over-π pulse combined with a small inhomogeneity below 0.1 allows a fractional uncertainty on a level of 10-18 in both Sr and Yb optical lattice clocks by canceling the collisional frequency shift.

  12. Collisional Scaling of the Energy Transfer in Drift-Wave Zonal Flow Turbulence. (United States)

    Schmid, B; Manz, P; Ramisch, M; Stroth, U


    The collisionality scaling of density and potential coupling together with zonal flow energy transfer and spectral power is investigated at the stellarator experiment TJ-K. With a poloidal probe array, consisting of 128 Langmuir probes, density and potential fluctuations are measured on four neighboring flux surfaces simultaneously over the complete poloidal circumference. By analyzing Reynolds stress and pseudo-Reynolds stress, it is found that, for increasing collisionality, the coupling between density and potential decreases which hinders the zonal flow drive. Also, as a consequence, the nonlinear energy transfer, as well as the zonal flow contribution to the complete turbulent spectrum, decreases the same way. This is in line with theoretical expectations and is a first experimental verification of the importance of collisionality for large-scale structure formation in magnetically confined toroidal plasmas.

  13. Construction of non-Markovian coarse-grained models employing the Mori-Zwanzig formalism and iterative Boltzmann inversion. (United States)

    Yoshimoto, Yuta; Li, Zhen; Kinefuchi, Ikuya; Karniadakis, George Em


    We propose a new coarse-grained (CG) molecular simulation technique based on the Mori-Zwanzig (MZ) formalism along with the iterative Boltzmann inversion (IBI). Non-Markovian dissipative particle dynamics (NMDPD) taking into account memory effects is derived in a pairwise interaction form from the MZ-guided generalized Langevin equation. It is based on the introduction of auxiliary variables that allow for the replacement of a non-Markovian equation with a Markovian one in a higher dimensional space. We demonstrate that the NMDPD model exploiting MZ-guided memory kernels can successfully reproduce the dynamic properties such as the mean square displacement and velocity autocorrelation function of a Lennard-Jones system, as long as the memory kernels are appropriately evaluated based on the Volterra integral equation using the force-velocity and velocity-velocity correlations. Furthermore, we find that the IBI correction of a pair CG potential significantly improves the representation of static properties characterized by a radial distribution function and pressure, while it has little influence on the dynamic processes. Our findings suggest that combining the advantages of both the MZ formalism and IBI leads to an accurate representation of both the static and dynamic properties of microscopic systems that exhibit non-Markovian behavior.

  14. Determination of collisional quenching rate coefficients of metastable nitrogen molecules by air pollutants (United States)

    Suzuki, Susumu; Itoh, Haruo


    It has already been investigated on the determination of the collisional quenching rate coefficients of the metastable nitrogen molecules N2(A^3σu^+ ) by some air pollutants [1] in our laboratory. In this report, we present the result on the collisional quenching rate coefficient of N2(A^3σu^+ ) by formaldehyde (CH2O) using a theoretical procedure that takes into account the reflection of metastables at the boundary. As far as we know, this report is the first result of the collisional quenching rate coefficients of N2(A^3σu^+ ) by CH2O. Formaldehyde is a colorless gas with the foul odor, and elements of the adhesive, paints, and preservative, etc. It is widely used for construction materials such as houses, because it is low cost. It is released from paint of construction materials in air, and, in that case, it is known as one of the causative agents of so-called ``Sick building syndrome'' to influence the human body harmfully even if it is a low concentration. The obtained collisional quenching rate coefficient of N2(A^3σu^+ ) by CH2O is (4.7±0.4) x 10-12 cm^3/s. Because the collisional quenching rate coefficient by CH2O is large, it is understood that CH2O receives energy easily from N2(A^3σu^+ ). In addition, we reports on the obtained collisional quenching rate coefficient of N2(A^3σu^+ ) by some air pollutants. [1] S. Suzuki, T.Suzuki and H.Itoh: Proc. of XXVIII ICPIG (Prague, Czech Republic), (2007) 1P01-40.

