Full Boltzmann equations for leptogenesis including scattering
Hahn-Woernle, F; Wong, Y Y Y
2009-01-01
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...
On the full Boltzmann equations for Leptogenesis
Garayoa, J; Pinto, T; Rius, N; Vives, O
2009-01-01
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...
On the full Boltzmann equations for leptogenesis
Energy Technology Data Exchange (ETDEWEB)
Garayoa, J.; Pastor, S.; Pinto, T.; Rius, N.; Vives, O., E-mail: garayoa@ific.uv.es, E-mail: pastor@ific.uv.es, E-mail: teguayco@gmail.com, E-mail: nuria@ific.uv.es, E-mail: vives@ific.uv.es [Depto. de Física Teórica and IFIC, Universidad de Valencia-CSIC, Edificio de Institutos de Paterna, Apt. 22085, 46071 Valencia (Spain)
2009-09-01
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.
A unified gas-kinetic scheme for continuum and rarefied flows IV: Full Boltzmann and model equations
Liu, Chang; Xu, Kun; Sun, Quanhua; Cai, Qingdong
2016-06-01
Fluid dynamic equations are valid in their respective modeling scales, such as the particle mean free path scale of the Boltzmann equation and the hydrodynamic scale of the Navier-Stokes (NS) equations. With a variation of the modeling scales, theoretically there should have a continuous spectrum of fluid dynamic equations. Even though the Boltzmann equation is claimed to be valid in all scales, many Boltzmann solvers, including direct simulation Monte Carlo method, require the cell resolution to the order of particle mean free path scale. Therefore, they are still single scale methods. In order to study multiscale flow evolution efficiently, the dynamics in the computational fluid has to be changed with the scales. A direct modeling of flow physics with a changeable scale may become an appropriate approach. The unified gas-kinetic scheme (UGKS) is a direct modeling method in the mesh size scale, and its underlying flow physics depends on the resolution of the cell size relative to the particle mean free path. The cell size of UGKS is not limited by the particle mean free path. With the variation of the ratio between the numerical cell size and local particle mean free path, the UGKS recovers the flow dynamics from the particle transport and collision in the kinetic scale to the wave propagation in the hydrodynamic scale. The previous UGKS is mostly constructed from the evolution solution of kinetic model equations. Even though the UGKS is very accurate and effective in the low transition and continuum flow regimes with the time step being much larger than the particle mean free time, it still has space to develop more accurate flow solver in the region, where the time step is comparable with the local particle mean free time. In such a scale, there is dynamic difference from the full Boltzmann collision term and the model equations. This work is about the further development of the UGKS with the implementation of the full Boltzmann collision term in the region
Ringe, Stefan; Oberhofer, Harald; Hille, Christoph; Matera, Sebastian; Reuter, Karsten
2016-08-01
The size-modified Poisson-Boltzmann (MPB) equation is an efficient implicit solvation model which also captures electrolytic solvent effects. It combines an account of the dielectric solvent response with a mean-field description of solvated finite-sized ions. We present a general solution scheme for the MPB equation based on a fast function-space-oriented Newton method and a Green's function preconditioned iterative linear solver. In contrast to popular multigrid solvers, this approach allows us to fully exploit specialized integration grids and optimized integration schemes. We describe a corresponding numerically efficient implementation for the full-potential density-functional theory (DFT) code FHI-aims. We show that together with an additional Stern layer correction the DFT+MPB approach can describe the mean activity coefficient of a KCl aqueous solution over a wide range of concentrations. The high sensitivity of the calculated activity coefficient on the employed ionic parameters thereby suggests to use extensively tabulated experimental activity coefficients of salt solutions for a systematic parametrization protocol.
Lattice Boltzmann solver of Rossler equation
Institute of Scientific and Technical Information of China (English)
GuangwuYAN; LiRUAN
2000-01-01
We proposed a lattice Boltzmann model for the Rossler equation. Using a method of multiscales in the lattice Boltzmann model, we get the diffusion reaction as a special case. If the diffusion effect disappeared, we can obtain the lattice Boltzmann solution of the Rossler equation on the mesescopic scale. The numerical results show the method can be used to simulate Rossler equation.
Quantum corrections for Boltzmann equation
Institute of Scientific and Technical Information of China (English)
M.; Levy; PETER
2008-01-01
We present the lowest order quantum correction to the semiclassical Boltzmann distribution function,and the equation satisfied by this correction is given. Our equation for the quantum correction is obtained from the conventional quantum Boltzmann equation by explicitly expressing the Planck constant in the gradient approximation,and the quantum Wigner distribution function is expanded in pow-ers of Planck constant,too. The negative quantum correlation in the Wigner dis-tribution function which is just the quantum correction terms is naturally singled out,thus obviating the need for the Husimi’s coarse grain averaging that is usually done to remove the negative quantum part of the Wigner distribution function. We also discuss the classical limit of quantum thermodynamic entropy in the above framework.
Hot electrons in superlattices: quantum transport versus Boltzmann equation
DEFF Research Database (Denmark)
Wacker, Andreas; Jauho, Antti-Pekka; Rott, S.;
1999-01-01
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...
Lattice Boltzmann equation for relativistic quantum mechanics.
Succi, Sauro
2002-03-15
Relativistic versions of the quantum lattice Boltzmann equation are discussed. It is shown that the inclusion of nonlinear interactions requires the standard collision operator to be replaced by a pair of dynamic fields coupling to the relativistic wave function in a way which can be described by a multicomponent complex lattice Boltzmann equation.
Comparison of Boltzmann Equations with Quantum Dynamics for Scalar Fields
Lindner, Manfred; Lindner, Manfred; Muller, Markus Michael
2006-01-01
Boltzmann equations are often used to study the thermal evolution of particle reaction networks. Prominent examples are the computation of the baryon asymmetry of the universe and the evolution of the quark-gluon plasma after relativistic heavy ion collisions. However, Boltzmann equations are only a classical approximation of the quantum thermalization process which is described by the so-called Kadanoff-Baym equations. This raises the question how reliable Boltzmann equations are as approximations to the full Kadanoff-Baym equations. Therefore, we present in this paper a detailed comparison between the Kadanoff-Baym and Boltzmann equations in the framework of a scalar Phi^4 quantum field theory in 3+1 space-time dimensions. The obtained numerical solutions reveal significant discrepancies in the results predicted by both types of equations. Most notably, apart from quantitative discrepancies, on a qualitative level the universality observed for the Kadanoff-Baym equations is severely restricted in the case o...
Kinetic Boltzmann, Vlasov and Related Equations
Sinitsyn, Alexander; Vedenyapin, Victor
2011-01-01
Boltzmann and Vlasov equations played a great role in the past and still play an important role in modern natural sciences, technique and even philosophy of science. Classical Boltzmann equation derived in 1872 became a cornerstone for the molecular-kinetic theory, the second law of thermodynamics (increasing entropy) and derivation of the basic hydrodynamic equations. After modifications, the fields and numbers of its applications have increased to include diluted gas, radiation, neutral particles transportation, atmosphere optics and nuclear reactor modelling. Vlasov equation was obtained in
An introduction to the theory of the Boltzmann equation
Harris, Stewart
2011-01-01
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
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)
2015-05-06
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.
The Non-Classical Boltzmann Equation, and Diffusion-Based Approximations to the Boltzmann Equation
Frank, Martin; Larsen, Edward W; Vasques, Richard
2014-01-01
We show that several diffusion-based approximations (classical diffusion or SP1, SP2, SP3) to the linear Boltzmann equation can (for an infinite, homogeneous medium) be represented exactly by a non-classical transport equation. As a consequence, we indicate a method to solve diffusion-based approximations to the Boltzmann equation via Monte Carlo, with only statistical errors - no truncation errors.
Raines, Alla
2015-01-01
Numerical solution of non-steady problems of supersonic inflow of a binary mixture of a rarefied gas on a normally posed wall with mirror and diffuse reflection laws is obtained on the basis of the kinetic Boltzmann equation for the model of hard sphere molecules. For calculation of collision integrals we apply the projection method, developed by Tcheremissine for a one-component gas and generalized by the author for a binary gas mixture in the case of cylindrical symmetry. We demonstrate a good qualitative agreement of our results with other authors for one-component gases.
Entropic lattice Boltzmann model for Burgers's equation.
Boghosian, Bruce M; Love, Peter; Yepez, Jeffrey
2004-08-15
Entropic lattice Boltzmann models are discrete-velocity models of hydrodynamics that possess a Lyapunov function. This feature makes them useful as nonlinearly stable numerical methods for integrating hydrodynamic equations. Over the last few years, such models have been successfully developed for the Navier-Stokes equations in two and three dimensions, and have been proposed as a new category of subgrid model of turbulence. In the present work we develop an entropic lattice Boltzmann model for Burgers's equation in one spatial dimension. In addition to its pedagogical value as a simple example of such a model, our result is actually a very effective way to simulate Burgers's equation in one dimension. At moderate to high values of viscosity, we confirm that it exhibits no trace of instability. At very small values of viscosity, however, we report the existence of oscillations of bounded amplitude in the vicinity of the shock, where gradient scale lengths become comparable with the grid size. As the viscosity decreases, the amplitude at which these oscillations saturate tends to increase. This indicates that, in spite of their nonlinear stability, entropic lattice Boltzmann models may become inaccurate when the ratio of gradient scale length to grid spacing becomes too small. Similar inaccuracies may limit the utility of the entropic lattice Boltzmann paradigm as a subgrid model of Navier-Stokes turbulence.
General relativistic Boltzmann equation, I: Covariant treatment
Debbasch, F.; van Leeuwen, W.A.
2009-01-01
This series of two articles aims at dissipating the rather dense haze existing in the present literature around the General Relativistic Boltzmann equation. In this first article, the general relativistic one-particle distribution function in phase space is defined as an average of delta functions.
THREE WAY DECOMPOSITION FOR THE BOLTZMANN EQUATION
Institute of Scientific and Technical Information of China (English)
Ilgis Ibragimov; Sergej Rjasanow
2009-01-01
The initial value problem for the spatially homogeneous Boltzmann equation is considered. A deterministic numerical scheme for this problem is developed by the use of the three way decomposition of the unknown function as well as of the collision integral. On this way, almost linear complexity of the algorithm is achieved. Some numerical examples are presented.
Consistent lattice Boltzmann equations for phase transitions.
Siebert, D N; Philippi, P C; Mattila, K K
2014-11-01
Unlike conventional computational fluid dynamics methods, the lattice Boltzmann method (LBM) describes the dynamic behavior of fluids in a mesoscopic scale based on discrete forms of kinetic equations. In this scale, complex macroscopic phenomena like the formation and collapse of interfaces can be naturally described as related to source terms incorporated into the kinetic equations. In this context, a novel athermal lattice Boltzmann scheme for the simulation of phase transition is proposed. The continuous kinetic model obtained from the Liouville equation using the mean-field interaction force approach is shown to be consistent with diffuse interface model using the Helmholtz free energy. Density profiles, interface thickness, and surface tension are analytically derived for a plane liquid-vapor interface. A discrete form of the kinetic equation is then obtained by applying the quadrature method based on prescribed abscissas together with a third-order scheme for the discretization of the streaming or advection term in the Boltzmann equation. Spatial derivatives in the source terms are approximated with high-order schemes. The numerical validation of the method is performed by measuring the speed of sound as well as by retrieving the coexistence curve and the interface density profiles. The appearance of spurious currents near the interface is investigated. The simulations are performed with the equations of state of Van der Waals, Redlich-Kwong, Redlich-Kwong-Soave, Peng-Robinson, and Carnahan-Starling.
Thermal creep problems by the discrete Boltzmann equation
Directory of Open Access Journals (Sweden)
L. Preziosi
1991-05-01
Full Text Available This paper deals with an initial-boundary value problem for the discrete Boltzmann equation confined between two moving walls at different temperature. A model suitable for the quantitative analysis of the initial boundary value problem and the relative existence theorem are given.
Boltzmann equations for neutrinos with flavor mixings
Yamada, Shoichi
2000-01-01
With a view of applications to the simulations of supernova explosion and proto neutron star cooling, we derive the Boltzmann equations for the neutrino transport with the flavor mixing based on the real time formalism of the nonequilibrium field theory and the gradient expansion of the Green function. The relativistic kinematics is properly taken into account. The advection terms are derived in the mean field approximation for the neutrino self-energy whiles the collision terms are obtained ...
Fluctuating lattice Boltzmann method for the diffusion equation.
Wagner, Alexander J; Strand, Kyle
2016-09-01
We derive a fluctuating lattice Boltzmann method for the diffusion equation. The derivation removes several shortcomings of previous derivations for fluctuating lattice Boltzmann methods for hydrodynamic systems. The comparative simplicity of this diffusive system highlights the basic features of this first exact derivation of a fluctuating lattice Boltzmann method.
A Fluctuating Lattice Boltzmann Method for the Diffusion Equation
Wagner, Alexander J
2016-01-01
We derive a fluctuating lattice Boltzmann method for the diffusion equation. The derivation removes several shortcomings of previous derivations for fluctuating lattice Boltzmann methods for hydrodynamic systems. The comparative simplicity of this diffusive system highlights the basic features of this first exact derivation of a fluctuating lattice Boltzmann method.
Energy Technology Data Exchange (ETDEWEB)
Stoenescu, M.L.
1977-06-01
The terms in Boltzmann kinetic equation corresponding to elastic short range collisions, inelastic excitational collisions, coulomb interactions and electric field acceleration are evaluated numerically for a standard distribution function minimizing the computational volume by expressing the terms as linear combinations with recalculable coefficients, of the distribution function and its derivatives. The present forms are suitable for spatial distribution calculations.
Non-Boltzmann behavior from the Boltzmann equation
Hagen, M.H.J.; Lowe, C.P.; Frenkel, D.
1995-01-01
We compute the stress autocorrelation function in a two- and three-dimensional system by using the lattice-Boltzmann method. The algebraic long-time behavior ∼t-d/2 in the stress correlation function is clearly observed. The amplitude of this tail is compared with the mode-coupling expression for
Soluble Boltzmann equations for internal state and Maxwell models
Futcher, E.; Hoare, M.R.; Hendriks, E.M.; Ernst, M.H.
1980-01-01
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 Ma
On kinetic Boltzmann equations and related hydrodynamic flows with dry viscosity
Directory of Open Access Journals (Sweden)
Nikolai N. Bogoliubov (Jr.
2007-01-01
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.
Analysis of Jeans instability from Boltzmann equation
Kremer, Gilberto M
2015-01-01
The dynamics of self-gravitating fluids is analyzed within the framework of a collisionless Boltzmann equation in the presence of gravitational fields and Poisson equation. The equilibrium distribution function takes into account the expansion of the Universe and a pressureless fluid in the matter dominated Universe. Without invoking Jeans "swindle" a dispersion relation is obtained by considering small perturbations of the equilibrium values of the distribution function and gravitational potential. The collapse criterion -- which happens in an unstable region where the solution grows exponentially with time -- is determined from the dispersion relation. The collapse criterion in a static Universe occurs when the wavenumber $k$ is smaller than the Jeans wavenumber $k_J$, which was the solution found by Jeans. For an expanding Universe it is shown that this criterion is $k\\leq\\sqrt{7/6}\\,k_J$. As a consequence the ratio of the mass contained in a sphere of diameter equal to the wavelength $\\lambda=2\\pi/k$ to t...
Gevrey Regularity for the Noncutoff Nonlinear Homogeneous Boltzmann Equation with Strong Singularity
Directory of Open Access Journals (Sweden)
Shi-you Lin
2014-01-01
Full Text Available The Cauchy problem of the nonlinear spatially homogeneous Boltzmann equation without angular cutoff is studied. By using analytic techniques, one proves the Gevrey regularity of the C∞ solutions in non-Maxwellian and strong singularity cases.
Full Eulerian lattice Boltzmann model for conjugate heat transfer.
Hu, Yang; Li, Decai; Shu, Shi; Niu, Xiaodong
2015-12-01
In this paper a full Eulerian lattice Boltzmann model is proposed for conjugate heat transfer. A unified governing equation with a source term for the temperature field is derived. By introducing the source term, we prove that the continuity of temperature and its normal flux at the interface is satisfied automatically. The curved interface is assumed to be zigzag lines. All physical quantities are recorded and updated on a Cartesian grid. As a result, any complicated treatment near the interface is avoided, which makes the proposed model suitable to simulate the conjugate heat transfer with complex interfaces efficiently. The present conjugate interface treatment is validated by several steady and unsteady numerical tests, including pure heat conduction, forced convection, and natural convection problems. Both flat and curved interfaces are also involved. The obtained results show good agreement with the analytical and/or finite volume results.
General relativistic Boltzmann equation, II: Manifestly covariant treatment
Debbasch, F.; van Leeuwen, W.A.
2009-01-01
In a preceding article we presented a general relativistic treatment of the derivation of the Boltzmann equation. The four-momenta occurring in this formalism were all on-shell four-momenta, verifying the mass-shell restriction p(2) = m(2)c(2). Due to this restriction, the resulting Boltzmann equati
General relativistic Boltzmann equation, II: Manifestly covariant treatment
Debbasch, F.; van Leeuwen, W.A.
2009-01-01
In a preceding article we presented a general relativistic treatment of the derivation of the Boltzmann equation. The four-momenta occurring in this formalism were all on-shell four-momenta, verifying the mass-shell restriction p(2) = m(2)c(2). Due to this restriction, the resulting Boltzmann
Axisymmetric multiphase lattice Boltzmann method for generic equations of state
Reijers, Sten Arjen; Gelderblom, Hanneke; Toschi, F.
2016-01-01
We present an axisymmetric lattice Boltzmann model based on the Kupershtokh et al. multiphase model that is capable of solving liquid–gas density ratios up to 103. Appropriate source terms are added to the lattice Boltzmann evolution equation to fully recover the axisymmetric multiphase conservation
Thermal equation of state for lattice Boltzmann gases
Institute of Scientific and Technical Information of China (English)
Ran Zheng
2009-01-01
The Galilean invaxiance and the induced thermo-hydrodynamics of the lattice Boltzmann Bhatnagar-Gross-Krook model axe proposed together with their rigorous theoretical background. From the viewpoint of group invariance,recovering the Galilean invariance for the isothermal lattice Boltzmann Bhatnagar-Gross-Krook equation (LBGKE) induces a new natural thermal-dynamical system, which is compatible with the elementary statistical thermodynamics.
Axisymmetric multiphase lattice Boltzmann method for generic equations of state
Reijers, S.A.; Gelderblom, H.; Toschi, F.
2016-01-01
We present an axisymmetric lattice Boltzmann model based on the Kupershtokh et al. multiphase model that is capable of solving liquid–gas density ratios up to 103. Appropriate source terms are added to the lattice Boltzmann evolution equation to fully recover the axisymmetric multiphase conservation
Boltzmann equations for a binary one-dimensional ideal gas.
Boozer, A D
2011-09-01
We consider a time-reversal invariant dynamical model of a binary ideal gas of N molecules in one spatial dimension. By making time-asymmetric assumptions about the behavior of the gas, we derive Boltzmann and anti-Boltzmann equations that describe the evolution of the single-molecule velocity distribution functions for an ensemble of such systems. We show that for a special class of initial states of the ensemble one can obtain an exact expression for the N-molecule velocity distribution function, and we use this expression to rigorously prove that the time-asymmetric assumptions needed to derive the Boltzmann and anti-Boltzmann equations hold in the limit of large N. Our results clarify some subtle issues regarding the origin of the time asymmetry of Boltzmann's H theorem.
Upper Maxwellian bounds for the spatially homogeneous Boltzmann equation
Gamba, I. M.; Panferov, V.; Villani, C.
2007-01-01
For the spatially homogeneous Boltzmann equation with cutoff hard potentials it is shown that solutions remain bounded from above, uniformly in time, by a Maxwellian distribution, provided the initial data have a Maxwellian upper bound. The main technique is based on a comparison principle that uses a certain dissipative property of the linear Boltzmann equation. Implications of the technique to propagation of upper Maxwellian bounds in the spatially-inhomogeneous case are discussed.
THE NON-CUTOFF BOLTZMANN EQUATION WITH POTENTIAL FORCE IN THE WHOLE SPACE
Institute of Scientific and Technical Information of China (English)
Yuanjie LEI
2014-01-01
This paper is concerned with the non-cutoff Boltzmann equation for full-range interactions with potential force in the whole space. We establish the global existence and optimal temporal convergence rates of classical solutions to the Cauchy problem when initial data is a small perturbation of the stationary solution. The analysis is based on the time-weighted energy method building also upon the recent studies of the non-cutoff Boltzmann equation in [1-3, 15] and the non-cutoff Vlasov-Poisson-Boltzmann system [6].
Analysis of spectral methods for the homogeneous Boltzmann equation
Filbet, Francis
2011-04-01
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.
Metamaterial characterization using Boltzmann's kinetic equation for electrons
DEFF Research Database (Denmark)
Novitsky, Andrey; Zhukovsky, Sergei; Novitsky, D.
2013-01-01
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...
Celebrating Cercignani's conjecture for the Boltzmann equation
Villani, Cédric
2011-01-01
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.
Nonaligned shocks for discrete velocity models of the Boltzmann equation
Directory of Open Access Journals (Sweden)
J. M. Greenberg
1991-05-01
Full Text Available At the conclusion of I. Bonzani's presentation on the existence of structured shock solutions to the six-velocity, planar, discrete Boltzmann equation (with binary and triple collisions, Greenberg asked whether such solutions were possible in directions e(α=(cosα ,sinα when α was not one of the particle flow directions. This question generated a spirited discussion but the question was still open at the conclusion of the conference. In this note the author will provide a partial resolution to the question raised above. Using formal perturbation arguments he will produce approximate solutions to the equation considered by Bonzani which represent traveling waves propagating in any direction e(α=(cosα ,sinα.