  15. Collisional transport across the magnetic field in drift-fluid models

    DEFF Research Database (Denmark)

    Madsen, Jens; Naulin, Volker; Nielsen, Anders Henry


    altering the drift-fluid energy integral. We demonstrate that the inclusion of collisional transport in drift-fluid models gives rise to diffusion of particle density, momentum, and pressures in drift-fluid turbulence models and, thereby, obviates the customary use of artificial diffusion in turbulence......Drift ordered fluid models are widely applied in studies of low-frequency turbulence in the edge and scrape-off layer regions of magnetically confined plasmas. Here, we show how collisional transport across the magnetic field is self-consistently incorporated into drift-fluid models without...

  16. Exponentially long Equilibration times in a 1-D Collisional Model of a classical gas

    DEFF Research Database (Denmark)

    Hjorth, Poul; Benettin, G.


    separation between the time scale for the vibration and the time scale associated with a typical binary collision in the gas. We consider here a simple 1-D model, and show how, when these time scales are well separated, the collisional dynamics is constrained by a many-particle adiabatic invariant....... The effect is that the collisional energy exchanges between the translational and the vibrational degrees of freedom are slowed down by an exponential factor (as Jeans conjectured). A metastable situation thus occurs, in which the fast vibrational degrees of freedom effectivly do not contribute...... the time scale for the evolution to statistical equilibrium. The theoretical analysis is supported by numerical examples....

  17. Integral equations

    CERN Document Server

    Tricomi, Francesco Giacomo


    This classic text on integral equations by the late Professor F. G. Tricomi, of the Mathematics Faculty of the University of Turin, Italy, presents an authoritative, well-written treatment of the subject at the graduate or advanced undergraduate level. To render the book accessible to as wide an audience as possible, the author has kept the mathematical knowledge required on the part of the reader to a minimum; a solid foundation in differential and integral calculus, together with some knowledge of the theory of functions is sufficient. The book is divided into four chapters, with two useful

  18. Langevin equation vs. kinetic equation. Sub diffusive behavior of charged particles in a stochastic magnetic field

    Energy Technology Data Exchange (ETDEWEB)

    Balescu, R.; Hai-Da Wang [Universite Libre de Bruxelles (Belgium); Misguich, J.H. [Association Euratom-CEA, Centre d`Etudes Nucleaires de Cadarache, 13 - Saint-Paul-lez-Durance (France). Dept. de Recherches sur la Fusion Controlee


    The running diffusion coefficient D(t) is evaluated for a system of charged particles undergoing the effect of a fluctuating magnetic field and of their mutual collisions. The latter coefficient can be expressed either in terms of the mean square displacement (MSD) of a test particle, or in terms of a correlation between a fluctuating distribution function and the magnetic field fluctuation. Using suitable simplifications, both problems are amenable to exact analytic solution. The conclusion is that the equivalence of the two approaches is by no means automatically guaranteed. A new type of object, the `hybrid kinetic equation` is constructed: it automatically ensures the equivalence with the Langevin results. A linear approximation to the hybrid kinetic equation yields an inexact behaviour, but represents an acceptable approximation in the strongly collisional limit. (author). 34 refs., 2 figs.

  19. Predictions of Thrombus Formation Using Lattice Boltzmann Method (Modeling of Adhesion Force for Particles to Wall)

    National Research Council Canada - National Science Library

    TAMAGAWA, Masaaki; MATSUO, Sumiaki


    ...) with Lattice Boltzmann Method (LBM), applying to orifice-pipe blood flow and flow around a cylinder, which is simple model of turbulent shear stress in the high speed rotary blood pumps and complicated geometry of medical fluid machines...