Wang, Yahui; Yan, Liming; Ma, Yu
2017-06-01
Applications of the transient Boltzmann transport equation (BTE) have undergone much investigation, such as radiative heat transfer and neutron transport. This paper provides a lattice Boltzmann model to efficiently resolve the multidimensional transient BTE. For a higher angular resolution, enough transport directions are considered while the transient BTE in each direction is treated as a conservation law equation and solved independently. Both macroscopic equations recovered from a Chapman-Enskog expansion and simulated results of typical benchmark problems show not only the second-order accuracy but also the flexibility and applicability of the proposed lattice Boltzmann model. This approach may contribute a powerful technique for the parallel simulation of large-scale engineering and some alternative perspectives for solving the nonlinear transport problem further.
Spinor Boltzmann Equation with Two Momenta at the Fermi Level
Institute of Scientific and Technical Information of China (English)
王正川
2012-01-01
Based on the formalism of Keldysh's nonequilibrium Green function, we establish a two momenta spinor Boltzmann equation for longitudinal scalar distribution function and transverse vector distribution function. The lon- gitudinal charge currents, transverse spin currents and the continuity equations satisfied by them are then studied, it indicates that both the charge currents and spin currents decay oscillately along with position, which is due to the momenta integral over the Fermi surface. We also compare our charge currents and spin currents with the corresponding results of one momentum spinor Boltzmann equation, the differences are obvious.
Lattice Boltzmann model for nonlinear convection-diffusion equations.
Shi, Baochang; Guo, Zhaoli
2009-01-01
A lattice Boltzmann model for convection-diffusion equation with nonlinear convection and isotropic-diffusion terms is proposed through selecting equilibrium distribution function properly. The model can be applied to the common real and complex-valued nonlinear evolutionary equations, such as the nonlinear Schrödinger equation, complex Ginzburg-Landau equation, Burgers-Fisher equation, nonlinear heat conduction equation, and sine-Gordon equation, by using a real and complex-valued distribution function and relaxation time. Detailed simulations of these equations are performed, and it is found that the numerical results agree well with the analytical solutions and the numerical solutions reported in previous studies.
Generalized Relativistic Chapman-Enskog Solution of the Boltzmann Equation
García-Perciante, A L; García-Colin, L S
2007-01-01
The Chapman-Enskog method of solution of the relativistic Boltzmann equation is generalized in order to admit a time-derivative term associated to the thermodynamic force in its first order solution. Both existence and uniqueness of such a solution are proved based on the standard theory of integral equations. The mathematical implications of the generalization here introduced are briefly explored.
Information Geometry Formalism for the Spatially Homogeneous Boltzmann Equation
Directory of Open Access Journals (Sweden)
Bertrand Lods
2015-06-01
Full Text Available Information Geometry generalizes to infinite dimension by modeling the tangent space of the relevant manifold of probability densities with exponential Orlicz spaces. We review here several properties of the exponential manifold on a suitable set Ɛ of mutually absolutely continuous densities. We study in particular the fine properties of the Kullback-Liebler divergence in this context. We also show that this setting is well-suited for the study of the spatially homogeneous Boltzmann equation if Ɛ is a set of positive densities with finite relative entropy with respect to the Maxwell density. More precisely, we analyze the Boltzmann operator in the geometric setting from the point of its Maxwell’s weak form as a composition of elementary operations in the exponential manifold, namely tensor product, conditioning, marginalization and we prove in a geometric way the basic facts, i.e., the H-theorem. We also illustrate the robustness of our method by discussing, besides the Kullback-Leibler divergence, also the property of Hyvärinen divergence. This requires us to generalize our approach to Orlicz–Sobolev spaces to include derivatives.
The Boltzmann equation near a rotational local Maxwellian
Kim, Chanwoo
2011-01-01
In rotationally symmetric domains, the Boltzmann equation with specular reflection boundary condition has a special type of equilibrium states called the rotational local Maxwellian which, unlike the uniform Maxwellian, has an additional term related to the angular momentum of the gas. In this paper, we consider the initial boundary value problem of the Boltzmann equation near the rotational local Maxwellian. Based on the L2-L1 framework of [12], we establish the global well-posedness and the convergence toward such equilibrium states.
Acoustic limit of the Boltzmann equation: classical solutions
Jang, Juhi; Jiang, Ning
2009-01-01
We study the acoustic limit from the Boltzmann equation in the framework of classical solutions. For a solution $F_\\varepsilon=\\mu +\\varepsilon \\sqrt{\\mu}f_\\varepsilon$ to the rescaled Boltzmann equation in the acoustic time scaling \\partial_t F_\\varepsilon +\\vgrad F_\\varepsilon =\\frac{1}{\\varepsilon} \\Q(F_\\varepsilon,F_\\varepsilon), inside a periodic box $\\mathbb{T}^3$, we establish the global-in-time uniform energy estimates of $f_\\varepsilon$ in $\\varepsilon$ and prove that $f_\\varepsilon$...
Axisymmetric multiphase Lattice Boltzmann method for generic equations of state
Reijers, Sten A; Toschi, Federico
2015-01-01
We present an axisymmetric lattice Boltzmann model based on the Kupershtokh et al. multiphase model that is capable of solving liquid-gas density ratios up to $10^3$. Appropriate source terms are added to the lattice Boltzmann evolution equation to fully recover the axisymmetric multiphase conservation equations. We validate the model by showing that a stationary droplet obeys the Young-Laplace law, comparing the second oscillation mode of a droplet with respect to an analytical solution and showing correct mass conservation of a propagating density wave.
Numerical solution of Boltzmann's equation
Energy Technology Data Exchange (ETDEWEB)
Sod, G.A.
1976-04-01
The numerical solution of Boltzmann's equation is considered for a gas model consisting of rigid spheres by means of Hilbert's expansion. If only the first two terms of the expansion are retained, Boltzmann's equation reduces to the Boltzmann-Hilbert integral equation. Successive terms in the Hilbert expansion are obtained by solving the same integral equation with a different source term. The Boltzmann-Hilbert integral equation is solved by a new very fast numerical method. The success of the method rests upon the simultaneous use of four judiciously chosen expansions; Hilbert's expansion for the distribution function, another expansion of the distribution function in terms of Hermite polynomials, the expansion of the kernel in terms of the eigenvalues and eigenfunctions of the Hilbert operator, and an expansion involved in solving a system of linear equations through a singular value decomposition. The numerical method is applied to the study of the shock structure in one space dimension. Numerical results are presented for Mach numbers of 1.1 and 1.6. 94 refs, 7 tables, 1 fig.
Celebrating Cercignani's conjecture for the Boltzmann equation
Desvillettes, Laurent; Villani, Cédric
2010-01-01
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.
Metamaterial characterization using Boltzmann's kinetic equation for electrons
DEFF Research Database (Denmark)
Novitsky, Andrey; Zhukovsky, Sergei; Novitsky, D.;
2013-01-01
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...... the resonant behavior of electronic response to an external electromagnetic field. We demonstrate the approach for planar and circular geometries of the metamolecules....
Physical scales in the Wigner-Boltzmann equation.
Nedjalkov, M; Selberherr, S; Ferry, D K; Vasileska, D; Dollfus, P; Querlioz, D; Dimov, I; Schwaha, P
2013-01-01
The Wigner-Boltzmann equation provides the Wigner single particle theory with interactions with bosonic degrees of freedom associated with harmonic oscillators, such as phonons in solids. Quantum evolution is an interplay of two transport modes, corresponding to the common coherent particle-potential processes, or to the decoherence causing scattering due to the oscillators. Which evolution mode will dominate depends on the scales of the involved physical quantities. A dimensionless formulation of the Wigner-Boltzmann equation is obtained, where these scales appear as dimensionless strength parameters. A notion called scaling theorem is derived, linking the strength parameters to the coupling with the oscillators. It is shown that an increase of this coupling is equivalent to a reduction of both the strength of the electric potential, and the coherence length. Secondly, the existence of classes of physically different, but mathematically equivalent setups of the Wigner-Boltzmann evolution is demonstrated.
Bazow, D; Heinz, U; Martinez, M; Noronha, J
2016-01-01
The dissipative dynamics of an expanding massless gas with constant cross section in a spatially flat Friedmann-Lema\\^itre-Robertson-Walker (FLRW) universe is studied. The mathematical problem of solving the full nonlinear relativistic Boltzmann equation is recast into an infinite set of nonlinear ordinary differential equations for the moments of the one-particle distribution function. Momentum-space resolution is determined by the number of non-hydrodynamic modes included in the moment hierarchy, i.e., by the truncation order. We show that in the FLRW spacetime the non-hydrodynamic modes decouple completely from the hydrodynamic degrees of freedom. This results in the system flowing as an ideal fluid while at the same time producing entropy. The solutions to the nonlinear Boltzmann equation exhibit transient tails of the distribution function with nontrivial momentum dependence. The evolution of this tail is not correctly captured by the relaxation time approximation nor by the linearized Boltzmann equation...
CORRECTIONS TO THE COLLISION TERM IN THE BGK BOLTZMANN EQUATION
Institute of Scientific and Technical Information of China (English)
FENG SHI-DE; REN RONG-CAI; CUI XIAO-PENG; JI ZHONG-ZHEN
2001-01-01
With the discrete method of the hexagonal cell and three different velocities of particle population in each cell,a two-dimensional lattice Boltzmann model is developed in this paper.[1,2] The collision operator in the Boltzmann equation is expanded to fourth order using the Taylor expansion.[3,4] With this model, good results have been obtained from the numerical simulation of the reflection phenomenon of the shock wave on the surface of an obstacle, and the numerical stability is also good. Thus the applicability of the D2Q19 model is verified.
A fast iterative scheme for the linearized Boltzmann equation
Wu, Lei; Zhang, Jun; Liu, Haihu; Zhang, Yonghao; Reese, Jason M.
2017-06-01
Iterative schemes to find steady-state solutions to the Boltzmann equation are efficient for highly rarefied gas flows, but can be very slow to converge in the near-continuum flow regime. In this paper, a synthetic iterative scheme is developed to speed up the solution of the linearized Boltzmann equation by penalizing the collision operator L into the form L = (L + Nδh) - Nδh, where δ is the gas rarefaction parameter, h is the velocity distribution function, and N is a tuning parameter controlling the convergence rate. The velocity distribution function is first solved by the conventional iterative scheme, then it is corrected such that the macroscopic flow velocity is governed by a diffusion-type equation that is asymptotic-preserving into the Navier-Stokes limit. The efficiency of this new scheme is assessed by calculating the eigenvalue of the iteration, as well as solving for Poiseuille and thermal transpiration flows. We find that the fastest convergence of our synthetic scheme for the linearized Boltzmann equation is achieved when Nδ is close to the average collision frequency. The synthetic iterative scheme is significantly faster than the conventional iterative scheme in both the transition and the near-continuum gas flow regimes. Moreover, due to its asymptotic-preserving properties, the synthetic iterative scheme does not need high spatial resolution in the near-continuum flow regime, which makes it even faster than the conventional iterative scheme. Using this synthetic scheme, with the fast spectral approximation of the linearized Boltzmann collision operator, Poiseuille and thermal transpiration flows between two parallel plates, through channels of circular/rectangular cross sections and various porous media are calculated over the whole range of gas rarefaction. Finally, the flow of a Ne-Ar gas mixture is solved based on the linearized Boltzmann equation with the Lennard-Jones intermolecular potential for the first time, and the difference
The Nonclassical Diffusion Approximation to the Nonclassical Linear Boltzmann Equation
Vasques, Richard
2015-01-01
We show that, by correctly selecting the probability distribution function $p(s)$ for a particle's distance-to-collision, the nonclassical diffusion equation can be represented exactly by the nonclassical linear Boltzmann equation for an infinite homogeneous medium. This choice of $p(s)$ preserves the $true$ mean-squared free path of the system, which sheds new light on the results obtained in previous work.
Classical Boltzmann equation and high-temperature QED
Brandt, F. T.; Ferreira, R. B.; Thuorst, J. F.
2015-01-01
The equivalence between thermal field theory and the Boltzmann transport equation is investigated at higher orders in the context of quantum electrodynamics. We compare the contributions obtained from the collisionless transport equation with the high temperature limit of the one-loop thermal Green's function. Our approach employs the representation of the thermal Green's functions in terms of forward scattering amplitudes. The general structure of these amplitudes clearly indicates that the ...
Analysis of Jeans instability from the Boltzmann equation
Kremer, Gilberto M.
2016-11-01
The dynamics of self-gravitating fluids is analyzed within the framework of a collisionless Boltzmann equation in the presence of gravitational fields and Poisson equation. Two cases are analyzed: a system with baryonic and dark matter in a static universe and a single system in an expanding universe. The amplitudes of the perturbed distribution functions are considered as a linear combination of the collision invariants of the Boltzmann equation. For the system of baryonic and dark matter, the Jeans mass of the combined system is smaller than the one of the single system indicating that a smaller mass is needed to initiate the collapse. For the single system in an expanding universe it is not necessary to make use of Jeans "swindle"and it shown that for small wavelengths the density contrast oscillates while for large wavelengths it grows with time and the Jeans instability emerges.
CMB spectral distortions as solutions to the Boltzmann equations
Ota, Atsuhisa
2016-01-01
We newly re-interpret cosmic microwave background spectral distortions as solutions to the Boltzmann equation at second order. This approach makes it possible to solve the equation of the momentum dependent temperature perturbations explicitly. In addition, we define higher order spectral distortions systematically, assuming that the collision term is linear in the photon distribution functions. For example, we find the linear Sunyaev-Zel'dovich effect whose momentum shape is different from the usual $y$ distortion, and show that the higher order spectral distortions are also generated as a result of the diffusion process in a language of higher order Boltzmann equations. The method may be applicable to a wider class of problems and has potential to give a general prescription to non-equilibrium physics.
Classical Boltzmann equation and high-temperature QED
Brandt, F. T.; Ferreira, R. B.; Thuorst, J. F.
2015-02-01
The equivalence between thermal field theory and the Boltzmann transport equation is investigated at higher orders in the context of quantum electrodynamics. We compare the contributions obtained from the collisionless transport equation with the high temperature limit of the one-loop thermal Green's function. Our approach employs the representation of the thermal Green's functions in terms of forward scattering amplitudes. The general structure of these amplitudes clearly indicates that the physics described by the leading high temperature limit of quantum electrodynamics can be obtained from the Boltzman transport equation. We also present some explicit examples of this interesting equivalence.
Classical Boltzmann equation and high-temperature QED
Brandt, F T; Thuorst, J F
2015-01-01
The equivalence between thermal field theory and the Boltzmann transport equation is investigated at higher orders in the context of Quantum Electrodynamics. We compare the contributions obtained from the collisionless transport equation with the high temperature limit of the one-loop thermal Green's function. Our approach employs the representation of the thermal Green's functions in terms of forward scattering amplitudes. The general structure of these amplitudes clearly indicates that the physics described by the leading high temperature limit of Quantum Electrodynamics can be obtained from the Boltzman transport equation. We also present some explicit examples of this interesting equivalence.
Green, B. I.; Vedula, Prakash
2013-07-01
An alternative approach for solution of the collisional Boltzmann equation for a lattice architecture is presented. In the proposed method, termed the collisional lattice Boltzmann method (cLBM), the effects of spatial transport are accounted for via a streaming operator, using a lattice framework, and the effects of detailed collisional interactions are accounted for using the full collision operator of the Boltzmann equation. The latter feature is in contrast to the conventional lattice Boltzmann methods (LBMs) where collisional interactions are modeled via simple equilibrium based relaxation models (e.g. BGK). The underlying distribution function is represented using weights and fixed velocity abscissas according to the lattice structure. These weights are evolved based on constraints on the evolution of generalized moments of velocity according to the collisional Boltzmann equation. It can be shown that the collision integral can be reduced to a summation of elementary integrals, which can be analytically evaluated. The proposed method is validated using studies of canonical microchannel Couette and Poiseuille flows (both body force and pressure driven) and the results are found to be in good agreement with those obtained from conventional LBMs and experiments where available. Unlike conventional LBMs, the proposed method does not involve any equilibrium based approximations and hence can be useful for simulation of highly nonequilibrium flows (for a range of Knudsen numbers) using a lattice framework.
Dynamics of annihilation. I. Linearized Boltzmann equation and hydrodynamics.
García de Soria, María Isabel; Maynar, Pablo; Schehr, Grégory; Barrat, Alain; Trizac, Emmanuel
2008-05-01
We study the nonequilibrium statistical mechanics of a system of freely moving particles, in which binary encounters lead either to an elastic collision or to the disappearance of the pair. Such a system of ballistic annihilation therefore constantly loses particles. The dynamics of perturbations around the free decay regime is investigated using the spectral properties of the linearized Boltzmann operator, which characterize linear excitations on all time scales. The linearized Boltzmann equation is solved in the hydrodynamic limit by a projection technique, which yields the evolution equations for the relevant coarse-grained fields and expressions for the transport coefficients. We finally present the results of molecular dynamics simulations that validate the theoretical predictions.
Efficient Asymptotic Preserving Deterministic methods for the Boltzmann Equation
2011-04-01
important role in modelling granular gases (Bobylev et al., 2000), charged particles in semiconductors (Markowich et al., 1989), neutron transport (Jin...1.6 The splitting approach 1 THE BOLTZMANN EQUATION • Granular gas models : particles undergo inelastic collisions . Energy is dissipated by the model ...A model for collision processes in gases i. small amplitute processes in charged and neutral one component systems. Phys. Rev., 94:511–525. 8
Asymptotic stability of the Boltzmann equation with Maxwell boundary conditions
Briant, Marc; Guo, Yan
2016-12-01
In a general C1 domain, we study the perturbative Cauchy theory for the Boltzmann equation with Maxwell boundary conditions with an accommodation coefficient α in (√{ 2 / 3 } , 1 ], and discuss this threshold. We consider polynomial or stretched exponential weights m (v) and prove existence, uniqueness and exponential trend to equilibrium around a global Maxwellian in Lx,v∞ (m). Of important note is the fact that the methods do not involve contradiction arguments.
LATTICE BOLTZMANN EQUATION MODEL IN THE CORIOLIS FIELD
Institute of Scientific and Technical Information of China (English)
FENG SHI-DE; MAO JIANG-YU; ZHANG QIONG
2001-01-01
In a large-scale field of rotational fluid, various unintelligible and surprising dynamic phenomena are produced due to the effect of the Coriolis force. The lattice Boltzmann equation (LBE) model in the Coriolis field is developed based on previous works.[1-4] Geophysical fluid dynamics equations are derived from the model. Numerical simulations have been made on an ideal atmospheric circulation of the Northern Hemisphere by using the model and they reproduce the Rossby wave motion well. Hence the applicability of the model is verified in both theory and experiment.
Lattice Boltzmann method for the fractional advection-diffusion equation.
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
2016-04-01
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.
Lattice Boltzmann method for the fractional advection-diffusion equation
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.
2016-04-01
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.
Numerical simulation for the Gross-Pitaevskii equation based on the lattice Boltzmann method
Wang, Huimin
2017-09-01
A lattice Boltzmann model for the Gross-Pitaevskii equation is proposed in this paper. Some numerical tests for one- and two-dimensional Gross-Pitaevskii equation have been conducted. The waves of the Gross-Pitaevskii equation are simulated. Numerical results show that the lattice Boltzmann method is an effective method for the wave of the Gross-Pitaevskii equation.
Lattice Boltzmann model for a steady radiative transfer equation.
Yi, Hong-Liang; Yao, Feng-Ju; Tan, He-Ping
2016-08-01
A complete lattice Boltzmann model (LBM) is proposed for the steady radiative transfer equation (RTE). The RTE can be regarded as a pure convection equation with a source term. To derive the expressions for the equilibrium distribution function and the relaxation time, an artificial isotropic diffusion term is introduced to form a convection-diffusion equation. When the dimensionless relaxation time has a value of 0.5, the lattice Boltzmann equation (LBE) is exactly applicable to the original steady RTE. We also perform a multiscale analysis based on the Chapman-Enskog expansion to recover the macroscopic RTE from the mesoscopic LBE. The D2Q9 model is used to solve the LBE, and the numerical results obtained by the LBM are comparable to the results obtained by other methods or analytical solutions, which demonstrates that the proposed model is highly accurate and stable in simulating multidimensional radiative transfer. In addition, we find that the convergence rate of the LBM depends on the transport properties of RTE: for diffusion-dominated RTE with a large optical thickness, the LBM shows a second-order convergence rate in space, while for convection-dominated RTE with a small optical thickness, a lower convergence rate is observed.
The Boltzmann Equation for a Multi-species Mixture Close to Global Equilibrium
Briant, Marc; Daus, Esther S.
2016-12-01
We study the Cauchy theory for a multi-species mixture, where the different species can have different masses, in a perturbative setting on the three dimensional torus. The ultimate aim of this work is to obtain the existence, uniqueness and exponential trend to equilibrium of solutions to the multi-species Boltzmann equation in {L^1_vL^∞_x(m)}, where {m˜ (1+ |v|^k)} is a polynomial weight. We prove the existence of a spectral gap for the linear multi-species Boltzmann operator allowing different masses, and then we establish a semigroup property thanks to a new explicit coercive estimate for the Boltzmann operator. Then we develop an {L^2-L^∞} theory à la Guo for the linear perturbed equation. Finally, we combine the latter results with a decomposition of the multi-species Boltzmann equation in order to deal with the full equation. We emphasize that dealing with different masses induces a loss of symmetry in the Boltzmann operator which prevents the direct adaptation of standard mono-species methods (for example Carleman representation, Povzner inequality). Of important note is the fact that all methods used and developed in this work are constructive. Moreover, they do not require any Sobolev regularity and the {L^1_vL^∞_x} framework is dealt with for any {k > k_0}, recovering the optimal physical threshold of finite energy {k_0=2} in the particular case of a multi-species hard spheres mixture with the same masses.
Energy Technology Data Exchange (ETDEWEB)
Zabadal, Jorge; Borges, Volnei; Van der Laan, Flavio T., E-mail: jorge.zabadal@ufrgs.br, E-mail: borges@ufrgs.br, E-mail: ftvdl@ufrgs.br [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: vinicius_ribeiro@uniritter.edu.br [Centro Universitario Ritter dos Reis (UNIRITTER), Porto Alegre, RS (Brazil); Santos, Marcio G., E-mail: phd.marcio@gmail.com [Universidade Federal do Rio Grande do Sul (UFRGS), Tramandai, RS (Brazil). Departamento Interdisciplinar do Campus Litoral Norte
2015-07-01
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)
Generalizing the Boltzmann equation in complex phase space.
Zadehgol, Abed
2016-08-01
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)JCTPAH0021-999110.1016/j.jcp.2014.06.053] and Zadehgol [Phys. Rev. E 91, 063311 (2015)PLEEE81539-375510.1103/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.
Generalizing the Boltzmann equation in complex phase space
Zadehgol, Abed
2016-08-01
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/j.jcp.2014.06.053] 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.
From Conformal Invariance towards Dynamical Symmetries of the Collisionless Boltzmann Equation
Directory of Open Access Journals (Sweden)
Stoimen Stoimenov
2015-09-01
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.
Interplay of Boltzmann equation and continuity equation for accelerated electrons in solar flares
Codispoti, Anna
2015-01-01
During solar flares a large amount of electrons are accelerated within the plasma present in the solar atmosphere. Accurate measurements of the motion of these electrons start becoming available from the analysis of hard X-ray imaging-spectroscopy observations. In this paper, we discuss the linearized perturbations of the Boltzmann kinetic equation describing an ensemble of electrons accelerated by the energy release occurring during solar flares. Either in the limit of high energy or at vanishing background temperature such an equation reduces to a continuity equation equipped with an extra force of stochastic nature. This stochastic force is actually described by the well known energy loss rate due to Coulomb collision with ambient particles, but, in order to match the collision kernel in the linearized Boltzmann equation it needs to be treated in a very specific manner. In the second part of the paper the derived continuity equation is solved with some hyperbolic techniques, and the obtained solution is wr...
Numerical Treatment of the Boltzmann Equation for Self-Propelled Particle Systems
Directory of Open Access Journals (Sweden)
Florian Thüroff
2014-11-01
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.
Global Solutions of the Boltzmann Equation Over {{R}^D} Near Global Maxwellians with Small Mass
Bardos, Claude; Gamba, Irene M.; Golse, François; Levermore, C. David
2016-09-01
We study the dynamics defined by the Boltzmann equation set in the Euclidean space {{R}^D} in the vicinity of global Maxwellians with finite mass. A global Maxwellian is a special solution of the Boltzmann equation for which the collision integral vanishes identically. In this setting, the dispersion due to the advection operator quenches the dissipative effect of the Boltzmann collision integral. As a result, the large time limit of solutions of the Boltzmann equation in this regime is given by noninteracting, freely transported states and can be described with the tools of scattering theory.
An Exact Solution to the Two-Particle Boltzmann Equation System for Maxwell Gases
Institute of Scientific and Technical Information of China (English)
布仁满都拉; 赵迎春
2012-01-01
An exact solution to the two-particle Boltzmann equation system for Maxwell gases is obtained with use of Bobylev approach.The relationship between the exact solution and the self-similar solution of the boltzmann equation is also given.
Acoustic equations of state for simple lattice Boltzmann velocity sets.
Viggen, Erlend Magnus
2014-07-01
The lattice Boltzmann (LB) method typically uses an isothermal equation of state. This is not sufficient to simulate a number of acoustic phenomena where the equation of state cannot be approximated as linear and constant. However, it is possible to implement variable equations of state by altering the LB equilibrium distribution. For simple velocity sets with velocity components ξ(iα)∈(-1,0,1) for all i, these equilibria necessarily cause error terms in the momentum equation. These error terms are shown to be either correctable or negligible at the cost of further weakening the compressibility. For the D1Q3 velocity set, such an equilibrium distribution is found and shown to be unique. Its sound propagation properties are found for both forced and free waves, with some generality beyond D1Q3. Finally, this equilibrium distribution is applied to a nonlinear acoustics simulation where both mechanisms of nonlinearity are simulated with good results. This represents an improvement on previous such simulations and proves that the compressibility of the method is still sufficiently strong even for nonlinear acoustics.
Boltzmann equation for the electron gas of a nondegenerate plasma
Gould, R. J.
1974-01-01
The collision terms in the Boltzmann equation associated with various processes are derived. For processes having a Fokker-Planck (F-P) limit, the associated F-P operator is derived by means of physical arguments to determine the form of the operator; its multiplying constant is fixed by calculating the total energy exchange rate and comparing with the rate computed by other means. In this manner, the F-P operator is derived for electron-ion scattering, electron-electron scattering in the high-velocity limit, electron-atom elastic scattering, Compton scattering, and the high-velocity limit for inelastic scattering. Other processes considered are bremsstrahlung, radiative-recombination, photoionization, collisional ionization of atoms, and suprathermal-particle ionization of atoms.
Electric Conductivity from the solution of the Relativistic Boltzmann Equation
Puglisi, A; Greco, V
2014-01-01
We present numerical results of electric conductivity $\\sigma_{el}$ of a fluid obtained solving the Relativistic Transport Boltzmann equation in a box with periodic boundary conditions. We compute $\\sigma_{el}$ using two methods: the definition itself, i.e. applying an external electric field, and the evaluation of the Green-Kubo relation based on the time evolution of the current-current correlator. We find a very good agreement between the two methods. We also compare numerical results with analytic formulas in Relaxation Time Approximation (RTA) where the relaxation time for $\\sigma_{el}$ is determined by the transport cross section $\\sigma_{tr}$, i.e. the differential cross section weighted with the collisional momentum transfer. We investigate the electric conductivity dependence on the microscopic details of the 2-body scatterings: isotropic and anisotropic cross-section, and massless and massive particles. We find that the RTA underestimates considerably $\\sigma_{el}$; for example at screening masses $...
Discretization of the velocity space in solution of the Boltzmann equation
Shan, X; Shan, Xiaowen; He, Xiaoyi
1998-01-01
We point out an equivalence between the discrete velocity method of solving the Boltzmann equation, of which the lattice Boltzmann equation method is a special example, and the approximations to the Boltzmann equation by a Hermite polynomial expansion. Discretizing the Boltzmann equation with a BGK collision term at the velocities that correspond to the nodes of a Hermite quadrature is shown to be equivalent to truncating the Hermite expansion of the distribution function to the corresponding order. The truncated part of the distribution has no contribution to the moments of low orders and is negligible at small Mach numbers. Higher order approximations to the Boltzmann equation can be achieved by using more velocities in the quadrature.
A Lattice Boltzmann Model and Simulation of KdV-Burgers Equation
Institute of Scientific and Technical Information of China (English)
ZHANGChao-Ying; TANHui-Li; LIUMu-Ren; KONGLing-Jiang
2004-01-01
A lattice Boltzmann model of KdV-Burgers equation is derived by using the single-relaxation form of the lattice Boltzmann equation. With the present model, we simulate the traveling-wave solutions, the solitary-wave solutions, and the sock-wave solutions of KdV-Burgers equation, and calculate the decay factor and the wavelength of the sock-wave solution, which has exponential decay. The numerical results agree with the analytical solutions quite well.
A New Lattice Boltzmann Model for KdV-Burgers Equation
Institute of Scientific and Technical Information of China (English)
MA Chang-Feng
2005-01-01
@@ A new lattice Boltzmann model with amending-function for KdV-Burgers equation, ut +uux - αuxx +βuxxx = 0,is presented by using the single-relaxation form of the lattice Boltzmann equation. Applying the proposed model,we simulate the solutions ofa kind of KdV-Burgers equations, and the numerical results agree with the analytical solutions quite well.
Distributional Monte Carlo Methods for the Boltzmann Equation
2013-03-01
become the first to possess non - Maxwellian distributions, and therefore become the only location where 112 collisions are required to be calculated... Maxwellian . . . . . . . . . . . . . . . . . 16 fMB Maxwell-Boltzmann Density . . . . . . . . . . . . . . . . . . . . . . . . 16 nMB Maxwell-Boltzmann...is equivalent to assuming that millions of actual particles all share the exact velocity vector. This assumption is non -physical in the sense that
An introduction to the Boltzmann equation and transport processes in gases
Energy Technology Data Exchange (ETDEWEB)
Kremer, Gilberto Medeiros [Universidade Federal do Parana, Curitiba (Brazil). Dept. de Fisica
2010-07-01
This book deals with the classical kinetic theory of gases. Its aim is to present the basic principles of this theory within an elementary framework and from a more rigorous approach based on the Boltzmann equation. The subjects are presented in a self-contained manner such that the readers can understand and learn some methods used in the kinetic theory of gases in order to investigate the Boltzmann equation. It is expected that this book could be useful as a textbook for students and researchers who are interested in the principles of the Boltzmann equation and in the methods used in the kinetic theory of gases. (orig.)
Multilevel Methods for the Poisson-Boltzmann Equation
Holst, Michael Jay
We consider the numerical solution of the Poisson -Boltzmann equation (PBE), a three-dimensional second order nonlinear elliptic partial differential equation arising in biophysics. This problem has several interesting features impacting numerical algorithms, including discontinuous coefficients representing material interfaces, rapid nonlinearities, and three spatial dimensions. Similar equations occur in various applications, including nuclear physics, semiconductor physics, population genetics, astrophysics, and combustion. In this thesis, we study the PBE, discretizations, and develop multilevel-based methods for approximating the solutions of these types of equations. We first outline the physical model and derive the PBE, which describes the electrostatic potential of a large complex biomolecule lying in a solvent. We next study the theoretical properties of the linearized and nonlinear PBE using standard function space methods; since this equation has not been previously studied theoretically, we provide existence and uniqueness proofs in both the linearized and nonlinear cases. We also analyze box-method discretizations of the PBE, establishing several properties of the discrete equations which are produced. In particular, we show that the discrete nonlinear problem is well-posed. We study and develop linear multilevel methods for interface problems, based on algebraic enforcement of Galerkin or variational conditions, and on coefficient averaging procedures. Using a stencil calculus, we show that in certain simplified cases the two approaches are equivalent, with different averaging procedures corresponding to different prolongation operators. We also develop methods for nonlinear problems based on a nonlinear multilevel method, and on linear multilevel methods combined with a globally convergent damped-inexact-Newton method. We derive a necessary and sufficient descent condition for the inexact-Newton direction, enabling the development of extremely
A Stability Notion for the viscous Shallow Water Lattice Boltzmann Equations
Banda, Mapundi K
2015-01-01
The stability of Lattice Boltzmann Equations modelling Shallow Water Equations in the special case of reduced gravity is investigated theoretically. A stability notion is defined as applied in incompressible Navier-Stokes equations in Banda, M. K., Yong, W.- A. and Klar, A: A stability notion for lattice Boltzmann equations. SIAM J. Sci. Comput. {\\bf 27(6)}, 2098-2111 (2006). It is found that to maintain stability a careful choice of the value of the reduced gravity must be made. The stability notion is employed to investigate different shallow water lattice Boltzmann Equations. The effect of the reduced gravity on the mechanism of instability is investigated. Results are tested using the Lattice Boltzmann Method for various values of the governing parameters of the flow. It is observed that even for the discrete model the reduced gravity has a significant effect on the stability.
Equations of motion of test particles for solving the spin-dependent Boltzmann-Vlasov equation
Xia, Yin; Li, Bao-An; Shen, Wen-Qing
2016-01-01
A consistent derivation of the equations of motion (EOMs) of test particles for solving the spin-dependent Boltzmann-Vlasov equation is presented. Though the obtained EOMs are general, they are particularly useful in simulating nucleon spinor transport in heavy-ion collisions at intermediate energies. It is shown that the nucleon transverse flow in heavy-ion collisions especially those involving polarized projectile and/or target nuclei depends strongly on the spin-orbit coupling. Future comparisons of model simulations with experimental data will help constrain the poorly known in-medium nucleon spin-orbit coupling relevant for understanding properties of rare isotopes and their astrophysical impacts.
High-order regularization in lattice-Boltzmann equations
Mattila, Keijo K.; Philippi, Paulo C.; Hegele, Luiz A.
2017-04-01
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.
Towards Full Aircraft Airframe Noise Prediction: Lattice Boltzmann Simulations
Khorrami, Mehdi R.; Fares, Ehab; Casalino, Damiano
2014-01-01
Computational results for an 18%-scale, semi-span Gulfstream aircraft model are presented. Exa Corporation's lattice Boltzmann PowerFLOW(trademark) solver was used to perform time-dependent simulations of the flow field associated with this high-fidelity aircraft model. The simulations were obtained for free-air at a Mach number of 0.2 with the flap deflected at 39 deg (landing configuration). We focused on accurately predicting the prominent noise sources at the flap tips and main landing gear for the two baseline configurations, namely, landing flap setting without and with gear deployed. Capitalizing on the inherently transient nature of the lattice Boltzmann formulation, the complex time-dependent flow features associated with the flap were resolved very accurately and efficiently. To properly simulate the noise sources over a broad frequency range, the tailored grid was very dense near the flap inboard and outboard tips. Extensive comparison of the computed time-averaged and unsteady surface pressures with wind tunnel measurements showed excellent agreement for the global aerodynamic characteristics and the local flow field at the flap inboard and outboard tips and the main landing gear. In particular, the computed fluctuating surface pressure field for the flap agreed well with the measurements in both amplitude and frequency content, indicating that the prominent airframe noise sources at the tips were captured successfully. Gear-flap interaction effects were remarkably well predicted and were shown to affect only the inboard flap tip, altering the steady and unsteady pressure fields in that region. The simulated farfield noise spectra for both baseline configurations, obtained using a Ffowcs-Williams and Hawkings acoustic analogy approach, were shown to be in close agreement with measured values.
On the effects of the reactive terms in the Boltzmann equation
Directory of Open Access Journals (Sweden)
C. J. Zamlutti
Full Text Available The effects of the production and loss mechanisms that affect the Boltzmann equations are considered by the inclusion of a reactive term. The necessary elements to develop a proper form for this term are revised and the current trends analyzed. Although no accurate theoretical treatment of the problem is possible due to the many body nature of it, important relations can be derived which, besides being representative of the quantitative aspects of the matter, are illustrative of the qualitative features of the phenomenon. The overall procedure is detailed in this revision.
Global classical solutions of the Boltzmann equation with long-range interactions
National Research Council Canada - National Science Library
Philip T. Gressman; Robert M. Strain; Richard V. Kadison
2010-01-01
This is a brief announcement of our recent proof of global existence and rapid decay to equilibrium of classical solutions to the Boltzmann equation without any angular cutoff, that is, for long-range interactions...
Equivalence Between Forward and Backward Boltzmann Equations in Multi-Component Medium
Institute of Scientific and Technical Information of China (English)
张竹林
2002-01-01
The author generalized the propagator function theory introduced first by Sigmund, and gave a explicitly proof of a equivalence between forward and backward Boltzmann equations in a multi-component medium by using the generalized propagator function theory.
An introduction to the Boltzmann equation and transport processes in gases
Kremer, Gilberto M; Colton, David
2010-01-01
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.
Computational Aeroacoustics Using the Generalized Lattice Boltzmann Equation Project
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...
Note on Invariance of One-Dimensional Lattice-Boltzmann Equation
Institute of Scientific and Technical Information of China (English)
RAN Zheng
2007-01-01
Invariance of the one-dimensional lattice Boltzmann model is proposed together with its rigorous theoretical background.It is demonstrated that the symmetry inherent in Navier-Stokes equations is not really recovered in the one-dimensional lattice Boltzmann equation (LBE),especially for shock calculation.Symmetry breaking may be the inherent cause for the non-physical oscillations in the vicinity of the shock for LBE calculation.
Bazow, D.; Denicol, G. S.; Heinz, U.; Martinez, M.; Noronha, J.
2016-12-01
The dissipative dynamics of an expanding massless gas with constant cross section in a spatially flat Friedmann-Lemaître-Robertson-Walker (FLRW) universe is studied. The mathematical problem of solving the full nonlinear relativistic Boltzmann equation is recast into an infinite set of nonlinear ordinary differential equations for the moments of the one-particle distribution function. Momentum-space resolution is determined by the number of nonhydrodynamic modes included in the moment hierarchy, i.e., by the truncation order. We show that in the FLRW spacetime the nonhydrodynamic modes decouple completely from the hydrodynamic degrees of freedom. This results in the system flowing as an ideal fluid while at the same time producing entropy. The solutions to the nonlinear Boltzmann equation exhibit transient tails of the distribution function with nontrivial momentum dependence. The evolution of this tail is not correctly captured by the relaxation time approximation nor by the linearized Boltzmann equation. However, the latter probes additional high-momentum details unresolved by the relaxation time approximation. While the expansion of the FLRW spacetime is slow enough for the system to move towards (and not away from) local thermal equilibrium, it is not sufficiently slow for the system to actually ever reach complete local equilibrium. Equilibration is fastest in the relaxation time approximation, followed, in turn, by kinetic evolution with a linearized and a fully nonlinear Boltzmann collision term.
Recovering Navier-Stokes Equations from Asymptotic Limits of the Boltzmann Gas Mixture Equation
Bianca, Carlo; Dogbe, Christian
2016-05-01
This paper is devoted to the derivation of macroscopic fluid dynamics from the Boltzmann mesoscopic dynamics of a binary mixture of hard-sphere gas particles. Specifically the hydrodynamics limit is performed by employing different time and space scalings. The paper shows that, depending on the magnitude of the parameters which define the scaling, the macroscopic quantities (number density, mean velocity and local temperature) are solutions of the acoustic equation, the linear incompressible Euler equation and the incompressible Navier-Stokes equation. The derivation is formally tackled by the recent moment method proposed by [C. Bardos, et al., J. Stat. Phys. 63 (1991) 323] and the results generalize the analysis performed in [C. Bianca, et al., Commun. Nonlinear Sci. Numer. Simulat. 29 (2015) 240].
A Stability notion for the viscous Shallow Water Discrete-Velocity Boltzmann Equations
Banda, Mapundi K.; Uoane, Tumelo R. A.
2015-01-01
The stability of Lattice Boltzmann Equations modelling Shallow Water Equations in the special case of reduced gravity is investigated theoretically. A stability notion is defined as applied in incompressible Navier-Stokes equations in Banda, M. K., Yong, W.- A. and Klar, A: A stability notion for lattice Boltzmann equations. SIAM J. Sci. Comput. {\\bf 27(6)}, 2098-2111 (2006). It is found that to maintain stability a careful choice of the value of the reduced gravity must be made. The stabilit...
Lattice Boltzmann Model for The Volume-Averaged Navier-Stokes Equations
Zhang, Jingfeng; Ouyang, Jie
2014-01-01
A numerical method, based on discrete lattice Boltzmann equation, is presented for solving the volume-averaged Navier-Stokes equations. With a modified equilibrium distribution and an additional forcing term, the volume-averaged Navier-Stokes equations can be recovered from the lattice Boltzmann equation in the limit of small Mach number by the Chapman-Enskog analysis and Taylor expansion. Due to its advantages such as explicit solver and inherent parallelism, the method appears to be more competitive with traditional numerical techniques. Numerical simulations show that the proposed model can accurately reproduce both the linear and nonlinear drag effects of porosity in the fluid flow through porous media.
Energy Technology Data Exchange (ETDEWEB)
Uchaikin, V V; Sibatov, R T, E-mail: vuchaikin@gmail.com, E-mail: ren_sib@bk.ru [Ulyanovsk State University, 432000, 42 Leo Tolstoy str., Ulyanovsk (Russian Federation)
2011-04-08
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.
A Combined MPI-CUDA Parallel Solution of Linear and Nonlinear Poisson-Boltzmann Equation
Directory of Open Access Journals (Sweden)
José Colmenares
2014-01-01
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.
Structural interactions in ionic liquids linked to higher-order Poisson-Boltzmann equations.
Blossey, R; Maggs, A C; Podgornik, R
2017-06-01
We present a derivation of generalized Poisson-Boltzmann equations starting from classical theories of binary fluid mixtures, employing an approach based on the Legendre transform as recently applied to the case of local descriptions of the fluid free energy. Under specific symmetry assumptions, and in the linearized regime, the Poisson-Boltzmann equation reduces to a phenomenological equation introduced by Bazant et al. [Phys. Rev. Lett. 106, 046102 (2011)]PRLTAO0031-900710.1103/PhysRevLett.106.046102, whereby the structuring near the surface is determined by bulk coefficients.
A New Regularization Mechanism for the Boltzmann Equation Without Cut-Off
Silvestre, Luis
2016-11-01
We apply recent results on regularity for general integro-differential equations to derive a priori estimates in Hölder spaces for the space homogeneous Boltzmann equation in the non cut-off case. We also show an a priori estimate in {L^∞} which applies in the space inhomogeneous case as well, provided that the macroscopic quantities remain bounded.
A Combined MPI-CUDA Parallel Solution of Linear and Nonlinear Poisson-Boltzmann Equation
Colmenares, José; Galizia, Antonella; Ortiz, Jesús; Clematis, Andrea; Rocchia, Walter
2014-01-01
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. PMID:25013789
A combined MPI-CUDA parallel solution of linear and nonlinear Poisson-Boltzmann equation.
Colmenares, José; Galizia, Antonella; Ortiz, Jesús; Clematis, Andrea; Rocchia, Walter
2014-01-01
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.
Discrete Boltzmann model of shallow water equations with polynomial equilibria
Meng, Jianping; Emerson, David R; Peng, Yong; Zhang, Jianmin
2016-01-01
A hierarchy of discrete Boltzmann model is proposed for simulating shallow water flows. By using the Hermite expansion and Gauss-Hermite quadrature, the conservation laws are automatically satisfied without extra effort. Moreover, the expansion order and quadrature can be chosen flexibly according to the problem for striking the balance of accuracy and efficiency. The models are then tested using the classical one-dimensional dam-breaking problem, and successes are found for both supercritical and subcritical flows.