  20. Stochastic partial differential equations

    CERN Document Server

    Chow, Pao-Liu


    Preliminaries Introduction Some Examples Brownian Motions and Martingales Stochastic Integrals Stochastic Differential Equations of Itô Type Lévy Processes and Stochastic IntegralsStochastic Differential Equations of Lévy Type Comments Scalar Equations of First Order Introduction Generalized Itô's Formula Linear Stochastic Equations Quasilinear Equations General Remarks Stochastic Parabolic Equations Introduction Preliminaries Solution of Stochastic Heat EquationLinear Equations with Additive Noise Some Regularity Properties Stochastic Reaction-Diffusion Equations Parabolic Equations with Grad

  1. Lattice Boltzmann phase-field modeling of thermocapillary flows in a confined microchannel (United States)

    Liu, Haihu; Valocchi, Albert J.; Zhang, Yonghao; Kang, Qinjun


    To understand how thermocapillary forces manipulate the droplet motion in a confined microchannel, a lattice Boltzmann phase-field model is developed to simulate immiscible thermocapillary flows with consideration of fluid-surface interactions. Based on our recent work of Liu et al., 2013 [54], an interfacial force of potential form is proposed to model the interfacial tension force and the Marangoni stress. As only the first-order derivatives are involved, the proposed interfacial force is easily combined with the wetting boundary condition to account for fluid-surface interactions. The hydrodynamic equations are solved using a multiple-relaxation-time algorithm with the interfacial force treated as a forcing term, while an additional convection-diffusion equation is solved by a passive-scalar approach to obtain the temperature field, which is coupled to the interfacial tension by an equation of state. The model is first validated against analytical solutions for the thermocapillary-driven convection in two superimposed fluids at negligibly small Reynolds and Marangoni numbers. It is then demonstrated to produce the correct equilibrium contact angle for a binary fluid with different viscosities when a constant interfacial tension is taken into account. Finally, we numerically simulate the thermocapillary flows for a microfluidic droplet adhering on a solid wall and subject to a simple shear flow when a laser is applied to locally heat the fluids, and investigate the influence of contact angle and fluid viscosity ratio on the droplet dynamical behavior. The droplet motion can be completely blocked provided that the contact angle exceeds a threshold value, below which the droplet motion successively undergoes four stages: constant velocity, deceleration, acceleration, and approximately constant velocity. When the droplet motion is completely blocked, three steady vortices are clearly visible, and the droplet is fully filled by two counter-rotating vortices with the

  2. Mechanistic slumber vs. statistical insomnia: the early history of Boltzmann's H-theorem (1868-1877) (United States)

    Badino, M.


    An intricate, long, and occasionally heated debate surrounds Boltzmann's H-theorem (1872) and his combinatorial interpretation of the second law (1877). After almost a century of devoted and knowledgeable scholarship, there is still no agreement as to whether Boltzmann changed his view of the second law after Loschmidt's 1876 reversibility argument or whether he had already been holding a probabilistic conception for some years at that point. In this paper, I argue that there was no abrupt statistical turn. In the first part, I discuss the development of Boltzmann's research from 1868 to the formulation of the H-theorem. This reconstruction shows that Boltzmann adopted a pluralistic strategy based on the interplay between a kinetic and a combinatorial approach. Moreover, it shows that the extensive use of asymptotic conditions allowed Boltzmann to bracket the problem of exceptions. In the second part I suggest that both Loschmidt's challenge and Boltzmann's response to it did not concern the H-theorem. The close relation between the theorem and the reversibility argument is a consequence of later investigations on the subject.

  3. Numerical Comparison of Solutions of Kinetic Model Equations

    Directory of Open Access Journals (Sweden)