Solution of time-dependent Boltzmann equation for electrons in non-thermal plasma
Energy Technology Data Exchange (ETDEWEB)
Trunec, D; Bonaventura, Z; Necas, D [Department of Physical Electronics, Faculty of Science, Masaryk University, Kotlarska 2, 611 37 Brno (Czech Republic)
2006-06-21
The time development of the electron distribution function and electron macroscopic parameters was studied by solving the time-dependent Boltzmann equation for low temperature plasma. A new technique for solving the time-dependent Boltzmann equation was developed. This technique is based on a multi-term approximation of the electron distribution function expansion in Legendre polynomials. The results for electron relaxation in Reid's ramp model and argon plasma are presented. The effect of negative mobility was studied and is discussed for argon plasma. Finally, the time-dependent Boltzmann equation was solved for pulsed microwave discharge in nitrogen. The accuracy of all results was confirmed by the Monte Carlo simulation.
An H Theorem for Boltzmann's Equation for the Yard-Sale Model of Asset Exchange
Boghosian, Bruce M.; Johnson, Merek; Marcq, Jeremy A.
2015-12-01
In recent work (Boghosian, Phys Rev E 89:042804-042825, 2014; Boghosian, Int J Mod Phys 25:1441008-1441015, 2014), Boltzmann and Fokker-Planck equations were derived for the "Yard-Sale Model" of asset exchange. For the version of the model without redistribution, it was conjectured, based on numerical evidence, that the time-asymptotic state of the model was oligarchy—complete concentration of wealth by a single individual. In this work, we prove that conjecture by demonstrating that the Gini coefficient, a measure of inequality commonly used by economists, is an H function of both the Boltzmann and Fokker-Planck equations for the model.
Institute of Scientific and Technical Information of China (English)
Du Gang; Liu Xiao-Yan; Han Ru-Qi
2006-01-01
A two-dimensional (2D) full band self-consistent ensemble Monte Carlo (MC) method for solving the quantum Boltzmann equation, including collision broadening and quantum potential corrections, is developed to extend the MC method to the study of nano-scale semiconductor devices with obvious quantum mechanical (QM) effects. The quantum effects both in real space and momentum space in nano-scale semiconductor devices can be simulated. The effective mobility in the inversion layer of n and p channel MOSFET is simulated and compared with experimental data to verify this method. With this method 50nm ultra thin body silicon on insulator MOSFET are simulated. Results indicate that this method can be used to simulate the 2D QM effects in semiconductor devices including tunnelling effect.
Flavoured quantum Boltzmann equations from cQPA
Fidler, Christian; Kainulainen, Kimmo; Rahkila, Pyry Matti
2011-01-01
We develop a Boltzmann-type quantum transport theory for interacting fermion and scalar fields including both flavour and particle-antiparticle mixing. Our formalism is based on the coherent quasiparticle approximation (cQPA) for the 2-point correlation functions, whose extended phase-space structure contains new spectral shells for flavour- and particle-antiparticle coherence. We derive explicit cQPA propagators and Feynman rules for the transport theory. In particular the nontrivial Wightman functions can be written as composite operators $\\sim {\\cal A} F {\\cal A}$, which generalize the usual Kadanoff-Baym ansatz. Our numerical results show that particle-antiparticle coherence can strongly influence CP-violating flavour mixing even for relatively slowly-varying backgrounds. Thus, unlike recently suggested, these correlations cannot be neglected when studying asymmetry generation due to time-varying mass transition, for example in electroweak-type baryogenesis models. Finally, we show that the cQPA coherence...
Kinetic Description for a Suspension of Inelastic Spheres - Boltzmann and BGK Equations
2007-11-02
Kinetic description for a suspension of inelastic spheres - Boltzmann and BGK equations Cedric Croizet and Renee Gatignol Laboratoire de Modelisation ...Organization Name(s) and Address(es) Laboratoire de Modelisation en Mecanique - Universite Pierre et Marie Curie (Paris 6) et CNRS UMR 7607 - 4) place
DEFF Research Database (Denmark)
Johannessen, Kim
2014-01-01
The exact solution to the one-dimensional Poisson–Boltzmann equation with asymmetric boundary conditions can be expressed in terms of the Jacobi elliptic functions. The boundary conditions determine the modulus of the Jacobi elliptic functions. The boundary conditions can not be solved analytically...
Ender, I. A.; Bakaleinikov, L. A.; Flegontova, E. Yu.; Gerasimenko, A. B.
2017-08-01
We have proposed an algorithm for the sequential construction of nonisotropic matrix elements of the collision integral, which are required to solve the nonlinear Boltzmann equation using the moments method. The starting elements of the matrix are isotropic and assumed to be known. The algorithm can be used for an arbitrary law of interactions for any ratio of the masses of colliding particles.
Global classical solution to the inelastic Boltzmann equation with potential force
Wang, Xiaolong; Jiang, Zhenglu
2017-08-01
In this paper, motivated by Guo [Commun. Math. Phys. 218(2), 293-313 (2001)], we construct the global classical solution to the inelastic Boltzmann equation with potential force by using the standard contraction mapping theorem. Moreover, the dispersion estimates of the solution are obtained.
Convergence Rate to Stationary Solutions for Boltzmann Equation with External Force
Institute of Scientific and Technical Information of China (English)
Seiji UKAI; Tong YANG; Huijiang ZHAO
2006-01-01
For the Boltzmann equation with an external force in the form of the gradient of a potential function in space variable, the stability of its stationary solutions as local Maxwellians was studied by S. Ukai et al. (2005) through the energy method. Based on this stability analysis and some techniques on analyzing the convergence rates to stationary solutions for the compressible Navier-Stokes equations, in this paper, we study the convergence rate to the above stationary solutions for the Boltzmann equation which is a fundamental equation in statistical physics for non-equilibrium rarefied gas. By combining the dissipation from the viscosity and heat conductivity on the fluid components and the dissipation on the non-fluid component through the celebrated H-theorem, a convergence rate of the same order as the one for the compressible Navier-Stokes is obtained by constructing some energy functionals.
Serov, S A
2013-01-01
In the article correct method for the kinetic Boltzmann equation asymptotic solution is formulated, the Hilbert's and Enskog's methods are discussed. The equations system of multicomponent non-equilibrium gas dynamics is derived, that corresponds to the first order in the approximate (asymptotic) method for solution of the system of kinetic Boltzmann equations. It is shown, that the velocity distribution functions of particles, obtained by the proposed method and by Enskog's method, within Enskog's approach, are equivalent up to the infinitesimal first order terms of the asymptotic expansion, but, generally speaking, differ in the next order. Interpretation of turbulent gas flow is proposed, as stratified on components gas flow, which is described by the derived equations system of multicomponent non-equilibrium gas dynamics.
A unified lattice Boltzmann model for some nonlinear partial differential equations
Energy Technology Data Exchange (ETDEWEB)
Chai Zhenhua [State Key Laboratory of Coal Combustion, Huazhong University of Science and Technology, Wuhan 430074 (China); Shi Baochang [Department of Mathematics, Huazhong University of Science and Technology, Wuhan 430074 (China)], E-mail: sbchust@126.com; Zheng Lin [Department of Mathematics, Huazhong University of Science and Technology, Wuhan 430074 (China)
2008-05-15
In this paper, a unified and novel lattice Boltzmann model is proposed for solving nonlinear partial differential equation that has the form DU{sub t} + {alpha}UU{sub x} + {beta}U{sup n}U{sub x} - {gamma}U{sub xx} + {delta} U{sub xxx} = F(x,t). Numerical results agree well with the analytical solutions and results derived by existing literature, which indicates the present model is satisfactory and efficient on solving nonlinear partial differential equations.
Steady detonation waves via the Boltzmann equation for a reacting mixture
Conforto, F; Schürrer, F; Ziegler, I
2003-01-01
Based on the Boltzmann equation, the detonation problem is dealt with on a mesoscopic level. The model is based on the assumption that ahead of a shock an explosive gas mixture is in meta stable equilibrium. Starting from the Von Neumann point the chemical reaction, initiated by the pressure jump, proceeds until the chemical equilibrium is reached. Numerical solutions of the derived macroscopic equations as well as the corresponding Hugoniot diagrams which reveal the physical relevance of the mathematical model are provided.
Generalized Poisson—Boltzmann Equation Taking into Account Ionic Interaction and Steric Effects
Liu, Xin-Min; Li, Hang; Li, Rui; Tian, Rui; Xu, Chen-Yang
2012-09-01
Generalized Poisson—Boltzmann equation which takes into account both ionic interaction in bulk solution and steric effects of adsorbed ions has been suggested. We found that, for inorganic cations adsorption on negatively charged surface, the steric effect is not significant for surface charge density 0.15 mol/l in bulk solution. We conclude that for most actual cases the original PB equation can give reliable result in describing inorganic cation adsorption.
Electric-field conditions for Landauer and Boltzmann-Drude conductance equations
Fenton, E. W.
1992-08-01
It is shown explicitly in a unified theory of conductance, for bulk metals and mesoscopic systems, that a Landauer type of conductance equation is compatible with a spatially localized continuous-q-spectrum electric field that is unidirectional, but not with a homogeneous q=0 field. The reverse field condition holds for the Boltzmann-Drude conductance equation for an inhomogeneous bulk metal that has no inelastic scattering. A Feynman-diagram form of Green-function theory shows explicitly the virtual processes and repeated quantum scattering from a single object that occur with Feynman path integrals. The distinction between repeated scattering of current and repeated one-electron scattering is important. For a mesoscopic system, infinite conduction would occur if scattering were to be exactly zero-there is no necessity for postulated contact potentials between lead wires and thermal reservoirs. This is because just in this translationally invariant case a q=0 electric field must occur, and for this the Landauer equation must be replaced by the Boltzmann-Drude equation with zero scattering. In contrast to the strong frequency dependence of the Boltzmann-Drude equation, it is shown that no frequency dependence of the conductance occurs in the Landauer type of equation for frequencies much smaller than the inverse of the electron transit time across the electric-field region.
Sels, Dries; Brosens, Fons
2013-10-01
The equation of motion for the reduced Wigner function of a system coupled to an external quantum system is presented for the specific case when the external quantum system can be modeled as a set of harmonic oscillators. The result is derived from the Wigner function formulation of the Feynman-Vernon influence functional theory. It is shown how the true self-energy for the equation of motion is connected with the influence functional for the path integral. Explicit expressions are derived in terms of the bare Wigner propagator. Finally, we show under which approximations the resulting equation of motion reduces to the Wigner-Boltzmann equation.
Formulating Weak Lensing from the Boltzmann Equation and Application to Lens-lens Couplings
Su, S -C
2014-01-01
The Planck mission has conclusively detected lensing of the Cosmic Microwave Background (CMB) radiation from foreground sources to an overall significance of greater than $25\\sigma$. The high precision of this measurement motivates the development of a more complete formulation of the calculation of this effect. While most effects on the CMB anisotropies are widely studied through direct solutions of the Boltzmann equation, the non-linear effect of CMB lensing is formulated through the solutions of the geodesic equation. In this paper, we present a new formalism to the calculation of the lensing effect by \\emph{directly solving the Boltzmann equation}, as we did in the calculation of the CMB anisotropies at recombination. In particular, we developed a diagrammatic approach to efficiently keep track of all the interaction terms and calculate all possible non-trivial correlations to arbitrary high orders. Using this formalism, we explicitly articulate the approximations required to recover the usual remapping a...
Analytical solution of the Boltzmann-Poisson equation and its application to MIS tunneling junctions
Institute of Scientific and Technical Information of China (English)
Zhang Li-Zhi; Wang Zheng-Chuan
2009-01-01
In order to consider quantum transport under the influence of an electron-electron (e-e) interaction in a mesoscopic conductor, the Boltzmann equation and Poisson equation are investigated jointly. The analytical expressions of the distribution function for the Boltzmann equation and the self-consistent average potential concerned with e-e interaction are obtained, and the dielectric function appearing in the self-consistent average potential is naturally generalized beyond the Thomas-Fermi approximation. Then we apply these results to the tunneling junctions of a metal-insulatorsemiconductor (MIS) in which the electrons are accumulated near the interface of the semiconductor, and we find that the e-e interaction plays an important role in the transport procedure of this system. The electronic density, electric current as well as screening Coulombic potential in this case are studied, and we reveal the time and position dependence of these physical quantities explicitly affected by the e-e interaction.
Li, Q; He, Y L; Wang, Y; Tao, W Q
2007-11-01
A coupled double-distribution-function lattice Boltzmann method is developed for the compressible Navier-Stokes equations. Different from existing thermal lattice Boltzmann methods, this method can recover the compressible Navier-Stokes equations with a flexible specific-heat ratio and Prandtl number. In the method, a density distribution function based on a multispeed lattice is used to recover the compressible continuity and momentum equations, while the compressible energy equation is recovered by an energy distribution function. The energy distribution function is then coupled to the density distribution function via the thermal equation of state. In order to obtain an adjustable specific-heat ratio, a constant related to the specific-heat ratio is introduced into the equilibrium energy distribution function. Two different coupled double-distribution-function lattice Boltzmann models are also proposed in the paper. Numerical simulations are performed for the Riemann problem, the double-Mach-reflection problem, and the Couette flow with a range of specific-heat ratios and Prandtl numbers. The numerical results are found to be in excellent agreement with analytical and/or other solutions.
Directory of Open Access Journals (Sweden)
Nilson C. Roberty
2011-01-01
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.
Robson, R E; Winkler, R; Sigeneger, F
2002-05-01
The Boltzmann equation corresponding to a general "multiterm" representation of the phase space distribution function f(r,c,t) for charged particles in a gas in an electric field was reformulated entirely in terms of spherical tensors f(l)(m) some time ago, and numerous applications, including extension to time varying and crossed electric and magnetic fields, have followed. However, these applications have, by and large, been limited to the hydrodynamic conditions that prevail in swarm experiments and the full potential of the tensor formalism has thus never been realized. This paper resumes the discussion in the context of the more general nonhydrodynamic situation. Geometries for which a simple Legendre polynomial expansion suffices to represent f are discussed briefly, but the emphasis is upon cylindrical geometry, where such simplification does not arise. In particular, we consider an axisymmetric cylindrical column of weakly ionized plasma, and derive an infinite hierarchy of integrodifferential equations for the expansion coefficients of the phase space distribution function, valid for both electrons and ions, and for all types of binary interaction with neutral gas molecules.
Energy Technology Data Exchange (ETDEWEB)
Merton, S.R. [Computational Physics Group, AWE, Aldermaston, Reading, Berkshire RG7 4PR (United Kingdom)], E-mail: simon.merton@awe.co.uk; Pain, C.C. [Computational Physics and Geophysics Group, Department of Earth Science and Engineering, Imperial College London, London SW7 2A7 (United Kingdom); Smedley-Stevenson, R.P. [Computational Physics Group, AWE, Aldermaston, Reading, Berkshire RG7 4PR (United Kingdom); Buchan, A.G.; Eaton, M.D. [Computational Physics and Geophysics Group, Department of Earth Science and Engineering, Imperial College London, London SW7 2A7 (United Kingdom)
2008-09-15
This paper describes the development of two optimal discontinuous finite element (FE) Riemann methods and their application to the one-speed Boltzmann transport equation in the steady-state. The proposed methods optimise the amount of dissipation applied in the streamline direction. This dissipation is applied within an element using a novel Riemann FE method, which is based on an analogy between control volume discretisation methods and finite element methods when integration by parts is applied to the transport terms. In one-dimension the optimal finite element solutions match the analytical solution exactly at each outlet node. Both schemes couple elements in space via a Riemann approach. The first of the two schemes is a Petrov-Galerkin (PG) method which introduces dissipation via the equation residual. The second scheme uses a streamline diffusion stabilisation term in the discretisation. These two methods provide a discontinuous Petrov-Galerkin (DPG) scheme that can stabilise an element across the full range of radiation regimes, obtaining robust solutions with suppressed oscillation. Three basis functions in angle of particle travel have been implemented in an optimal DPG Riemann solver, which include the P{sub N} (spherical harmonic), S{sub N} (discrete ordinate) and LW{sub N} (linear octahedral wavelet) angular expansions. These methods are applied to a series of demanding two-dimensional radiation transport problems.
Barbaroux, Jean-Marie; Hundertmark, Dirk; Ried, Tobias; Vugalter, Semjon
2017-08-01
It has long been suspected that the non-cutoff Boltzmann operator has similar coercivity properties to the fractional Laplacian. This has led to the hope that the homogenous Boltzmann equation enjoys similar regularity properties to the heat equation with a fractional Laplacian. In particular, the weak solution of the fully nonlinear non-cutoff homogenous Boltzmann equation with initial datum in {L^1_2(R^d)\\cap L log L(R^d)}, i.e., finite mass, energy and entropy, should immediately become Gevrey regular for strictly positive times. We prove this conjecture for Maxwellian molecules.
Diffusive Boltzmann equation, its fluid dynamics, Couette flow and Knudsen layers
Abramov, Rafail V
2016-01-01
In the current work we propose a diffusive modification of the Boltzmann equation. This naturally leads to the corresponding diffusive fluid dynamics equations, which we numerically investigate in a simple Couette flow setting. This diffusive modification is based on the assumption of the "imperfect" model collision term, which is unable to track all collisions in the corresponding real gas particle system. The effect of missed collisions is then modeled by an appropriately scaled long-term homogenization process of the particle dynamics. The corresponding diffusive fluid dynamics equations are produced in a standard way by closing the hierarchy of the moment equations using either the Euler or the Grad closure. In the numerical experiments with the Couette flow, we discover that the diffusive Euler equations behave similarly to the conventional Navier-Stokes equations, while the diffusive Grad equations additionally exhibit Knudsen-like velocity boundary layers. We compare the simulations with the correspond...
A novel protocol for linearization of the Poisson-Boltzmann equation
Tsekov, R
2014-01-01
A new protocol for linearization of the Poisson-Boltzmann equation is proposed and the resultant electrostatic equation coincides formally with the Debye-Huckel equation, the solution of which is well known for many electrostatic problems. The protocol is examined on the example of electrostatically stabilized nano-bubbles and it is shown that stable nano-bubbles could be present in aqueous solutions of anionic surfactants near the critical temperature, if the surface potential is constant. At constant surface charge non nano-bubbles could exist.
Global solutions in the critical Besov space for the non-cutoff Boltzmann equation
Morimoto, Yoshinori; Sakamoto, Shota
2016-10-01
The Boltzmann equation is studied without the cutoff assumption. Under a perturbative setting, a unique global solution of the Cauchy problem of the equation is established in a critical Chemin-Lerner space. In order to analyze the collisional term of the equation, a Chemin-Lerner norm is combined with a non-isotropic norm with respect to a velocity variable, which yields an a priori estimate for an energy estimate. Together with local existence following from commutator estimates and the Hahn-Banach extension theorem, the desired solution is obtained.
Energy Technology Data Exchange (ETDEWEB)
EL Safadi, M
2007-03-15
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)
Phonon Boltzmann equation-based discrete unified gas kinetic scheme for multiscale heat transfer
Guo, Zhaoli
2016-01-01
Numerical prediction of multiscale heat transfer is a challenging problem due to the wide range of time and length scales involved. In this work a discrete unified gas kinetic scheme (DUGKS) is developed for heat transfer in materials with different acoustic thickness based on the phonon Boltzmann equation. With discrete phonon direction, the Boltzmann equation is discretized with a second-order finite-volume formulation, in which the time-step is fully determined by the Courant-Friedrichs-Lewy (CFL) condition. The scheme has the asymptotic preserving (AP) properties for both diffusive and ballistic regimes, and can present accurate solutions in the whole transition regime as well. The DUGKS is a self-adaptive multiscale method for the capturing of local transport process. Numerical tests for both heat transfers with different Knudsen numbers are presented to validate the current method.
Xie, Dexuan; Jiang, Yi
2016-10-01
The nonlocal dielectric approach has been studied for more than forty years but only limited to water solvent until the recent work of Xie et al. (2013) [20]. As the development of this recent work, in this paper, a nonlocal modified Poisson-Boltzmann equation (NMPBE) is proposed to incorporate nonlocal dielectric effects into the classic Poisson-Boltzmann equation (PBE) for protein in ionic solvent. The focus of this paper is to present an efficient finite element algorithm and a related software package for solving NMPBE. Numerical results are reported to validate this new software package and demonstrate its high performance for protein molecules. They also show the potential of NMPBE as a better predictor of electrostatic solvation and binding free energies than PBE.
A Fokker-Planck Model of the Boltzmann Equation with Correct Prandtl Number for Polyatomic Gases
Mathiaud, J.; Mieussens, L.
2017-09-01
We propose an extension of the Fokker-Planck model of the Boltzmann equation to get a correct Prandtl number in the Compressible Navier-Stokes asymptotics for polyatomic gases. This is obtained by replacing the diffusion coefficient (which is the equilibrium temperature) by a non diagonal temperature tensor, like the Ellipsoidal-Statistical model is obtained from the Bathnagar-Gross-Krook model of the Boltzmann equation, and by adding a diffusion term for the internal energy. Our model is proved to satisfy the properties of conservation and a H-theorem. A Chapman-Enskog analysis shows how to compute the transport coefficients of our model. Some numerical tests are performed to illustrate that a correct Prandtl number can be obtained.
A Fokker-Planck Model of the Boltzmann Equation with Correct Prandtl Number for Polyatomic Gases
Mathiaud, J.; Mieussens, L.
2017-07-01
We propose an extension of the Fokker-Planck model of the Boltzmann equation to get a correct Prandtl number in the Compressible Navier-Stokes asymptotics for polyatomic gases. This is obtained by replacing the diffusion coefficient (which is the equilibrium temperature) by a non diagonal temperature tensor, like the Ellipsoidal-Statistical model is obtained from the Bathnagar-Gross-Krook model of the Boltzmann equation, and by adding a diffusion term for the internal energy. Our model is proved to satisfy the properties of conservation and a H-theorem. A Chapman-Enskog analysis shows how to compute the transport coefficients of our model. Some numerical tests are performed to illustrate that a correct Prandtl number can be obtained.
Premnath, Kannan N; Banerjee, Sanjoy
2008-01-01
Several applications exist in which lattice Boltzmann methods (LBM) are used to compute stationary states of fluid motions, particularly those driven or modulated by external forces. Standard LBM, being explicit time-marching in nature, requires a long time to attain steady state convergence, particularly at low Mach numbers due to the disparity in characteristic speeds of propagation of different quantities. In this paper, we present a preconditioned generalized lattice Boltzmann equation (GLBE) with forcing term to accelerate steady state convergence to flows driven by external forces. The use of multiple relaxation times in the GLBE allows enhancement of the numerical stability. Particular focus is given in preconditioning external forces, which can be spatially and temporally dependent. In particular, correct forms of moment-projections of source/forcing terms are derived such that they recover preconditioned Navier-Stokes equations with non-uniform external forces. As an illustration, we solve an extende...
Adaptive Finite Element Modeling Techniques for the Poisson-Boltzmann Equation
Holst, Michael; Yu, Zeyun; Zhou, Yongcheng; Zhu, Yunrong
2010-01-01
We develop an efficient and reliable adaptive finite element method (AFEM) for the nonlinear Poisson-Boltzmann equation (PBE). We first examine the regularization technique of Chen, Holst, and Xu; this technique made possible the first a priori pointwise estimates and the first complete solution and approximation theory for the Poisson-Boltzmann equation. It also made possible the first provably convergent discretization of the PBE, and allowed for the development of a provably convergent AFEM for the PBE. However, in practice the regularization turns out to be numerically ill-conditioned. In this article, we examine a second regularization, and establish a number of basic results to ensure that the new approach produces the same mathematical advantages of the original regularization, without the ill-conditioning property. We then design an AFEM scheme based on the new regularized problem, and show that the resulting AFEM scheme is accurate and reliable, by proving a contraction result for the error. This res...
Asinari, Pietro
2010-01-01
The homogeneous isotropic Boltzmann equation (HIBE) is a fundamental dynamic model for many applications in thermodynamics, econophysics and sociodynamics. Despite recent hardware improvements, the solution of the Boltzmann equation remains extremely challenging from the computational point of view, in particular by deterministic methods (free of stochastic noise). This work aims to improve a deterministic direct method recently proposed [V.V. Aristov, Kluwer Academic Publishers, 2001] for solving the HIBE with a generic collisional kernel and, in particular, for taking care of the late dynamics of the relaxation towards the equilibrium. Essentially (a) the original problem is reformulated in terms of particle kinetic energy (exact particle number and energy conservation during microscopic collisions) and (b) the computation of the relaxation rates is improved by the DVM-like correction, where DVM stands for Discrete Velocity Model (ensuring that the macroscopic conservation laws are exactly satisfied). Both ...
A new lattice Boltzmann equation to simulate density-driven convection of carbon dioxide
Allen, Rebecca
2013-01-01
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
A new exact solution of the relativistic Boltzmann equation and its hydrodynamic limit
Denicol, Gabriel S; Martinez, Mauricio; Noronha, Jorge; Strickland, Michael
2014-01-01
We present an exact solution of the relativistic Boltzmann equation for a system undergoing boost-invariant longitudinal and azimuthally symmetric transverse flow ("Gubser flow"). The resulting exact non-equilibrium dynamics is compared to 1st- and 2nd-order relativistic hydrodynamic approximations for various shear viscosity to entropy density ratios. This novel solution can be used to test the validity and accuracy of different hydrodynamic approximations in conditions similar to those generated in relativistic heavy-ion collisions.
Inhomogeneous relativistic Boltzmann equation near vacuum in the Robertson-Walker space-time
Takou, Etienne
2016-01-01
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. The collision kernel considered here is for the hard potentials case and the background space-time in which the study is done is the Robertson-Walker space-time. Unique global (in time) mild solution is obtained in a suitable weighted space.
The Relativistic Boltzmann Equation on Bianchi Type I Space Time for Hard Potentials
Noutchegueme, Norbert; Takou, Etienne; Tchuengue, E. Kamdem
2017-08-01
In this paper, we consider the Cauchy problem for the spatially homogeneous relativistic Boltzmann equation with small initial data. The collision kernel considered here is for a hard potentials case. The background space-time in which the study is done is the Bianchi type I space-time. Under certain conditions made on the scattering kernel and on the metric, a uniqueness global (in time) solution is obtained in a suitable weighted functional space.
Deterministic numerical solutions of the Boltzmann equation using the fast spectral method
Wu, Lei; White, Craig; Scanlon, Thomas J.; Reese, Jason M.; Zhang, Yonghao
2013-10-01
The Boltzmann equation describes the dynamics of rarefied gas flows, but the multidimensional nature of its collision operator poses a real challenge for its numerical solution. In this paper, the fast spectral method [36], originally developed by Mouhot and Pareschi for the numerical approximation of the collision operator, is extended to deal with other collision kernels, such as those corresponding to the soft, Lennard-Jones, and rigid attracting potentials. The accuracy of the fast spectral method is checked by comparing our numerical solutions of the space-homogeneous Boltzmann equation with the exact Bobylev-Krook-Wu solutions for a gas of Maxwell molecules. It is found that the accuracy is improved by replacing the trapezoidal rule with Gauss-Legendre quadrature in the calculation of the kernel mode, and the conservation of momentum and energy are ensured by the Lagrangian multiplier method without loss of spectral accuracy. The relax-to-equilibrium processes of different collision kernels with the same value of shear viscosity are then compared; the numerical results indicate that different forms of the collision kernels can be used as long as the shear viscosity (not only the value, but also its temperature dependence) is recovered. An iteration scheme is employed to obtain stationary solutions of the space-inhomogeneous Boltzmann equation, where the numerical errors decay exponentially. Four classical benchmarking problems are investigated: the normal shock wave, and the planar Fourier/Couette/force-driven Poiseuille flows. For normal shock waves, our numerical results are compared with a finite difference solution of the Boltzmann equation for hard sphere molecules, experimental data, and molecular dynamics simulation of argon using the realistic Lennard-Jones potential. For planar Fourier/Couette/force-driven Poiseuille flows, our results are compared with the direct simulation Monte Carlo method. Excellent agreements are observed in all test cases
EXISTENCE OF INFINITE ENERGY SOLUTION TO THE INELASTIC BOLTZMANN EQUATION WITH EXTERNAL FORCE
Institute of Scientific and Technical Information of China (English)
Wei Jinbo; Zhang Xianwen
2012-01-01
In this paper,the Cauchy problem for the inelastic Boltzmann equation with external force is considered in the case of initial data with infinite energy.More precisely,under the assumptions on the bicharacteristic generated by external force,we prove the global existence of solution for small initial data compared to the local Maxwellian exp{-p|x-v|2},which has infinite mass and energy.
Gas kinetic algorithm for flows in Poiseuille-like microchannels using Boltzmann model equation
Institute of Scientific and Technical Information of China (English)
LI; Zhihui; ZHANG; Hanxin; FU; Song
2005-01-01
The gas-kinetic unified algorithm using Boltzmann model equation have been extended and developed to solve the micro-scale gas flows in Poiseuille-like micro-channels from Micro-Electro-Mechanical Systems (MEMS). The numerical modeling of the gas kinetic boundary conditions suitable for micro-scale gas flows is presented. To test the present method, the classical Couette flows with various Knudsen numbers, the gas flows from short microchannels like plane Poiseuille and the pressure-driven gas flows in two-dimensional short microchannels have been simulated and compared with the approximate solutions of the Boltzmann equation, the related DSMC results, the modified N-S solutions with slip-flow boundary theory, the gas-kinetic BGK-Burnett solutions and the experimental data. The comparisons show that the present gas-kinetic numerical algorithm using the mesoscopic Boltzmann simplified velocity distribution function equation can effectively simulate and reveal the gas flows in microchannels. The numerical experience indicates that this method may be a powerful tool in the numerical simulation of micro-scale gas flows from MEMS.
Full-deautonomisation of a lattice equation
Willox, R.; Mase, T.; Ramani, A.; Grammaticos, B.
2016-07-01
In this letter we report on the unexpected possibility of applying the full-deautonomisation approach we recently proposed for predicting the algebraic entropy of second-order birational mappings, to discrete lattice equations. Moreover, we show, on two examples, that the full-deautonomisation technique can in fact also be successfully applied to reductions of these lattice equations to mappings with orders higher than 2. In particular, we apply this technique to a recently discovered lattice equation that has confined singularities while being nonintegrable, and we show that our approach accurately predicts this nonintegrable character. Finally, we demonstrate how our method can even be used to predict the algebraic entropy for some nonconfining higher order mappings.
Bouchard, Hugo; Bielajew, Alex
2015-07-07
To establish a theoretical framework for generalizing Monte Carlo transport algorithms by adding external electromagnetic fields to the Boltzmann radiation transport equation in a rigorous and consistent fashion. Using first principles, the Boltzmann radiation transport equation is modified by adding a term describing the variation of the particle distribution due to the Lorentz force. The implications of this new equation are evaluated by investigating the validity of Fano's theorem. Additionally, Lewis' approach to multiple scattering theory in infinite homogeneous media is redefined to account for the presence of external electromagnetic fields. The equation is modified and yields a description consistent with the deterministic laws of motion as well as probabilistic methods of solution. The time-independent Boltzmann radiation transport equation is generalized to account for the electromagnetic forces in an additional operator similar to the interaction term. Fano's and Lewis' approaches are stated in this new equation. Fano's theorem is found not to apply in the presence of electromagnetic fields. Lewis' theory for electron multiple scattering and moments, accounting for the coupling between the Lorentz force and multiple elastic scattering, is found. However, further investigation is required to develop useful algorithms for Monte Carlo and deterministic transport methods. To test the accuracy of Monte Carlo transport algorithms in the presence of electromagnetic fields, the Fano cavity test, as currently defined, cannot be applied. Therefore, new tests must be designed for this specific application. A multiple scattering theory that accurately couples the Lorentz force with elastic scattering could improve Monte Carlo efficiency. The present study proposes a new theoretical framework to develop such algorithms.
Denicol, Gabriel S; Martinez, Mauricio; Noronha, Jorge; Strickland, Michael
2014-01-01
We present an exact solution to the Boltzmann equation which describes a system undergoing boost-invariant longitudinal and azimuthally symmetric radial expansion for arbitrary shear viscosity to entropy density ratio. This new solution is constructed by considering the conformal map between Minkowski space and the direct product of three dimensional de Sitter space with a line. The resulting solution respects SO(3)_q x SO(1,1) x Z_2 symmetry. We compare the exact kinetic solution with exact solutions of the corresponding macroscopic equations with the same symmetry that were obtained from the kinetic theory in ideal and second-order viscous hydrodynamic approximations.
Solution Poisson-Boltzmann equation: Application in the Human Neuron Membrane
Soares, M A G; Cortez, C M
2008-01-01
With already demonstrated in previous work the equations that describe the space dependence of the electric potential are determined by the solution of the equation of Poisson-Boltzmann. In this work we consider these solutions for the membrane of the human neuron, using a model simplified for this structure considering the distribution of electrolytes in each side of the membrane, as well as the effect of glycocalyx and the lipidic bilayer. It was assumed that on both sides of the membrane the charges are homogeneously distributed and that the potential depends only on coordinate z.
From Newton's Law to the Linear Boltzmann Equation Without Cut-Off
Ayi, Nathalie
2017-01-01
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.
From Newton's Law to the Linear Boltzmann Equation Without Cut-Off
Ayi, Nathalie
2017-03-01
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.
Theory of the Lattice Boltzmann Equation: Symmetry properties of Discrete Velocity Sets
Rubinstein, Robert; Luo, Li-Shi
2007-01-01
In the lattice Boltzmann equation, continuous particle velocity space is replaced by a finite dimensional discrete set. The number of linearly independent velocity moments in a lattice Boltzmann model cannot exceed the number of discrete velocities. Thus, finite dimensionality introduces linear dependencies among the moments that do not exist in the exact continuous theory. Given a discrete velocity set, it is important to know to exactly what order moments are free of these dependencies. Elementary group theory is applied to the solution of this problem. It is found that by decomposing the velocity set into subsets that transform among themselves under an appropriate symmetry group, it becomes relatively straightforward to assess the behavior of moments in the theory. The construction of some standard two- and three-dimensional models is reviewed from this viewpoint, and procedures for constructing some new higher dimensional models are suggested.
Perturbative and non-perturbative aspects non-Abelian Boltzmann-Langevin equation
Energy Technology Data Exchange (ETDEWEB)
Boedeker, Dietrich. E-mail: bodeker@physik.uni-bielefeld.de
2002-12-30
We study the Boltzmann-Langevin equation which describes the dynamics of hot Yang-Mills fields with typical momenta of order of the magnetic screening scale g{sup 2}T. It is transformed into a path integral and Feynman rules are obtained. We find that the leading log Langevin equation can be systematically improved in a well behaved expansion in log(1/g){sup -1}. The result by Arnold and Yaffe that the leading log Langevin equation is still valid at next-to-leading-log order is confirmed. We also confirm their result for the next-to-leading-log damping coefficient, or color conductivity, which is shown to be gauge fixing independent for a certain class of gauges. The frequency scale g{sup 2}T does not contribute to this result, but it does contribute, by power counting, to the transverse gauge field propagator. Going beyond a perturbative expansion we find 1-loop ultraviolet divergences which cannot be removed by renormalizing the parameters in the Boltzmann-Langevin equation.
Asinari, P.
2011-03-01
Boltzmann equation is one the most powerful paradigms for explaining transport phenomena in fluids. Since early fifties, it received a lot of attention due to aerodynamic requirements for high altitude vehicles, vacuum technology requirements and nowadays, micro-electro-mechanical systems (MEMs). Because of the intrinsic mathematical complexity of the problem, Boltzmann himself started his work by considering first the case when the distribution function does not depend on space (homogeneous case), but only on time and the magnitude of the molecular velocity (isotropic collisional integral). The interest with regards to the homogeneous isotropic Boltzmann equation goes beyond simple dilute gases. In the so-called econophysics, a Boltzmann type model is sometimes introduced for studying the distribution of wealth in a simple market. Another recent application of the homogeneous isotropic Boltzmann equation is given by opinion formation modeling in quantitative sociology, also called socio-dynamics or sociophysics. The present work [1] aims to improve the deterministic method for solving homogenous isotropic Boltzmann equation proposed by Aristov [2] by two ideas: (a) the homogeneous isotropic problem is reformulated first in terms of particle kinetic energy (this allows one to ensure exact particle number and energy conservation during microscopic collisions) and (b) a DVM-like correction (where DVM stands for Discrete Velocity Model) is adopted for improving the relaxation rates (this allows one to satisfy exactly the conservation laws at macroscopic level, which is particularly important for describing the late dynamics in the relaxation towards the equilibrium).
Fokker-Planck Equation for Boltzmann-type and Active Particles transfer probability approach
Trigger, S A
2002-01-01
Fokker-Planck equation with the velocity-dependent coefficients is considered for various isotropic systems on the basis of probability transition (PT) approach. This method provides the self-consistent and universal description of friction and diffusion for Brownian particles. Renormalization of the friction coefficient is shown to occur for two dimensional (2-D) and three dimensional (3-D) cases, due to the tensorial character of diffusion. The specific forms of PT are calculated for the Boltzmann-type of collisions and for the absorption-type of collisions (the later are typical for dusty plasmas and some other systems). Validity of the Einstein's relation for the Boltzmann-type collisions is proved for the velocity-dependent friction and diffusion coefficients. For non-Boltzmann collisions, such as, e.g., absorption collisions, the Einstein relation is violated, although some other relations (determined by the structure of PT) can exist. The collecting part of the ion drag force in a dusty plasma, arising...
Energy Technology Data Exchange (ETDEWEB)
Ayissi, Raoul Domingo, E-mail: raoulayissi@yahoo.fr; Noutchegueme, Norbert, E-mail: nnoutch@yahoo.fr [Department of Mathematics, Faculty of Science, University of Yaounde I, P.O. Box 812, Yaounde (Cameroon)
2015-01-15
Global solutions regular for the Einstein-Boltzmann equation on a magnetized Bianchi type-I cosmological model with the cosmological constant are investigated. We suppose that the metric is locally rotationally symmetric. The Einstein-Boltzmann equation has been already considered by some authors. But, in general Bancel and Choquet-Bruhat [Ann. Henri Poincaré XVIII(3), 263 (1973); Commun. Math. Phys. 33, 83 (1973)], they proved only the local existence, and in the case of the nonrelativistic Boltzmann equation. Mucha [Global existence of solutions of the Einstein-Boltzmann equation in the spatially homogeneous case. Evolution equation, existence, regularity and singularities (Banach Center Publications, Institute of Mathematics, Polish Academy of Science, 2000), Vol. 52] obtained a global existence result, for the relativistic Boltzmann equation coupled with the Einstein equations and using the Yosida operator, but confusing unfortunately with the nonrelativistic case. Noutchegueme and Dongho [Classical Quantum Gravity 23, 2979 (2006)] and Noutchegueme, Dongho, and Takou [Gen. Relativ. Gravitation 37, 2047 (2005)], have obtained a global solution in time, but still using the Yosida operator and considering only the uncharged case. Noutchegueme and Ayissi [Adv. Stud. Theor. Phys. 4, 855 (2010)] also proved a global existence of solutions to the Maxwell-Boltzmann system using the characteristic method. In this paper, we obtain using a method totally different from those used in the works of Noutchegueme and Dongho [Classical Quantum Gravity 23, 2979 (2006)], Noutchegueme, Dongho, and Takou [Gen. Relativ. Gravitation 37, 2047 (2005)], Noutchegueme and Ayissi [Adv. Stud. Theor. Phys. 4, 855 (2010)], and Mucha [Global existence of solutions of the Einstein-Boltzmann equation in the spatially homogeneous case. Evolution equation, existence, regularity and singularities (Banach Center Publications, Institute of Mathematics, Polish Academy of Science, 2000), Vol. 52] the
Measure Valued Solutions to the Spatially Homogeneous Boltzmann Equation Without Angular Cutoff
Morimoto, Yoshinori; Wang, Shuaikun; Yang, Tong
2016-12-01
A uniform approach is introduced to study the existence of measure valued solutions to the homogeneous Boltzmann equation for both hard potential with finite energy, and soft potential with finite or infinite energy, by using Toscani metric. Under the non-angular cutoff assumption on the cross-section, the solutions obtained are shown to be in the Schwartz space in the velocity variable as long as the initial data is not a single Dirac mass without any extra moment condition for hard potential, and with the boundedness on moments of any order for soft potential.
On measure solutions of the Boltzmann equation, part I: Moment production and stability estimates
Lu, Xuguang; Mouhot, Clément
The spatially homogeneous Boltzmann equation with hard potentials is considered for measure valued initial data having finite mass and energy. We prove the existence of weak measure solutions, with and without angular cutoff on the collision kernel; the proof in particular makes use of an approximation argument based on the Mehler transform. Moment production estimates in the usual form and in the exponential form are obtained for these solutions. Finally for the Grad angular cutoff, we also establish uniqueness and strong stability estimate on these solutions.
Heinz, U; Denicol, G S; Martinez, M; Nopoush, M; Noronha, J; Ryblewski, R; Strickland, M
2015-01-01
Several recent results are reported from work aiming to improve the quantitative precision of relativistic viscous fluid dynamics for relativistic heavy-ion collisions. The dense matter created in such collisions expands in a highly anisotropic manner. Due to viscous effects this also renders the local momentum distribution anisotropic. Optimized hydrodynamic approaches account for these anisotropies already at leading order in a gradient expansion. Recently discovered exact solutions of the relativistic Boltzmann equation in anisotropically expanding systems provide a powerful testbed for such improved hydrodynamic approximations. We present the latest status of our quest for a formulation of relativistic viscous fluid dynamics that is optimized for applications to relativistic heavy-ion collisions.
Numerical solution of the Boltzmann equation for the shock wave in a gas mixture
Raines, A A
2014-01-01
We study the structure of a shock wave for a two-, three- and four-component gas mixture on the basis of numerical solution of the Boltzmann equation for the model of hard sphere molecules. For the evaluation of collision integrals we use the Conservative Projection Method developed by F.G. Tscheremissine which we extended to gas mixtures in cylindrical coordinates. The transition from the upstream to downstream uniform state is presented by macroscopic values and distribution functions. The obtained results were compared with numerical and experimental results of other authors.
A novel incompressible finite-difference lattice Boltzmann equation for particle-laden flow
Institute of Scientific and Technical Information of China (English)
Sheng Chen; Zhaohui Liu; Baochang Shi; Zhu He; Chuguang Zheng
2005-01-01
In this paper, we propose a novel incompressible finite-difference lattice Boltzmann Equation (FDLBE). Because source terms that reflect the interaction between phases can be accurately described, the new model is suitable for simulating two-way coupling incompressible multiphase flow.The 2-D particle-laden flow over a backward-facing step is chosen as a test case to validate the present method. Favorable results are obtained and the present scheme is shown to have good prospects in practical applications.
Solution of the Boltzmann Equation for Electrons in Laser-Heated Metals
Pietanza, L. D.; Colonna, G.; Capitelli, M.
2005-05-01
A kinetic study of the electron relaxation dynamic inside a noble metal film (Ag) subjected to a femtosecond laser pulse has been performed. A time dependent numerical algorithm for the solution of the Boltzmann equations for electrons and phonons inside the film has been developped, considering electron-electron and electron-phonon collisions and the laser perturbation. The dependence of electron-electron and electron-phonon characteristic time-scales on the screening parameter values has been investigated. Electron-electron relaxation times are also compared with experimental data obtained through time-resolved two-photon photoemission technique.