    A. A. Frolova


    Full Text Available The collision integral approximation by different model equations has created a whole new trend in the theory of rarefied gas. One widely used model is the Shakhov model (S-model obtained by expansion of inverse collisions integral in a series of Hermite polynomials up to the third order. Using the same expansion with another value of free parameters leads to a linearized ellipsoidal statistical model (ESL.Both model equations (S and ESL have the same properties, as they give the correct relaxation of non-equilibrium stress tensor components and heat flux vector, the correct Prandtl number at the transition to the hydrodynamic regime and do not guarantee the positivity of the distribution function.The article presents numerical comparison of solutions of Shakhov equation, ESL- model and full Boltzmann equation in the four Riemann problems for molecules of hard spheres.We have considered the expansion of two gas flows, contact discontinuity, the problem of the gas counter-flows and the problem of the shock wave structure. For the numerical solution of the kinetic equations the method of discrete ordinates is used.The comparison shows that solution has a weak sensitivity to the form of collision operator in the problem of expansions of two gas flows and results obtained by the model and the kinetic Boltzmann equations coincide.In the problem of the contact discontinuity the solution of model equations differs from full kinetic solutions at the point of the initial discontinuity. The non-equilibrium stress tensor has the maximum errors, the error of the heat flux is much smaller, and the ESL - model gives the exact value of the extremum of heat flux.In the problems of gas counter-flows and shock wave structure the model equations give significant distortion profiles of heat flux and non-equilibrium stress tensor components in front of the shock waves. This behavior is due to fact that in the models under consideration there is no dependency of the

  4. Dynamo transformation of the collisional R-T in a weakly ionized ...

    Indian Academy of Sciences (India)

    where the interstellar neutrals undergo charge exchange collisions with ions in the solar wind [1]. The presence of neutrals in plasmas can drive new instability in many ways. For ... to be based on the dynamo principle of a.c. generator. However, a pertinent question remains to be addressed: what happens to the collisional ...

  5. Poloidal potential in the low-collisionality regime in a nonaxisymmetric torus

    Energy Technology Data Exchange (ETDEWEB)

    Hastings, D.E.; Tolliver, J.S.


    The poloidal potential is calculated numerically in the low-collisionality regime for nonaxisymmetric tori such as stellarators and bumpy tori. It is found that even fairly deep into the superbanana regime, the poloidal potential retains the simple azimuthal dependence of the plateau regime. 12 refs., 8 figs.

  6. Collisional effect on the Weibel instability in the limit of high plasma ...

    Indian Academy of Sciences (India)

    Abstract. The Weibel instability (WI) of relativistic electron beam (REB) penetrating an infinite collisional plasma was studied in the following models: (i) REB model, where the total equilibrium distribution function f0(p ) is approximated by nonrelativistic back- ground electron and REB distribution functions and (ii) relativistic ...

  7. Comparison of continuous and discontinuous collisional bumpers: Dimensionally scaled impact experiments into single wire meshes (United States)

    Hoerz, Friedrich; Cintala, Mark; See, Thomas; Bernhard, Ronald; Cardenas, Frank; Davidson, William; Haynes, Jerry


    An experimental inquiry into the utility of discontinuous bumpers was conducted to investigate the collisional outcomes of impacts into single grid-like targets and to compare the results with more traditional bumper designs that employ continuous sheet stock. We performed some 35 experiments using 6.3 and 3.2 mm diameter spherical soda-lime glass projectiles at low velocities (less than 2.5 km/s) and 13 at velocities between 5 and 6 km/s, using 3.2 mm spheres only. The thrust of the experiments related to the characterization of collisional fragments as a function of target thickness or areal shield mass of both bumper designs. The primary product of these experiments was witness plates that record the resulting population of collisional fragments. Substantial interpretive and predictive insights into bumper performance were obtained. All qualitative observations (on the witness plates) and detailed measurements of displaced masses seem simply and consistently related only to bumper mass available for interaction with the impactor. This renders the grid bumper into the superior shield design. These findings present evidence that discontinuous bumpers are a viable concept for collisional shields, possibly superior to continuous geometries.