Fares, Ehab; Duda, Benjamin; Khorrami, Mehdi R.
2016-01-01
Unsteady flow computations are presented for a Gulfstream aircraft model in landing configuration, i.e., flap deflected 39deg and main landing gear deployed. The simulations employ the lattice Boltzmann solver PowerFLOW(Trademark) to simultaneously capture the flow physics and acoustics in the near field. Sound propagation to the far field is obtained using a Ffowcs Williams and Hawkings acoustic analogy approach. Two geometry representations of the same aircraft are analyzed: an 18% scale, high-fidelity, semi-span model at wind tunnel Reynolds number and a full-scale, full-span model at half-flight Reynolds number. Previously published and newly generated model-scale results are presented; all full-scale data are disclosed here for the first time. Reynolds number and geometrical fidelity effects are carefully examined to discern aerodynamic and aeroacoustic trends with a special focus on the scaling of surface pressure fluctuations and farfield noise. An additional study of the effects of geometrical detail on farfield noise is also documented. The present investigation reveals that, overall, the model-scale and full-scale aeroacoustic results compare rather well. Nevertheless, the study also highlights that finer geometrical details that are typically not captured at model scales can have a non-negligible contribution to the farfield noise signature.
Leptogenesis with heavy neutrino flavours: from density matrix to Boltzmann equations
Blanchet, Steve; Di Bari, Pasquale; Marzola, Luca
2011-01-01
Leptogenesis with heavy neutrino flavours is discussed within a density matrix formalism. We write the density matrix equation that describes the generation of the matter-antimatter asymmetry, for an arbitrary choice of the right-handed (RH) neutrino masses. For hierarchical RH neutrino masses lying in the fully flavoured regimes, the density matrix equation reduces to multiple-stage Boltzmann equations. In this case we recover and extend results previously derived within a quantum state collapse description. We confirm the generic existence of phantom terms, which are not washed out at production and contribute to the flavoured asymmetries proportionally to the initial RH neutrino abundances. Even in the N_1-dominated scenario they can give rise to lepton flavour asymmetries much larger than the baryon asymmetry with potential applications. We also confirm that there is a (orthogonal) component in the asymmetry produced by the heavier RH neutrinos which completely escapes the washout from the lighter RH neut...
Tervo, J; Frank, M; Herty, M
2016-01-01
The paper considers a coupled system of linear Boltzmann transport equation (BTE), and its Continuous Slowing Down Approximation (CSDA). This system can be used to model the relevant transport of particles used e.g. in dose calculation in radiation therapy. The evolution of charged particles (e.g. electrons and positrons) are in practice often modelled using the CSDA version of BTE because of the so-called forward peakedness of scattering events contributing to the particle fluencies (or particle densities), which causes severe problems for numerical methods. First, we prove the existence and uniqueness of solutions, under sufficient criteria and in appropriate $L^2$-based spaces, of a single (particle) CSDA-equation by using two complementary techniques, the Lions-Lax-Milgram Theorem (variational approach), and the theory evolution operators (semigroup approach). The necessary a priori estimates are shown. In addition, we prove the corresponding results and estimates for the system of coupled transport equat...
Continuous surface force based lattice Boltzmann equation method for simulating thermocapillary flow
Zheng, Lin; Zhai, Qinglan
2014-01-01
In this paper, we extend a lattice Boltzmann equation (LBE) with continuous surface fore (CSF) to simulate thermocapillary flows. The model is designed on our previous CSF LBE for athermal two phase flow, in which the interfacial tension forces and the Marangoni stresses as the results of the interface interactions between different phases are described by a conception of CSF. In this model, the sharp interfaces between different phases are separated by a narrow transition layers, and the kinetics and morphology evolution of phase separation would be characterized by an order parameter visa Cahn-Hilliard equation which is solved in the frame work of LBE. The scalar convection-diffusion equation for temperature field is also solved by thermal LBE. The models are validated by thermal two layered Poiseuille flow, and a two superimposed planar fluids at negligibly small Reynolds and Marangoni numbers for the thermocapillary driven convection, which have analytical solutions for the velocity and temperature. Then ...
Cobos, Agustín C.; Poma, Ana L.; Alvarez, Guillermo D.; Sanz, Darío E.
2016-10-01
We introduce an alternative method to calculate the steady state solution of the angular photon flux after a numerical evolution of the time-dependent Boltzmann transport equation (BTE). After a proper discretization the transport equation was converted into an ordinary system of differential equations that can be iterated as a weighted Richardson algorithm. As a different approach, in this work the time variable regulates the iteration process and convergence criteria is based on physical parameters. Positivity and convergence was assessed from first principles and a modified Courant-Friedrichs-Lewy condition was devised to guarantee convergence. The Penelope Monte Carlo method was used to test the convergence and accuracy of our approach for different phase space discretizations. Benchmarking was performed by calculation of total fluence and photon spectra in different one-dimensional geometries irradiated with 60Co and 6 MV photon beams and radiological applications were devised.
Directory of Open Access Journals (Sweden)
G Boroni
2017-03-01
Full Text Available Lattice Boltzmann Method (LBM has shown great potential in fluid simulations, but performance issues and difficulties to manage complex boundary conditions have hindered a wider application. The upcoming of Graphic Processing Units (GPU Computing offered a possible solution for the performance issue, and methods like the Immersed Boundary (IB algorithm proved to be a flexible solution to boundaries. Unfortunately, the implicit IB algorithm makes the LBM implementation in GPU a non-trivial task. This work presents a fully parallel GPU implementation of LBM in combination with IB. The fluid-boundary interaction is implemented via GPU kernels, using execution configurations and data structures specifically designed to accelerate each code execution. Simulations were validated against experimental and analytical data showing good agreement and improving the computational time. Substantial reductions of calculation rates were achieved, lowering down the required time to execute the same model in a CPU to about two magnitude orders.
Wang, Huimin
2017-01-01
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.
Harris, Robert C; Boschitsch, Alexander H; Fenley, Marcia O
2017-08-08
Many researchers compute surface maps of the electrostatic potential (φ) with the Poisson-Boltzmann (PB) equation to relate the structural information obtained from X-ray and NMR experiments to biomolecular functions. Here we demonstrate that the usual method of obtaining these surface maps of φ, by interpolating from neighboring grid points on the solution grid generated by a PB solver, generates large errors because of the large discontinuity in the dielectric constant (and thus in the normal derivative of φ) at the surface. The Cartesian Poisson-Boltzmann solver contains several features that reduce the numerical noise in surface maps of φ: First, CPB introduces additional mesh points at the Cartesian grid/surface intersections where the PB equation is solved. This procedure ensures that the solution for interior mesh points only references nodes on the interior or on the surfaces; similarly for exterior points. Second, for added points on the surface, a second order least-squares reconstruction (LSR) is implemented that analytically incorporates the discontinuities at the surface. LSR is used both during the solution phase to compute φ at the surface and during postprocessing to obtain φ, induced charges, and ionic pressures. Third, it uses an adaptive grid where the finest grid cells are located near the molecular surface.
Sensitivities to parameterization in the size-modified Poisson-Boltzmann equation.
Harris, Robert C; Boschitsch, Alexander H; Fenley, Marcia O
2014-02-21
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.
Thermal transport at the nanoscale: A Fourier's law vs. phonon Boltzmann equation study
Kaiser, J.; Feng, T.; Maassen, J.; Wang, X.; Ruan, X.; Lundstrom, M.
2017-01-01
Steady-state thermal transport in nanostructures with dimensions comparable to the phonon mean-free-path is examined. Both the case of contacts at different temperatures with no internal heat generation and contacts at the same temperature with internal heat generation are considered. Fourier's law results are compared to finite volume method solutions of the phonon Boltzmann equation in the gray approximation. When the boundary conditions are properly specified, results obtained using Fourier's law without modifying the bulk thermal conductivity are in essentially exact quantitative agreement with the phonon Boltzmann equation in the ballistic and diffusive limits. The errors between these two limits are examined in this paper. For the four cases examined, the error in the apparent thermal conductivity as deduced from a correct application of Fourier's law is less than 6%. We also find that the Fourier's law results presented here are nearly identical to those obtained from a widely used ballistic-diffusive approach but analytically much simpler. Although limited to steady-state conditions with spatial variations in one dimension and to a gray model of phonon transport, the results show that Fourier's law can be used for linear transport from the diffusive to the ballistic limit. The results also contribute to an understanding of how heat transport at the nanoscale can be understood in terms of the conceptual framework that has been established for electron transport at the nanoscale.
Düring, Bertram
2015-01-01
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.
A New 2D-Transport, 1D-Diffusion Approximation of the Boltzmann Transport equation
Energy Technology Data Exchange (ETDEWEB)
Larsen, Edward
2013-06-17
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.
Jungemann, C.; Pham, A. T.; Meinerzhagen, B.; Ringhofer, C.; Bollhöfer, M.
2006-07-01
The Boltzmann equation for transport in semiconductors is projected onto spherical harmonics in such a way that the resultant balance equations for the coefficients of the distribution function times the generalized density of states can be discretized over energy and real spaces by box integration. This ensures exact current continuity for the discrete equations. Spurious oscillations of the distribution function are suppressed by stabilization based on a maximum entropy dissipation principle avoiding the H transformation. The derived formulation can be used on arbitrary grids as long as box integration is possible. The approach works not only with analytical bands but also with full band structures in the case of holes. Results are presented for holes in bulk silicon based on a full band structure and electrons in a Si NPN bipolar junction transistor. The convergence of the spherical harmonics expansion is shown for a device, and it is found that the quasiballistic transport in nanoscale devices requires an expansion of considerably higher order than the usual first one. The stability of the discretization is demonstrated for a range of grid spacings in the real space and bias points which produce huge gradients in the electron density and electric field. It is shown that the resultant large linear system of equations can be solved in a memory efficient way by the numerically robust package ILUPACK.
AQUASOL: An efficient solver for the dipolar Poisson-Boltzmann-Langevin equation.
Koehl, Patrice; Delarue, Marc
2010-02-14
The Poisson-Boltzmann (PB) formalism is among the most popular approaches to modeling the solvation of molecules. It assumes a continuum model for water, leading to a dielectric permittivity that only depends on position in space. In contrast, the dipolar Poisson-Boltzmann-Langevin (DPBL) formalism represents the solvent as a collection of orientable dipoles with nonuniform concentration; this leads to a nonlinear permittivity function that depends both on the position and on the local electric field at that position. The differences in the assumptions underlying these two models lead to significant differences in the equations they generate. The PB equation is a second order, elliptic, nonlinear partial differential equation (PDE). Its response coefficients correspond to the dielectric permittivity and are therefore constant within each subdomain of the system considered (i.e., inside and outside of the molecules considered). While the DPBL equation is also a second order, elliptic, nonlinear PDE, its response coefficients are nonlinear functions of the electrostatic potential. Many solvers have been developed for the PB equation; to our knowledge, none of these can be directly applied to the DPBL equation. The methods they use may adapt to the difference; their implementations however are PBE specific. We adapted the PBE solver originally developed by Holst and Saied [J. Comput. Chem. 16, 337 (1995)] to the problem of solving the DPBL equation. This solver uses a truncated Newton method with a multigrid preconditioner. Numerical evidences suggest that it converges for the DPBL equation and that the convergence is superlinear. It is found however to be slow and greedy in memory requirement for problems commonly encountered in computational biology and computational chemistry. To circumvent these problems, we propose two variants, a quasi-Newton solver based on a simplified, inexact Jacobian and an iterative self-consistent solver that is based directly on the PBE
Institute of Scientific and Technical Information of China (English)
YUAN Hongjun; YAN Han
2009-01-01
The existence and uniqueness of the solutions for the Boltzmann equations with measures as initial value are still an open problem which is posed by P. L. Lions (2000). The aim of this paper is to discuss the Cauchy problem of the system of discrete Boltzmann equations of the form etf∫i+(∫mii)x=Qi(∫1,∫2,…,∫n), (mi1, i=1,…,n) with non-negative finite Radon measures as initial conditions. In particular, the existence and uniqueness of BV solutions for the above problem are obtained.
A multi-term solution of the space-time Boltzmann equation for electrons in gaseous and liquid Argon
Boyle, G J; Tattersall, W J; McEachran, R P; White, R D
2015-01-01
In a recent paper [1] the scattering and transport of excess electrons in liquid argon in the hydrodynamic regime was investigated, generalizing the seminal works of Lekner and Cohen [2,3] with modern scattering theory techniques and kinetic theory. In this paper, the discussion is extended to the non-hydrodynamic regime through the development of a full multi-term space-time solution of Boltzmann's equation for electron transport in gases and liquids using a novel operator-splitting method. A Green's function formalism is considered that enables flexible adaptation to various experimental systems. The spatio-temporal evolution of electrons in liquids in the hydrodynamic regime is studied for a benchmark model Percus-Yevick liquid as well as for liquid argon. The temporal evolution of Franck-Hertz oscillations are observed for liquids, with striking differences in the spatio-temporal development of the velocity distribution function components between the uncorrelated gas and true liquid approximations in arg...
Botello-Smith, Wesley M.; Luo, Ray
2016-01-01
Continuum solvent models have been widely used in biomolecular modeling applications. Recently much attention has been given to inclusion of implicit membrane into existing continuum Poisson-Boltzmann solvent models to extend their applications to membrane systems. Inclusion of an implicit membrane complicates numerical solutions of the underlining Poisson-Boltzmann equation due to the dielectric inhomogeneity on the boundary surfaces of a computation grid. This can be alleviated by the use of the periodic boundary condition, a common practice in electrostatic computations in particle simulations. The conjugate gradient and successive over-relaxation methods are relatively straightforward to be adapted to periodic calculations, but their convergence rates are quite low, limiting their applications to free energy simulations that require a large number of conformations to be processed. To accelerate convergence, the Incomplete Cholesky preconditioning and the geometric multi-grid methods have been extended to incorporate periodicity for biomolecular applications. Impressive convergence behaviors were found as in the previous applications of these numerical methods to tested biomolecules and MMPBSA calculations. PMID:26389966
Arnab, Sarkar; Manjeet, Singh
2017-02-01
We report spectroscopic studies on plasma electron number density of laser-induced plasma produced by ns-Nd:YAG laser light pulses on an aluminum sample in air at atmospheric pressure. The effect of different laser energy and the effect of different laser wavelengths were compared. The experimentally observed line profiles of neutral aluminum have been used to extract the excitation temperature using the Boltzmann plot method, whereas the electron number density has been determined from the Stark broadened as well as using the Saha-Boltzmann equation (SBE). Each approach was also carried out by using the Al emission line and Mg emission lines. It was observed that the SBE method generated a little higher electron number density value than the Stark broadening method, but within the experimental uncertainty range. Comparisons of N e determined by the two methods show the presence of a linear relation which is independent of laser energy or laser wavelength. These results show the applicability of the SBE method for N e determination, especially when the system does not have any pure emission lines whose electron impact factor is known. Also use of Mg lines gives superior results than Al lines.
Farrell, Patricio; Koprucki, Thomas; Fuhrmann, Jürgen
2017-10-01
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.
Energy Technology Data Exchange (ETDEWEB)
Hagelaar, G J M; Pitchford, L C [Centre de Physique des Plasmas et de leurs Applications de Toulouse, Universite Paul Sabatier, 118 route de Narbonne, 31062 Toulouse Cedex 9 (France)
2005-11-15
Fluid models of gas discharges require the input of transport coefficients and rate coefficients that depend on the electron energy distribution function. Such coefficients are usually calculated from collision cross-section data by solving the electron Boltzmann equation (BE). In this paper we present a new user-friendly BE solver developed especially for this purpose, freely available under the name BOLSIG+, which is more general and easier to use than most other BE solvers available. The solver provides steady-state solutions of the BE for electrons in a uniform electric field, using the classical two-term expansion, and is able to account for different growth models, quasi-stationary and oscillating fields, electron-neutral collisions and electron-electron collisions. We show that for the approximations we use, the BE takes the form of a convection-diffusion continuity-equation with a non-local source term in energy space. To solve this equation we use an exponential scheme commonly used for convection-diffusion problems. The calculated electron transport coefficients and rate coefficients are defined so as to ensure maximum consistency with the fluid equations. We discuss how these coefficients are best used in fluid models and illustrate the influence of some essential parameters and approximations.
LATTICE BOLTZMANN METHOD SIMULATIONS FOR MULTIPHASE FLUIDS WITH REDICH-KWONG EQUATION OF STATE
Institute of Scientific and Technical Information of China (English)
WEI Yi-kun; QIAN Yue-hong
2011-01-01
In this article we state that the compression factor of the Redlich-Kwong Equation Of State (EOS) is smaller than that of van der Waals EOS.The Redlich-Kwong EOS is in better agreement with experimental data on coexistence curves at the critical point than the van der Waals EOS.We implement the Redlich-Kwong EOS in the Lattice Boltzmann Method (LBM) simulations via a pseudo-potential approach.We propose a new force,which can obtain computational stationary and reach larger density ratio.As a result,multi-phase flows with large density ratio (up to 1012 in the stationary case) can be simulated.We perform four numerical simulations,which are respectively related to single liquid droplet,vapor-liquid separation,surface tension and liquid coalescence of two droplets.
Multiple-component lattice Boltzmann equation for fluid-filled vesicles in flow.
Halliday, I; Lishchuk, S V; Spencer, T J; Pontrelli, G; Care, C M
2013-02-01
We document the derivation and implementation of extensions to a two-dimensional, multicomponent lattice Boltzmann equation model, with Laplace law interfacial tension. The extended model behaves in such a way that the boundary between its immiscible drop and embedding fluid components can be shown to describe a vesicle of constant volume bounded by a membrane with conserved length, specified interface compressibility, bending rigidity, preferred curvature, and interfacial tension. We describe how to apply this result to several, independent vesicles. The extended scheme is completely Eulerian, and it represents a two-way coupled vesicle membrane and flow within a single framework. Unlike previous methods, our approach dispenses entirely with the need explicitly to track the membrane, or boundary, and makes no use whatsoever of computationally expensive and intricate interface tracking and remeshing. Validation data are presented, which demonstrate the utility of the method in the simulation of the flow of high volume fraction suspensions of deformable objects.
Mirzadeh, Mohammad; Squires, Todd; Gibou, Frederic
2010-11-01
We present a finite difference discretization of the non-linear Poisson-Boltzmann (PB) equation over complex geometries that has second order accurracy. The level-set method is adopted to represent the interface and Octree (in three dimensions) or Quadtree (in two dimensions) data stuructures are used to generate adaptive grids. Such an approach garanties that the finest grid resolution is located near the interface where EDL forms and creates very large electric field. Several numerical experiments are carried which indicate the second order accuracy both in the case of Dirichlet and Neumann boundary conditions in L2 and L∞ norms. Finally, we use our method to study the electrostatic interaction of double layers between charged particles in an unbounded bulk electrolyte as well as in a channel where the channel width is of the order of Debye length.
Wu, Lei; Liu, Haihu; Zhang, Yonghao; Reese, Jason M.
2015-08-01
The Boltzmann equation with an arbitrary intermolecular potential is solved by the fast spectral method. As examples, noble gases described by the Lennard-Jones potential are considered. The accuracy of the method is assessed by comparing both transport coefficients with variational solutions and mass/heat flow rates in Poiseuille/thermal transpiration flows with results from the discrete velocity method. The fast spectral method is then applied to Fourier and Couette flows between two parallel plates, and the influence of the intermolecular potential on various flow properties is investigated. It is found that for gas flows with the same rarefaction parameter, differences in the heat flux in Fourier flow and the shear stress in Couette flow are small. However, differences in other quantities such as density, temperature, and velocity can be very large.
Exponential Runge-Kutta schemes for inhomogeneous Boltzmann equations with high order of accuracy
Li, Qin
2012-01-01
We consider the development of exponential methods for the robust time discretization of space inhomogeneous Boltzmann equations in stiff regimes. Compared to the space homogeneous case, or more in general to the case of splitting based methods, studied in Dimarco Pareschi (SIAM J. Num. Anal. 2011) a major difficulty is that the local Maxwellian equilibrium state is not constant in a time step and thus needs a proper numerical treatment. We show how to derive asymptotic preserving (AP) schemes of arbitrary order and in particular using the Shu-Osher representation of Runge-Kutta methods we explore the monotonicity properties of such schemes, like strong stability preserving (SSP) and positivity preserving. Several numerical results confirm our analysis.
Fillion-Gourdeau, F; Herrmann, H J; Mendoza, M; Palpacelli, S; Succi, S
2013-10-18
We point out a formal analogy between the Dirac equation in Majorana form and the discrete-velocity version of the Boltzmann kinetic equation. By a systematic analysis based on the theory of operator splitting, this analogy is shown to turn into a concrete and efficient computational method, providing a unified treatment of relativistic and nonrelativistic quantum mechanics. This might have potentially far-reaching implications for both classical and quantum computing, because it shows that, by splitting time along the three spatial directions, quantum information (Dirac-Majorana wave function) propagates in space-time as a classical statistical process (Boltzmann distribution).
Noutchegueme, N; Noutchegueme, Norbert; Tetsadjio, Mesmin Erick
2003-01-01
We prove, for the relativistic Boltzmann equation in the homogeneous case, on the Minkowski space-time, a global in time existence and uniqueness theorem. The method we develop extends to the cases of some curved space-times such as the flat Robertson-Walker space-time and some Bianchi type I space-times.
Dujko, S.; Ebert, U.; White, R.D.; Petrović, Z.L.
2010-01-01
A comprehensive investigation of electron transport in N$_{2}$-O$_{2}$ mixtures has been carried out using a multi term theory for solving the Boltzmann equation and Monte Carlo simulation technique instead of conventional two-term theory often employed in plasma modeling community. We focus on the
Improved lattice Boltzmann modeling of binary flow based on the conservative Allen-Cahn equation
Ren, Feng; Song, Baowei; Sukop, Michael C.; Hu, Haibao
2016-08-01
The primary and key task of binary fluid flow modeling is to track the interface with good accuracy, which is usually challenging due to the sharp-interface limit and numerical dispersion. This article concentrates on further development of the conservative Allen-Cahn equation (ACE) [Geier et al., Phys. Rev. E 91, 063309 (2015), 10.1103/PhysRevE.91.063309] under the framework of the lattice Boltzmann method (LBM), with incorporation of the incompressible hydrodynamic equations [Liang et al., Phys. Rev. E 89, 053320 (2014), 10.1103/PhysRevE.89.053320]. Utilizing a modified equilibrium distribution function and an additional source term, this model is capable of correctly recovering the conservative ACE through the Chapman-Enskog analysis. We also simulate four phase-tracking benchmark cases, including one three-dimensional case; all show good accuracy as well as low numerical dispersion. By coupling the incompressible hydrodynamic equations, we also simulate layered Poiseuille flow and the Rayleigh-Taylor instability, illustrating satisfying performance in dealing with complex flow problems, e.g., high viscosity ratio, high density ratio, and high Reynolds number situations. The present work provides a reliable and efficient solution for binary flow modeling.
Continuous surface force based lattice Boltzmann equation method for simulating thermocapillary flow
Zheng, Lin; Zheng, Song; Zhai, Qinglan
2016-02-01
In this paper, we extend a lattice Boltzmann equation (LBE) with continuous surface force (CSF) to simulate thermocapillary flows. The model is designed on our previous CSF LBE for athermal two phase flow, in which the interfacial tension forces and the Marangoni stresses as the results of the interface interactions between different phases are described by a conception of CSF. In this model, the sharp interfaces between different phases are separated by a narrow transition layers, and the kinetics and morphology evolution of phase separation would be characterized by an order parameter via Cahn-Hilliard equation which is solved in the frame work of LBE. The scalar convection-diffusion equation for temperature field is resolved by thermal LBE. The models are validated by thermal two layered Poiseuille flow, and two superimposed planar fluids at negligibly small Reynolds and Marangoni numbers for the thermocapillary driven convection, which have analytical solutions for the velocity and temperature. Then thermocapillary migration of two/three dimensional deformable droplet are simulated. Numerical results show that the predictions of present LBE agreed with the analytical solution/other numerical results.
Continuous surface force based lattice Boltzmann equation method for simulating thermocapillary flow
Energy Technology Data Exchange (ETDEWEB)
Zheng, Lin, E-mail: lz@njust.edu.cn [School of Energy and Power Engineering, Nanjing University of Science and Technology, Nanjing 210094 (China); Zheng, Song [School of Mathematics and Statistics, Zhejiang University of Finance and Economics, Hangzhou 310018 (China); Zhai, Qinglan [School of Economics Management and Law, Chaohu University, Chaohu 238000 (China)
2016-02-05
In this paper, we extend a lattice Boltzmann equation (LBE) with continuous surface force (CSF) to simulate thermocapillary flows. The model is designed on our previous CSF LBE for athermal two phase flow, in which the interfacial tension forces and the Marangoni stresses as the results of the interface interactions between different phases are described by a conception of CSF. In this model, the sharp interfaces between different phases are separated by a narrow transition layers, and the kinetics and morphology evolution of phase separation would be characterized by an order parameter via Cahn–Hilliard equation which is solved in the frame work of LBE. The scalar convection–diffusion equation for temperature field is resolved by thermal LBE. The models are validated by thermal two layered Poiseuille flow, and two superimposed planar fluids at negligibly small Reynolds and Marangoni numbers for the thermocapillary driven convection, which have analytical solutions for the velocity and temperature. Then thermocapillary migration of two/three dimensional deformable droplet are simulated. Numerical results show that the predictions of present LBE agreed with the analytical solution/other numerical results. - Highlights: • A CSF LBE to thermocapillary flows. • Thermal layered Poiseuille flows. • Thermocapillary migration.
Ion strength limit of computed excess functions based on the linearized Poisson-Boltzmann equation.
Fraenkel, Dan
2015-12-05
The linearized Poisson-Boltzmann (L-PB) equation is examined for its κ-range of validity (κ, Debye reciprocal length). This is done for the Debye-Hückel (DH) theory, i.e., using a single ion size, and for the SiS treatment (D. Fraenkel, Mol. Phys. 2010, 108, 1435), which extends the DH theory to the case of ion-size dissimilarity (therefore dubbed DH-SiS). The linearization of the PB equation has been claimed responsible for the DH theory's failure to fit with experiment at > 0.1 m; but DH-SiS fits with data of the mean ionic activity coefficient, γ± (molal), against m, even at m > 1 (κ > 0.33 Å(-1) ). The SiS expressions combine the overall extra-electrostatic potential energy of the smaller ion, as central ion-Ψa>b (κ), with that of the larger ion, as central ion-Ψb>a (κ); a and b are, respectively, the counterion and co-ion distances of closest approach. Ψa>b and Ψb>a are derived from the L-PB equation, which appears to conflict with their being effective up to moderate electrolyte concentrations (≈1 m). However, the L-PB equation can be valid up to κ ≥ 1.3 Å(-1) if one abandons the 1/κ criterion for its effectiveness and, instead, use, as criterion, the mean-field electrostatic interaction potential of the central ion with its ion cloud, at a radial distance dividing the cloud charge into two equal parts. The DH theory's failure is, thus, not because of using the L-PB equation; the lethal approximation is assigning a single size to the positive and negative ions. © 2015 Wiley Periodicals, Inc.
Collins, Kimberlee C.; Maznev, Alexei A.; Tian, Zhiting; Esfarjani, Keivan; Nelson, Keith A.; Chen, Gang
2013-09-01
The relaxation of an one-dimensional transient thermal grating (TTG) in a medium with phonon-mediated thermal transport is analyzed within the framework of the Boltzmann transport equation (BTE), with the goal of extracting phonon mean free path (MFP) information from TTG measurements of non-diffusive phonon transport. Both gray-medium (constant MFP) and spectrally dependent MFP models are considered. In the gray-medium approximation, an analytical solution is derived. For large TTG periods compared to the MFP, the model yields an exponential decay of grating amplitude with time in agreement with Fourier's heat diffusion equation, and at shorter periods, phonon transport transitions to the ballistic regime, with the decay becoming strongly non-exponential. Spectral solutions are obtained for Si and PbSe at 300 K using phonon dispersion and lifetime data from density functional theory calculations. The spectral decay behaviors are compared to several approximate models: a single MFP solution, a frequency-integrated gray-medium model, and a "two-fluid" BTE solution. We investigate the utility of using the approximate models for the reconstruction of phonon MFP distributions from non-diffusive TTG measurements.
Boltzmann-equation simulations of radio-frequency-driven, low-temperature plasmas
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Drallos, P.J.; Riley, M.E.
1995-01-01
We present a method for the numerical solution of the Boltzmann equation (BE) describing plasma electrons. We apply the method to a capacitively-coupled, radio-frequency-driven He discharge in parallel-plate (quasi-1D) geometry which contains time scales for physical processes spanning six orders of magnitude. Our BE solution procedure uses the method of characteristics for the Vlasov operator with interpolation in phase space at early time, allowing storage of the distribution function on a fixed phase-space grid. By alternating this BE method with a fluid description of the electrons, or with a novel time-cycle-average equation method, we compute the periodic steady state of a He plasma by time evolution from startup conditions. We find that the results compare favorably with measured current-voltage, plasma density, and ``cited state densities in the ``GEC`` Reference Cell. Our atomic He model includes five levels (some are summed composites), 15 electronic transitions, radiation trapping, and metastable-metastable collisions.
Khajepor, Sorush; Chen, Baixin
2016-01-01
A method is developed to analytically and consistently implement cubic equations of state into the recently proposed multipseudopotential interaction (MPI) scheme in the class of two-phase lattice Boltzmann (LB) models [S. Khajepor, J. Wen, and B. Chen, Phys. Rev. E 91, 023301 (2015)]10.1103/PhysRevE.91.023301. An MPI forcing term is applied to reduce the constraints on the mathematical shape of the thermodynamically consistent pseudopotentials; this allows the parameters of the MPI forces to be determined analytically without the need of curve fitting or trial and error methods. Attraction and repulsion parts of equations of state (EOSs), representing underlying molecular interactions, are modeled by individual pseudopotentials. Four EOSs, van der Waals, Carnahan-Starling, Peng-Robinson, and Soave-Redlich-Kwong, are investigated and the results show that the developed MPI-LB system can satisfactorily recover the thermodynamic states of interest. The phase interface is predicted analytically and controlled via EOS parameters independently and its effect on the vapor-liquid equilibrium system is studied. The scheme is highly stable to very high density ratios and the accuracy of the results can be enhanced by increasing the interface resolution. The MPI drop is evaluated with regard to surface tension, spurious velocities, isotropy, dynamic behavior, and the stability dependence on the relaxation time.
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Siewert, C.E. [North Carolina State Univ., Dept. Mathematics, Raleigh, NC (United States)
2002-10-01
A synthetic-kernel model (CES model) of the linearized Boltzmann equation is used along with an analytical discrete-ordinates method (ADO) to solve three fundamental problems concerning flow of a rarefied gas in a plane channel. More specifically, the problems of Couette flow, Poiseuille flow and thermal-creep flow are solved in terms of the CES model equation for an arbitrary mixture of specular and diffuse reflection at the walls confining the flow, and numerical results for the basic quantities of interest are reported. The comparisons made with results derived from solutions based on computationally intensive methods applied to the linearized Boltzmann equation are used to conclude that the CES model can be employed with confidence to improve the accuracy of results available from simpler approximations such as the BGK model or the S model. (author)
ADAPTIVE FINITE ELEMENT MODELING TECHNIQUES FOR THE POISSON-BOLTZMANN EQUATION
HOLST, MICHAEL; MCCAMMON, JAMES ANDREW; YU, ZEYUN; ZHOU, YOUNGCHENG; ZHU, YUNRONG
2011-01-01
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
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Ortega J, R.; Valle G, E. del [IPN-ESFM, 07738 Mexico D.F. (Mexico)]. e-mail: roj@correo.azc.uam.mx
2003-07-01
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)
On anisotropy function in crystal growth simulations using Lattice Boltzmann equation
Younsi, Amina
2016-01-01
In this paper, we present the ability of the Lattice Boltzmann (LB) equation, usually applied to simulate fluid flows, to simulate various shapes of crystals. Crystal growth is modeled with a phase-field model for a pure substance, numerically solved with a LB method in 2D and 3D. This study focuses on the anisotropy function that is responsible for the anisotropic surface tension between the solid phase and the liquid phase. The anisotropy function involves the unit normal vectors of the interface, defined by gradients of phase-field. Those gradients have to be consistent with the underlying lattice of the LB method in order to avoid unwanted effects of numerical anisotropy. Isotropy of the solution is obtained when the directional derivatives method, specific for each lattice, is applied for computing the gradient terms. With the central finite differences method, the phase-field does not match with its rotation and the solution is not any more isotropic. Next, the method is applied to simulate simultaneous...
Shrinkage of bubbles and drops in the lattice Boltzmann equation method for nonideal gases
Zheng, Lin; Lee, Taehun; Guo, Zhaoli; Rumschitzki, David
2014-03-01
One characteristic of multiphase lattice Boltzmann equation (LBE) methods is that the interfacial region has a finite (i.e., noninfinitesimal) thickness known as a diffuse interface. In simulations of, e.g., bubble or drop dynamics, for problems involving nonideal gases, one frequently observes that the diffuse interface method produces a spontaneous, nonphysical shrinkage of the bubble or drop radius. In this paper, we analyze in detail a single-fluid two-phase model and use a LBE model for nonideal gases in order to explain this fundamental problem. For simplicity, we only investigate the static bubble or droplet problem. We find that the method indeed produces a density shift, bubble or droplet shrinkage, as well as a critical radius below which the bubble or droplet eventually vanishes. Assuming that the ratio between the interface thickness D and the initial bubble or droplet radius r0 is small, we analytically show the existence of this density shift, bubble or droplet radius shrinkage, and critical bubble or droplet survival radius. Numerical results confirm our analysis. We also consider droplets on a solid surface with different curvatures, contact angles, and initial droplet volumes. Numerical results show that the curvature, contact angle, and the initial droplet volume have an effect on this spontaneous shrinkage process, consistent with the survival criterion.
Numerical solutions of the semiclassical Boltzmann ellipsoidal-statistical kinetic model equation
Yang, Jaw-Yen; Yan, Chin-Yuan; Huang, Juan-Chen; Li, Zhihui
2014-01-01
Computations of rarefied gas dynamical flows governed by the semiclassical Boltzmann ellipsoidal-statistical (ES) kinetic model equation using an accurate numerical method are presented. The semiclassical ES model was derived through the maximum entropy principle and conserves not only the mass, momentum and energy, but also contains additional higher order moments that differ from the standard quantum distributions. A different decoding procedure to obtain the necessary parameters for determining the ES distribution is also devised. The numerical method in phase space combines the discrete-ordinate method in momentum space and the high-resolution shock capturing method in physical space. Numerical solutions of two-dimensional Riemann problems for two configurations covering various degrees of rarefaction are presented and various contours of the quantities unique to this new model are illustrated. When the relaxation time becomes very small, the main flow features a display similar to that of ideal quantum gas dynamics, and the present solutions are found to be consistent with existing calculations for classical gas. The effect of a parameter that permits an adjustable Prandtl number in the flow is also studied. PMID:25104904
Halliday, I.; Lishchuk, S. V.; Spencer, T. J.; Burgin, K.; Schenkel, T.
2017-10-01
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.
Kernels of the linear Boltzmann equation for spherical particles and rough hard sphere particles.
Khurana, Saheba; Thachuk, Mark
2013-10-28
Kernels for the collision integral of the linear Boltzmann equation are presented for several cases. First, a rigorous and complete derivation of the velocity kernel for spherical particles is given, along with reductions to the smooth, rigid sphere case. This combines and extends various derivations for this kernel which have appeared previously in the literature. In addition, the analogous kernel is derived for the rough hard sphere model, for which a dependence upon both velocity and angular velocity is required. This model can account for exchange between translational and rotational degrees of freedom. Finally, an approximation to the exact rough hard sphere kernel is presented which averages over the rotational degrees of freedom in the system. This results in a kernel depending only upon velocities which retains a memory of the exchange with rotational states. This kernel tends towards the smooth hard sphere kernel in the limit when translational-rotational energy exchange is attenuated. Comparisons are made between the smooth and approximate rough hard sphere kernels, including their dependence upon velocity and their eigenvalues.
Lattice Boltzmann method for convection-diffusion equations with general interfacial conditions
Hu, Zexi; Huang, Juntao; Yong, Wen-An
2016-04-01
In this work, we propose an interfacial scheme accompanying the lattice Boltzmann method for convection-diffusion equations with general interfacial conditions, including conjugate conditions with or without jumps in heat and mass transfer, continuity of macroscopic variables and normal fluxes in ion diffusion in porous media with different porosity, and the Kapitza resistance in heat transfer. The construction of this scheme is based on our boundary schemes [Huang and Yong, J. Comput. Phys. 300, 70 (2015), 10.1016/j.jcp.2015.07.045] for Robin boundary conditions on straight or curved boundaries. It gives second-order accuracy for straight interfaces and first-order accuracy for curved ones. In addition, the new scheme inherits the advantage of the boundary schemes in which only the current lattice nodes are involved. Such an interfacial scheme is highly desirable for problems with complex geometries or in porous media. The interfacial scheme is numerically validated with several examples. The results show the utility of the constructed scheme and very well support our theoretical predications.
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Buchan, Andrew G., E-mail: andrew.buchan@imperial.ac.uk [Applied Modelling and Computational Group, Department of Earth Science and Engineering, Imperial College of Science, Technology and Medicine (United Kingdom); Merton, Simon R. [AWE, Aldermaston, Reading RG7 4PR (United Kingdom); Pain, Christopher C. [Applied Modelling and Computational Group, Department of Earth Science and Engineering, Imperial College of Science, Technology and Medicine (United Kingdom); Smedley-Stevenson, Richard P. [AWE, Aldermaston, Reading RG7 4PR (United Kingdom)
2011-05-15
In this paper a method for resolving the various boundary conditions (BCs) for the first order Boltzmann transport equation (BTE) is described. The approach has been formulated to resolve general BCs using an arbitrary angular approximation method within any weighted residual finite element formulation. The method is based on a Riemann decomposition which is used to decompose the particles' angular dependence into in-coming and out-going information through a surface. This operation recasts the flux into a Riemann space which is used directly to remove any incoming information, and thus satisfy void boundary conditions. The method is then extended by its coupling with a set of mapping operators that redirect the outgoing flux to form incoming images resembling other specified boundary conditions. These operators are based on Galerkin projections and are defined to enable reflective and diffusive (white) BCs to be resolved. A small number of numerical examples are then presented to demonstrate the method's ability in resolving void, reflective and white BCs. These examples have been chosen in order to show the method working for arbitrary angled surfaces. Furthermore, as the method has been designed for an arbitrary angular approximation, both S{sub N} and P{sub N} calculations are presented.
Li, Zhihui; Wu, Junlin; Ma, Qiang; Jiang, Xinyu; Zhang, Hanxin
2014-12-01
Based on the Gas-Kinetic Unified Algorithm (GKUA) directly solving the Boltzmann model equation, the effect of rotational non-equilibrium is investigated recurring to the kinetic Rykov model with relaxation property of rotational degrees of freedom. The spin movement of diatomic molecule is described by moment of inertia, and the conservation of total angle momentum is taken as a new Boltzmann collision invariant. The molecular velocity distribution function is integrated by the weight factor on the internal energy, and the closed system of two kinetic controlling equations is obtained with inelastic and elastic collisions. The optimization selection technique of discrete velocity ordinate points and numerical quadrature rules for macroscopic flow variables with dynamic updating evolvement are developed to simulate hypersonic flows, and the gas-kinetic numerical scheme is constructed to capture the time evolution of the discretized velocity distribution functions. The gas-kinetic boundary conditions in thermodynamic non-equilibrium and numerical procedures are studied and implemented by directly acting on the velocity distribution function, and then the unified algorithm of Boltzmann model equation involving non-equilibrium effect is presented for the whole range of flow regimes. The hypersonic flows involving non-equilibrium effect are numerically simulated including the inner flows of shock wave structures in nitrogen with different Mach numbers of 1.5-Ma-25, the planar ramp flow with the whole range of Knudsen numbers of 0.0009-Kn-10 and the three-dimensional re-entering flows around tine double-cone body.
Fiorentini, Mattia; Bonini, Nicola
2016-08-01
We present a first-principles computational approach to calculate thermoelectric transport coefficients via the exact solution of the linearized Boltzmann transport equation, also including the effect of nonequilibrium phonon populations induced by a temperature gradient. We use density functional theory and density functional perturbation theory for an accurate description of the electronic and vibrational properties of a system, including electron-phonon interactions; carriers' scattering rates are computed using standard perturbation theory. We exploit Wannier interpolation (both for electronic bands and electron-phonon matrix elements) for an efficient sampling of the Brillouin zone, and the solution of the Boltzmann equation is achieved via a fast and stable conjugate gradient scheme. We discuss the application of this approach to n -doped silicon. In particular, we discuss a number of thermoelectric properties such as the thermal and electrical conductivities of electrons, the Lorenz number and the Seebeck coefficient, including the phonon drag effect, in a range of temperatures and carrier concentrations. This approach gives results in good agreement with experimental data and provides a detailed characterization of the nature and the relative importance of the individual scattering mechanisms. Moreover, the access to the exact solution of the Boltzmann equation for a realistic system provides a direct way to assess the accuracy of different flavors of relaxation time approximation, as well as of models that are popular in the thermoelectric community to estimate transport coefficients.
Energy Technology Data Exchange (ETDEWEB)
Li, Zhihui; Ma, Qiang [Hypervelocity Aerodynamics Institute, China Aerodynamics Research and Development Center, P.O.Box 211, Mianyang 621000, China and National Laboratory for Computational Fluid Dynamics, No.37 Xueyuan Road, Beijing 100191 (China); Wu, Junlin; Jiang, Xinyu [Hypervelocity Aerodynamics Institute, China Aerodynamics Research and Development Center, P.O.Box 211, Mianyang 621000 (China); Zhang, Hanxin [National Laboratory for Computational Fluid Dynamics, No.37 Xueyuan Road, Beijing 100191 (China)
2014-12-09
Based on the Gas-Kinetic Unified Algorithm (GKUA) directly solving the Boltzmann model equation, the effect of rotational non-equilibrium is investigated recurring to the kinetic Rykov model with relaxation property of rotational degrees of freedom. The spin movement of diatomic molecule is described by moment of inertia, and the conservation of total angle momentum is taken as a new Boltzmann collision invariant. The molecular velocity distribution function is integrated by the weight factor on the internal energy, and the closed system of two kinetic controlling equations is obtained with inelastic and elastic collisions. The optimization selection technique of discrete velocity ordinate points and numerical quadrature rules for macroscopic flow variables with dynamic updating evolvement are developed to simulate hypersonic flows, and the gas-kinetic numerical scheme is constructed to capture the time evolution of the discretized velocity distribution functions. The gas-kinetic boundary conditions in thermodynamic non-equilibrium and numerical procedures are studied and implemented by directly acting on the velocity distribution function, and then the unified algorithm of Boltzmann model equation involving non-equilibrium effect is presented for the whole range of flow regimes. The hypersonic flows involving non-equilibrium effect are numerically simulated including the inner flows of shock wave structures in nitrogen with different Mach numbers of 1.5-Ma-25, the planar ramp flow with the whole range of Knudsen numbers of 0.0009-Kn-10 and the three-dimensional re-entering flows around tine double-cone body.