  8. Dependence of intermittent density fluctuations on collisionality in TJ-K

    Energy Technology Data Exchange (ETDEWEB)

    Reuther, Kyle; Garland, Stephen; Ramisch, Mirko [Institut fuer Grenzflaechenverfahrenstechnikund Plasmatechnologie, Universitaet Stuttgart (Germany); Manz, Peter [Physik-Department E28, Technische Universitaet Muenchen, Garching (Germany)


    Particle and heat transport losses due to edge turbulence are well known phenomena commonly seen in toroidal magnetic confinement devices. Furthermore in the scrape-off layer (SOL), turbulent density fluctuations are often observed to be intermittent and dominate particle transport to the vessel walls. In the adiabatic limit (small collisionality), of the two-field Hasegawa-Wakatani model, simulated turbulent density fluctuations are observed to couple to potential fluctuations and exhibit Gaussian behavior. However, in the hydrodynamic limit (large collisionality) the density and potential decouple. As a result, the density becomes passively advected, evolves towards the vorticity, and exhibits intermittent behavior. The relationship between collisionality and intermittency is investigated experimentally at the stellarator TJ-K. To vary the plasma collisionality, which is related to electron density and temperature, parameters such as gas type, neutral gas pressure, magnetic field, and heating power are varied. Radial profiles of plasma density, temperature, floating potential, and vorticity are recorded via a scanning 7-tip Langmuir probe array. First results are presented.

  9. Hydraulic fracture conductivity: effects of rod-shaped proppant from lattice-Boltzmann simulations and lab tests (United States)

    Osiptsov, Andrei A.


    The goal of this study is to evaluate the conductivity of random close packings of non-spherical, rod-shaped proppant particles under the closure stress using numerical simulation and lab tests, with application to the conductivity of hydraulic fractures created in subterranean formation to stimulate production from oil and gas reservoirs. Numerical simulations of a steady viscous flow through proppant packs are carried out using the lattice Boltzmann method for the Darcy flow regime. The particle packings were generated numerically using the sequential deposition method. The simulations are conducted for packings of spheres, ellipsoids, cylinders, and mixtures of spheres with cylinders at various volumetric concentrations. It is demonstrated that cylinders provide the highest permeability among the proppants studied. The dependence of the nondimensional permeability (scaled by the equivalent particle radius squared) on porosity obtained numerically is well approximated by the power-law function: K /Rv2 = 0.204ϕ4.58 in a wide range of porosity: 0.3 ≤ ϕ ≤ 0.7. Lattice-Boltzmann simulations are cross-verified against finite-volume simulations using Navier-Stokes equations for inertial flow regime. Correlations for the normalized beta-factor as a function of porosity and normalized permeability are presented as well. These formulae are in a good agreement with the experimental measurements (including packings of rod-shaped particles) and existing laboratory data, available in the porosity range 0.3 ≤ ϕ ≤ 0.5. Comparison with correlations by other authors is also given.

  10. An Explicit Formulation of Singularity-Free Dynamic Equations of Mechanical Systems in Lagrangian Form---Part Two: Multibody Systems

    Directory of Open Access Journals (Sweden)

    Pål Johan From


    Full Text Available This paper presents the explicit dynamic equations of multibody mechanical systems. This is the second paper on this topic. In the first paper the dynamics of a single rigid body from the Boltzmann--Hamel equations were derived. In this paper these results are extended to also include multibody systems. We show that when quasi-velocities are used, the part of the dynamic equations that appear from the partial derivatives of the system kinematics are identical to the single rigid body case, but in addition we get terms that come from the partial derivatives of the inertia matrix, which are not present in the single rigid body case. We present for the first time the complete and correct derivation of multibody systems based on the Boltzmann--Hamel formulation of the dynamics in Lagrangian form where local position and velocity variables are used in the derivation to obtain the singularity-free dynamic equations. The final equations are written in global variables for both position and velocity. The main motivation of these papers is to allow practitioners not familiar with differential geometry to implement the dynamic equations of rigid bodies without the presence of singularities. Presenting the explicit dynamic equations also allows for more insight into the dynamic structure of the system. Another motivation is to correct some errors commonly found in the literature. Unfortunately, the formulation of the Boltzmann-Hamel equations used here are presented incorrectly. This has been corrected by the authors, but we present here, for the first time, the detailed mathematical details on how to arrive at the correct equations. We also show through examples that using the equations presented here, the dynamics of a single rigid body is reduced to the standard equations on a Lagrangian form, for example Euler's equations for rotational motion and Euler--Lagrange equations for free motion.

  11. A correlation for the collector efficiency of Brownian particles in clean-bed filtration in sphere packings by a Lattice-Boltzmann method. (United States)

    Long, Wei; Hilpert, Markus


    In this paper, we develop a new correlation for the clean-bed filter coefficient (lambda0) for Brownian particles, for which diffusion is the main deposition mechanism. The correlation is based on numerical Lattice-Boltzmann (LB) simulations in random packings of spheres of uniform diameter. We use LB methods to solve the Navier-Stokes equation for flow and then the advection-diffusion equation for particle transport. We determine a correlation for an "equivalent" single-collector diffusion efficiency, etaD, so that we can compare our predictions to "true" single-collector correlations stemming from unit-cell modeling approaches. We compared our new correlation to experiments on the filtration of latex particles. For small particle diameters, 50 nm etaG and nI terms from unit-cell correlations to our etaD model. The resulting correlation predicts experiments with latex particles of dp > 300 nm well.

  12. Flow visualisation of downhill skiers using the lattice Boltzmann method (United States)

    Asai, Takeshi; Hong, Sungchan; Ijuin, Koichi


    In downhill alpine skiing, skiers often exceed speeds of 120 km h-1, with air resistance substantially affecting the overall race times. To date, studies on air resistance in alpine skiing have used wind tunnels and actual skiers to examine the relationship between the gliding posture and magnitude of drag and for the design of skiing equipment. However, these studies have not revealed the flow velocity distribution and vortex structure around the skier. In the present study, computational fluid dynamics are employed with the lattice Boltzmann method to derive the relationship between total drag and the flow velocity around a downhill skier in the full-tuck position. Furthermore, the flow around the downhill skier is visualised, and its vortex structure is examined. The results show that the total drag force in the downhill skier model is 27.0 N at a flow velocity of 15 m s-1, increasing to 185.8 N at 40 m s-1. From analysis of the drag distribution and the flow profile, the head, upper arms, lower legs, and thighs (including buttocks) are identified as the major sources of drag on a downhill skier. Based on these results, the design of suits and equipment for reducing the drag from each location should be the focus of research and development in ski equipment. This paper describes a pilot study that introduces undergraduate students of physics or engineering into this research field. The results of this study are easy to understand for undergraduate students.

  13. A Boltzmann Constant Determination Based on Johnson Noise Thermometry. (United States)

    Flowers-Jacobs, N E; Pollarolo, A; Coakley, K J; Fox, A E; Rogalla, H; Tew, W L; Benz, S P


    A value for the Boltzmann constant was measured electronically using an improved version of the Johnson Noise Thermometry (JNT) system at the National Institute of Standards and Technology (NIST), USA. This system is different from prior ones, including those from the 2011 determination at NIST and both 2015 and 2017 determinations at the National Institute of Metrology (NIM), China. As in all three previous determinations, the main contribution to the combined uncertainty is the statistical uncertainty in the noise measurement, which is mitigated by accumulating and integrating many weeks of cross-correlated measured data. The second major uncertainty contribution also still results from variations in the frequency response of the ratio of the measured spectral noise of the two noise sources, the sense resistor at the triple-point of water and the superconducting quantum voltage noise source. In this paper, we briefly describe the major differences between our JNT system and previous systems, in particular the input circuit and approach we used to match the frequency responses of the two noise sources. After analyzing and integrating 49 days of accumulated data, we determined a value: k = 1.380 642 9(69)×10-23 J/K with a relative standard uncertainty of 5.0×10-6 and relative offset -4.05×10-6 from the CODATA 2014 recommended value.

  14. Recursive regularization step for high-order lattice Boltzmann methods (United States)

    Coreixas, Christophe; Wissocq, Gauthier; Puigt, Guillaume; Boussuge, Jean-François; Sagaut, Pierre


    A lattice Boltzmann method (LBM) with enhanced stability and accuracy is presented for various Hermite tensor-based lattice structures. The collision operator relies on a regularization step, which is here improved through a recursive computation of nonequilibrium Hermite polynomial coefficients. In addition to the reduced computational cost of this procedure with respect to the standard one, the recursive step allows to considerably enhance the stability and accuracy of the numerical scheme by properly filtering out second- (and higher-) order nonhydrodynamic contributions in under-resolved conditions. This is first shown in the isothermal case where the simulation of the doubly periodic shear layer is performed with a Reynolds number ranging from 104 to 106, and where a thorough analysis of the case at Re=3 ×104 is conducted. In the latter, results obtained using both regularization steps are compared against the Bhatnagar-Gross-Krook LBM for standard (D2Q9) and high-order (D2V17 and D2V37) lattice structures, confirming the tremendous increase of stability range of the proposed approach. Further comparisons on thermal and fully compressible flows, using the general extension of this procedure, are then conducted through the numerical simulation of Sod shock tubes with the D2V37 lattice. They confirm the stability increase induced by the recursive approach as compared with the standard one.

  15. Large-scale lattice-Boltzmann simulations over lambda networks (United States)

    Saksena, R.; Coveney, P. V.; Pinning, R.; Booth, S.

    Amphiphilic molecules are of immense industrial importance, mainly due to their tendency to align at interfaces in a solution of immiscible species, e.g., oil and water, thereby reducing surface tension. Depending on the concentration of amphiphiles in the solution, they may assemble into a variety of morphologies, such as lamellae, micelles, sponge and cubic bicontinuous structures exhibiting non-trivial rheological properties. The main objective of this work is to study the rheological properties of very large, defect-containing gyroidal systems (of up to 10243 lattice sites) using the lattice-Boltzmann method. Memory requirements for the simulation of such large lattices exceed that available to us on most supercomputers and so we use MPICH-G2/MPIg to investigate geographically distributed domain decomposition simulations across HPCx in the UK and TeraGrid in the US. Use of MPICH-G2/MPIg requires the port-forwarder to work with the grid middleware on HPCx. Data from the simulations is streamed to a high performance visualisation resource at UCL (London) for rendering and visualisation. Lighting the Blue Touchpaper for UK e-Science - Closing Conference of ESLEA Project March 26-28 2007 The George Hotel, Edinburgh, UK

  16. A Boltzmann constant determination based on Johnson noise thermometry (United States)

    Flowers-Jacobs, N. E.; Pollarolo, A.; Coakley, K. J.; Fox, A. E.; Rogalla, H.; Tew, W. L.; Benz, S. P.


    A value for the Boltzmann constant was measured electronically using an improved version of the Johnson Noise Thermometry (JNT) system at the National Institute of Standards and Technology (NIST), USA. This system is different from prior ones, including those from the 2011 determination at NIST and both 2015 and 2017 determinations at the National Institute of Metrology (NIM), China. As in all three previous determinations, the main contribution to the combined uncertainty is the statistical uncertainty in the noise measurement, which is mitigated by accumulating and integrating many weeks of cross-correlated measured data. The second major uncertainty contribution also still results from variations in the frequency response of the ratio of the measured spectral noise of the two noise sources, the sense resistor at the triple-point of water and the superconducting quantum voltage noise source. In this paper, we briefly describe the major differences between our JNT system and previous systems, in particular the input circuit and approach we used to match the frequency responses of the two noise sources. After analyzing and integrating 50 d of accumulated data, we determined a value: k~=1.380 642 9(69)× {{10}-23} J K-1 with a relative standard uncertainty of 5.0× {{10}-6} and relative offset -4.05× {{10}-6} from the CODATA 2014 recommended value.

  17. Equivalence of restricted Boltzmann machines and tensor network states (United States)

    Chen, Jing; Cheng, Song; Xie, Haidong; Wang, Lei; Xiang, Tao


    The restricted Boltzmann machine (RBM) is one of the fundamental building blocks of deep learning. RBM finds wide applications in dimensional reduction, feature extraction, and recommender systems via modeling the probability distributions of a variety of input data including natural images, speech signals, and customer ratings, etc. We build a bridge between RBM and tensor network states (TNS) widely used in quantum many-body physics research. We devise efficient algorithms to translate an RBM into the commonly used TNS. Conversely, we give sufficient and necessary conditions to determine whether a TNS can be transformed into an RBM of given architectures. Revealing these general and constructive connections can cross fertilize both deep learning and quantum many-body physics. Notably, by exploiting the entanglement entropy bound of TNS, we can rigorously quantify the expressive power of RBM on complex data sets. Insights into TNS and its entanglement capacity can guide the design of more powerful deep learning architectures. On the other hand, RBM can represent quantum many-body states with fewer parameters compared to TNS, which may allow more efficient classical simulations.

  18. Stability and stabilization of the lattice Boltzmann method. (United States)

    Brownlee, R A; Gorban, A N; Levesley, J


    We revisit the classical stability versus accuracy dilemma for the lattice Boltzmann methods (LBM). Our goal is a stable method of second-order accuracy for fluid dynamics based on the lattice Bhatnager-Gross-Krook method (LBGK). The LBGK scheme can be recognized as a discrete dynamical system generated by free flight and entropic involution. In this framework the stability and accuracy analysis are more natural. We find the necessary and sufficient conditions for second-order accurate fluid dynamics modeling. In particular, it is proven that in order to guarantee second-order accuracy the distribution should belong to a distinguished surface--the invariant film (up to second order in the time step). This surface is the trajectory of the (quasi)equilibrium distribution surface under free flight. The main instability mechanisms are identified. The simplest recipes for stabilization add no artificial dissipation (up to second order) and provide second-order accuracy of the method. Two other prescriptions add some artificial dissipation locally and prevent the system from loss of positivity and local blowup. Demonstration of the proposed stable LBGK schemes are provided by the numerical simulation of a one-dimensional (1D) shock tube and the unsteady 2D flow around a square cylinder up to Reynolds number Re approximately 20,000.

  19. Real-time keypoint recognition using restricted Boltzmann machine. (United States)

    Yuan, Miaolong; Tang, Huajin; Li, Haizhou


    Feature point recognition is a key component in many vision-based applications, such as vision-based robot navigation, object recognition and classification, image-based modeling, and augmented reality. Real-time performance and high recognition rates are of crucial importance to these applications. In this brief, we propose a novel method for real-time keypoint recognition using restricted Boltzmann machine (RBM). RBMs are generative models that can learn probability distributions of many different types of data including labeled and unlabeled data sets. Due to the inherent noise of the training data sets, we use an RBM to model statistical distributions of the training data. Furthermore, the learned RBM can be used as a competitive classifier to recognize the keypoints in real-time during the tracking stage, thus making it advantageous to be employed in applications that require real-time performance. Experiments have been conducted under a variety of conditions to demonstrate the effectiveness and generalization of the proposed approach.

  20. Volumetric lattice Boltzmann simulation for blood flow in aorta arteries (United States)

    Deep, Debanjan; Yu, Huidan (Whitney); Teague, Shawn


    Complicated moving boundaries pose a major challenge in computational fluid dynamics for complex flows, especially in the biomechanics of both blood flow in the cardiovascular system and air flow in the respiratory system where the compliant nature of the vessels can have significant effects on the flow rate and wall shear stress. We develop a computation approach to treat arbitrarily moving boundaries using a volumetric representation of lattice Boltzmann method, which distributes fluid particles inside lattice cells. A volumetric bounce-back procedure is applied in the streaming step while momentum exchange between the fluid and moving solid boundary are accounted for in the collision sub-step. Additional boundary-induced migration is introduced to conserve fluid mass as the boundary moves across fluid cells. The volumetric LBM (VLBM) is used to simulate blood flow in both normal and dilated aorta arteries. We first compare flow structure and pressure distribution in steady state with results from Navier-Stokes based solver and good agreements are achieved. Then we focus on wall stress within the aorta for different heart pumping condition and present quantitative measurement of wall shear and normal stress.