Wu, Zhen; Zhang, Xian; Zhou, Chunjiao; Pang, Jing-Lin; Zhang, Panyue
2017-02-22
Single-molecule aluminum salt AlCl3, medium polymerized polyaluminum chloride (PAC), and high polymerized polyaluminum chloride (HPAC) were prepared in a laboratory. The characteristics and coagulation properties of these prepared aluminum salts were investigated. The Langmuir, Freundlich, and Sips adsorption isotherms were first used to describe the adsorption neutralization process in coagulation, and the Boltzmann equation was used to fit the reaction kinetics of floc growth in flocculation. It was novel to find that the experimental data fitted well with the Sips and Boltzmann equation, and the significance of parameters in the equations was discussed simultaneously. Through the Sips equation, the adsorption neutralization reaction was proved to be spontaneous and the adsorption neutralization capacity was HPAC > PAC > AlCl3. Sips equation also indicated that the zeta potential of water samples would reach a limit with the increase of coagulant dosage, and the equilibrium zeta potential values were 30.25, 30.23, and 27.25 mV for AlCl3, PAC, and HPAC, respectively. The lower equilibrium zeta potential value of HPAC might be the reason why the water sample was not easy to achieve restabilization at a high coagulant dosage. Through the Boltzmann equation modeling, the maximum average floc size formed by AlCl3, PAC, and HPAC were 196.0, 188.0, and 203.6 μm, respectively, and the halfway time of reactions were 31.23, 17.08, and 9.55 min, respectively. The HPAC showed the strongest floc formation ability and the fastest floc growth rate in the flocculation process, which might be caused by the stronger adsorption and bridging functions of Alb and Alc contained in HPAC.
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: hamedoun@hotmail.com [Institute of Nanomaterials and Nanotechnology, MAScIR, Rabat (Morocco); El Kenz, A. [LMPHE (URAC 12), Faculté des Sciences, Université Mohammed V-Agdal, Rabat (Morocco)
2014-08-25
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.
Chakavorty, Arghya; Li, Lin; Alexov, Emil
2016-10-30
Macromolecular interactions are essential for understanding numerous biological processes and are typically characterized by the binding free energy. Important component of the binding free energy is the electrostatics, which is frequently modeled via the solutions of the Poisson-Boltzmann Equations (PBE). However, numerous works have shown that the electrostatic component (ΔΔGelec ) of binding free energy is very sensitive to the parameters used and modeling protocol. This prompted some researchers to question the robustness of PBE in predicting ΔΔGelec . We argue that the sensitivity of the absolute ΔΔGelec calculated with PBE using different input parameters and definitions does not indicate PBE deficiency, rather this is what should be expected. We show how the apparent sensitivity should be interpreted in terms of the underlying changes in several numerous and physical parameters. We demonstrate that PBE approach is robust within each considered force field (CHARMM-27, AMBER-94, and OPLS-AA) once the corresponding structures are energy minimized. This observation holds despite of using two different molecular surface definitions, pointing again that PBE delivers consistent results within particular force field. The fact that PBE delivered ΔΔGelec values may differ if calculated with different modeling protocols is not a deficiency of PBE, but natural results of the differences of the force field parameters and potential functions for energy minimization. In addition, while the absolute ΔΔGelec values calculated with different force field differ, their ordering remains practically the same allowing for consistent ranking despite of the force field used. © 2016 Wiley Periodicals, Inc.
Xie, Yang; Ying, Jinyong; Xie, Dexuan
2017-03-30
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: smpbs.math.uwm.edu © 2017 Wiley Periodicals, Inc. © 2017 Wiley Periodicals, Inc.
Lattice Boltzmann Model for the Coupled Korteweg-de Vries Equations%耦合KdV方程组的格子Boltzmann模型
Institute of Scientific and Technical Information of China (English)
王慧敏; 刘艳红
2012-01-01
用格子Boltzmann方法研究耦合KdV方程组.构建耦合KdV方程组的格子Boltzmann模型并进行了数值实验,同时将格子Boltzmann解与其他传统数值方法得到的数值解进行比较.结果表明,格子Boltzmann方法是一种求解耦合KdV方程组的有效方法.%We used the lattice Boltzmann method to study the coupled Korteweg-de Vries equations. We constructed a lattice Boltzmann model for the coupled Korteweg-de Vries equations, and performed numerical experiments. Comparing the lattice Boltzmann solution with other numerical solutions, we obtained that the lattice Boltzmann method is an effective method for simulating the coupled Korteweg-de Vries equations.
Frydel, Derek
2011-06-21
We incorporate ion polarizabilities into the Poisson-Boltzmann equation by modifying the effective dielectric constant and the Boltzmann distribution of ions. The extent of the polarizability effects is controlled by two parameters, γ(1) and γ(2); γ(1) determines the polarization effects in a dilute system and γ(2) regulates the dependence of the polarizability effects on the concentration of ions. For a polarizable ion in an aqueous solution γ(1) ≈ 0.01 and the polarizability effects are negligible. The conditions where γ(1) and/or γ(2) are large and the polarizability is relevant involve the low dielectric constant media, high surface charge, and/or large ionic concentrations. © 2011 American Institute of Physics
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Lloyd, S. A. M.; Ansbacher, W. [Department of Physics and Astronomy, University of Victoria, Victoria, British Columbia V8W 3P6 (Canada); Department of Physics and Astronomy, University of Victoria, Victoria, British Columbia V8W 3P6 (Canada) and Department of Medical Physics, British Columbia Cancer Agency-Vancouver Island Centre, Victoria, British Columbia V8R 6V5 (Canada)
2013-01-15
Purpose: Acuros external beam (Acuros XB) is a novel dose calculation algorithm implemented through the ECLIPSE treatment planning system. The algorithm finds a deterministic solution to the linear Boltzmann transport equation, the same equation commonly solved stochastically by Monte Carlo methods. This work is an evaluation of Acuros XB, by comparison with Monte Carlo, for dose calculation applications involving high-density materials. Existing non-Monte Carlo clinical dose calculation algorithms, such as the analytic anisotropic algorithm (AAA), do not accurately model dose perturbations due to increased electron scatter within high-density volumes. Methods: Acuros XB, AAA, and EGSnrc based Monte Carlo are used to calculate dose distributions from 18 MV and 6 MV photon beams delivered to a cubic water phantom containing a rectangular high density (4.0-8.0 g/cm{sup 3}) volume at its center. The algorithms are also used to recalculate a clinical prostate treatment plan involving a unilateral hip prosthesis, originally evaluated using AAA. These results are compared graphically and numerically using gamma-index analysis. Radio-chromic film measurements are presented to augment Monte Carlo and Acuros XB dose perturbation data. Results: Using a 2% and 1 mm gamma-analysis, between 91.3% and 96.8% of Acuros XB dose voxels containing greater than 50% the normalized dose were in agreement with Monte Carlo data for virtual phantoms involving 18 MV and 6 MV photons, stainless steel and titanium alloy implants and for on-axis and oblique field delivery. A similar gamma-analysis of AAA against Monte Carlo data showed between 80.8% and 87.3% agreement. Comparing Acuros XB and AAA evaluations of a clinical prostate patient plan involving a unilateral hip prosthesis, Acuros XB showed good overall agreement with Monte Carlo while AAA underestimated dose on the upstream medial surface of the prosthesis due to electron scatter from the high-density material. Film measurements
Lattice Boltzmann equation calculation of internal, pressure-driven turbulent flow
Hammond, L A; Care, C M; Stevens, A
2002-01-01
We describe a mixing-length extension of the lattice Boltzmann approach to the simulation of an incompressible liquid in turbulent flow. The method uses a simple, adaptable, closure algorithm to bound the lattice Boltzmann fluid incorporating a law-of-the-wall. The test application, of an internal, pressure-driven and smooth duct flow, recovers correct velocity profiles for Reynolds number to 1.25 x 10 sup 5. In addition, the Reynolds number dependence of the friction factor in the smooth-wall branch of the Moody chart is correctly recovered. The method promises a straightforward extension to other curves of the Moody chart and to cylindrical pipe flow.
Conjugate heat and mass transfer in the lattice Boltzmann equation method.
Li, Like; Chen, Chen; Mei, Renwei; Klausner, James F
2014-04-01
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
A Discussion on Whether 15-20C Are All Skin Nuclei via Isospin-dependent Boltzmann-Langevin Equation
Institute of Scientific and Technical Information of China (English)
CHEN Yu; ZHANG Feng-Shou; SU Jun
2009-01-01
A new attempt of calculation for the total reaction cross sections (σR) has been carried out within the isospin-dependent Boltzmann-Langevin equation in the intermediate energy heavy-ion collision of isotopes of G. The σR of both stable and exotic nuclei are reproduced rather well. The incident energy and isospin dependencies of σR have been investigated. It is found that the isospin effect is comparatively remarkable at intermediate energy. It is also found that ~(15-18)C are neutron skin nuclei but for ~(19)C and ~(20)C we cannot draw a conclusion whether they have halo structures.
Di, Shaoyan; Shen, Lei; Chang, Pengying; Zhao, Kai; Lu, Tiao; Du, Gang; Liu, Xiaoyan
2017-04-01
A deterministic time-dependent Boltzmann transport equation (BTE) solver is employed to carry out a comparison work among 10 nm double-gate n-type MOSFETs with channel materials of Si, In0.53Ga0.47As, and GaSb in different surface orientations. Results show that the GaSb device has the highest drive current, while scattering affects carrier transport in the Si device the most. The InGaAs device exhibits the highest injection velocity but suffers from the density of state (DOS) bottleneck seriously.
Hu, Kainan; Geng, Shaojuan
2016-01-01
A decoupled scheme based on the Hermite expansion to construct lattice Boltzmann models for the compressible Navier-Stokes equations with arbitrary specific heat ratio is proposed. The local equilibrium distribution function including the rotational velocity of particle is decoupled into two parts, i.e. the local equilibrium distribution function of the translational velocity of particle and that of the rotational velocity of particle. From these two local equilibrium functions, two lattice Boltzmann models are derived via the Hermite expansion, namely one is in relation to the translational velocity and the other is connected with the rotational velocity. Accordingly, the distribution function is also decoupled. After this, the evolution equation is decoupled into the evolution equation of the translational velocity and that of the rotational velocity. The two evolution equations evolve separately. The lattice Boltzmann models used in the scheme proposed by this work are constructed via the Hermite expansion...
The inverse problem based on a full dispersive wave equation
Institute of Scientific and Technical Information of China (English)
Gegentana Bao; Naranmandula Bao
2012-01-01
The inverse problem for harmonic waves and wave packets was studied based on a full dispersive wave equation. First, a full dispersive wave equation which describes wave propagation in nondissipative microstructured linear solids is established based on the Mindlin theory, and the dispersion characteristics are discussed. Second, based on the full dispersive wave equation, an inverse problem for determining the four unknown coefficients of wave equa- tion is posed in terms of the frequencies and corresponding wave numbers of four different harmonic waves, and the inverse problem is demonstrated with rigorous mathematical theory. Research proves that the coefficients of wave equation related to material properties can be uniquely determined in cases of normal and anomalous dispersions by measuring the frequen- cies and corresponding wave numbers of four different harmonic waves which propagate in a nondissipative microstructured linear solids.
Ying, Jinyong
2016-01-01
The size-modified Poisson-Boltzmann equation (SMPBE) is one important variant of the popular dielectric model, the Poisson-Boltzmann equation (PBE), to reflect ionic size effects in the prediction of electrostatics for a biomolecule in an ionic solvent. In this paper, a new SMPBE hybrid solver is developed using a solution decomposition, the Schwartz's overlapped domain decomposition, finite element, and finite difference. It is then programmed as a software package in C, Fortran, and Python based on the state-of-the-art finite element library DOLFIN from the FEniCS project. This software package is well validated on a Born ball model with analytical solution and a dipole model with a known physical properties. Numerical results on six proteins with different net charges demonstrate its high performance. Finally, this new SMPBE hybrid solver is shown to be numerically stable and convergent in the calculation of electrostatic solvation free energy for 216 biomolecules and binding free energy for a DNA-drug com...
Energy Technology Data Exchange (ETDEWEB)
Mirza, Anwar M. [Department of Computer Science, National University of Computer and Emerging Sciences, NUCES-FAST, A.K. Brohi Road, H-11, Islamabad (Pakistan)], E-mail: anwar.m.mirza@gmail.com; Iqbal, Shaukat [Faculty of Computer Science and Engineering, Ghulam Ishaq Khan (GIK) Institute of Engineering Science and Technology, Topi-23460, Swabi (Pakistan)], E-mail: shaukat@giki.edu.pk; Rahman, Faizur [Department of Physics, Allama Iqbal Open University, H-8 Islamabad (Pakistan)
2007-07-15
A spatially adaptive grid-refinement approach has been investigated to solve the even-parity Boltzmann transport equation. A residual based a posteriori error estimation scheme has been utilized for checking the approximate solutions for various finite element grids. The local particle balance has been considered as an error assessment criterion. To implement the adaptive approach, a computer program ADAFENT (adaptive finite elements for neutron transport) has been developed to solve the second order even-parity Boltzmann transport equation using K{sup +} variational principle for slab geometry. The program has a core K{sup +} module which employs Lagrange polynomials as spatial basis functions for the finite element formulation and Legendre polynomials for the directional dependence of the solution. The core module is called in by the adaptive grid generator to determine local gradients and residuals to explore the possibility of grid refinements in appropriate regions of the problem. The a posteriori error estimation scheme has been implemented in the outer grid refining iteration module. Numerical experiments indicate that local errors are large in regions where the flux gradients are large. A comparison of the spatially adaptive grid-refinement approach with that of uniform meshing approach for various benchmark cases confirms its superiority in greatly enhancing the accuracy of the solution without increasing the number of unknown coefficients. A reduction in the local errors of the order of 10{sup 2} has been achieved using the new approach in some cases.
Lahiri, T.; Pal Majumder, T.; Ghosh, N. K.
2014-07-01
Commercialization of ferroelectric liquid crystal displays (FLCDs) suffers from mechanical and electro-convective instabilities. Impurity ions play a pivotal role in the latter case, and therefore we developed a mean-field type model to understand the complex role of space charges, particularly ions in a ferroelectric liquid crystal. Considering an effective ion-chirality relation, we obtained a modified Poisson-Boltzmann equation for ions dissolved into a chiral solvent like the ferroelectric smectic phase. A nonuniform director profile induced by the mean electrostatic potential of the ions is then calculated by solving an Euler-Lagrange equation for a helically twisted smectic state. A combination of effects resulting from molecular chirality and an electrostatically driven twist created by the ions seems to produce this nonuniform fluctuation in the director orientation. Finally, both theoretical and experimental points of view are presented on the prediction of this mean-field model.
Lattice Boltzmann approach for complex nonequilibrium flows.
Montessori, A; Prestininzi, P; La Rocca, M; Succi, S
2015-10-01
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.
Coarse-grained transport of a turbulent flow via moments of the Reynolds-averaged Boltzmann equation
Abramov, Rafail V
2015-01-01
Here we introduce new coarse-grained variables for a turbulent flow in the form of moments of its Reynolds-averaged Boltzmann equation. With the exception of the collision moments, the transport equations for the new variables are identical to the usual moment equations, and thus naturally lend themselves to the variety of already existing closure methods. Under the anelastic turbulence approximation, we derive equations for the Reynolds-averaged turbulent fluctuations around the coarse-grained state. We show that the global relative entropy of the coarse-grained state is bounded from above by the Reynolds average of the fine-grained global relative entropy, and thus obeys the time decay bound of Desvillettes and Villani. This is similar to what is observed in the rarefied gas dynamics, which makes the Grad moment closure a good candidate for truncating the hierarchy of the coarse-grained moment equations. We also show that, under additional assumptions on the form of the coarse-grained collision terms, one a...
Halliday, I; Lishchuk, S V; Spencer, T J; Pontrelli, G; Evans, P C
2016-08-01
We present a method for applying a class of velocity-dependent forces within a multicomponent lattice Boltzmann equation simulation that is designed to recover continuum regime incompressible hydrodynamics. This method is applied to the problem, in two dimensions, of constraining to uniformity the tangential velocity of a vesicle membrane implemented within a recent multicomponent lattice Boltzmann simulation method, which avoids the use of Lagrangian boundary tracers. The constraint of uniform tangential velocity is carried by an additional contribution to an immersed boundary force, which we derive here from physical arguments. The result of this enhanced immersed boundary force is to apply a physically appropriate boundary condition at the interface between separated lattice fluids, defined as that region over which the phase-field varies most rapidly. Data from this enhanced vesicle boundary method are in agreement with other data obtained using related methods [e.g., T. Krüger, S. Frijters, F. Günther, B. Kaoui, and J. Harting, Eur. Phys. J. 222, 177 (2013)10.1140/epjst/e2013-01834-y] and underscore the importance of a correct vesicle membrane condition.
Research and Application of Dynamic Equation for Full Frontal Impact
Institute of Scientific and Technical Information of China (English)
Sheng Tian∗
2015-01-01
Full frontal impact theory needs researching and exploring to satisfy the primary safety design of occupant restraint system, avoiding the increasingly “engineering” trend in order to develop and design safety vehicle. After occupant restraint system is simulated by using linear elastic stiffness k, the occupant⁃vehicle frontal rigid barrier impact model is established. Dynamic equation of dummy chest coupling vehicle is built for full frontal impact based on ordinary vehicle deceleration by Hooke law, and the equation is solved by comparing coefficient and satisfying boundary qualifications. While relative vehicle characteristic parameters are kept unchanging, the actual vehicle deceleration is fitted to the simplified equivalent square wave ( ESW ) , tipped equivalent square wave (TESW) and equivalent dual trapezoids wave (EDTW). Phase angle ϕ and amplitude A of dynamic equations based on ESW, TESW and EDTW are calculated and deduced. The results show that:the dynamic equation of dummy chest coupling vehicle can be well utilized to instruct the primary safety design of full frontal impact for objective vehicle to satisfy chest deceleration demands and the equation based on TESW is best for this design.
Xiong, Yuan
2014-04-28
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.
Langevin Poisson-Boltzmann equation: point-like ions and water dipoles near a charged surface.
Gongadze, Ekaterina; van Rienen, Ursula; Kralj-Iglič, Veronika; Iglič, Aleš
2011-06-01
Water ordering near a charged membrane surface is important for many biological processes such as binding of ligands to a membrane or transport of ions across it. In this work, the mean-field Poisson-Boltzmann theory for point-like ions, describing an electrolyte solution in contact with a planar charged surface, is modified by including the orientational ordering of water. Water molecules are considered as Langevin dipoles, while the number density of water is assumed to be constant everywhere in the electrolyte solution. It is shown that the dielectric permittivity of an electrolyte close to a charged surface is decreased due to the increased orientational ordering of water dipoles. The dielectric permittivity close to the charged surface is additionally decreased due to the finite size of ions and dipoles.
Stable Equilibrium Based on Lévy Statistics:A Linear Boltzmann Equation Approach
Barkai, Eli
2004-06-01
To obtain further insight on possible power law generalizations of Boltzmann equilibrium concepts, we consider stochastic collision models. The models are a generalization of the Rayleigh collision model, for a heavy one dimensional particle M interacting with ideal gas particles with a mass mlaw equilibrium. We show, under certain conditions, that the velocity distribution function of the heavy particle is Lévy stable, the Maxwellian distribution being a special case. We demonstrate our results with numerical examples. The relation of the power law equilibrium obtained here to thermodynamics is discussed. In particular we compare between two models: a thermodynamic and an energy scaling approaches. These models yield insight into questions like the meaning of temperature for power law equilibrium, and into the issue of the universality of the equilibrium (i.e., is the width of the generalized Maxwellian distribution functions obtained here, independent of coupling constant to the bath).
Li, Yunqi; Zhao, Qin; Huang, Qingrong
2014-01-30
A combination of turbidimetric titration, a sigmoidal Boltzmann equation approach and Monte Carlo simulation has been used to study the complex coacervation in serum albumin and pectin mixtures. The effects of the mass ratio of protein to polysaccharide on the critical pH values, the probability of complex coacervation and the electrostatic interaction from charge patches in serum albumin were investigated. Turbidimetric titration results showed an optimum pH for complex coacervation (pHm), which corresponded to the maximum turbidity in the protein/polysaccharide mixture. The pHm monotonically decreased as the ratio decreased, and could be fitted using the sigmoidal Boltzmann equation. It suggests that pHm could be a good ordering parameter to characterize the phase behavior associated with protein/polysaccharide complex coacervation. Qualitative understanding of pHm by taking into account the minimization of electrostatic interaction, as well as quantitative matching of pHm according to the concept of charge neutralization were both achieved. Our results suggest that the serum albumin/pectin complexes were ultimately neutralized by the partial charges originated from the titratable residues in protein and polysaccharide chains at pHm. The Monte Carlo simulation provided consistent phase boundaries for complex coacervation in the same system, and the intermolecular association strength was determined to be several kBT below the given ionic strength. The strongest binding site in the protein is convergent to the largest positive charge patch if pure electrostatic interaction was considered. Further inclusion of contribution from excluded volume resulted in the binding site distribution over five different positive charge patches at different protein/polysaccharide ratios and pH values. Copyright © 2013 Elsevier Ltd. All rights reserved.
Institute of Scientific and Technical Information of China (English)
无
2009-01-01
The Boltzmann simplified velocity distribution function equation describing the gas transfer phenomena from various flow regimes will be explored and solved numerically in this study. The discrete velocity ordinate method of the gas kinetic theory is studied and applied to simulate the complex multi-scale flows. Based on the uncoupling technique on molecular movement and colliding in the DSMC method, the gas-kinetic finite difference scheme is constructed to directly solve the discrete velocity distribution functions by extending and applying the unsteady time-splitting method from computational fluid dynamics. The Gauss-type discrete velocity numerical quadrature technique for different Mach number flows is developed to evaluate the macroscopic flow parameters in the physical space. As a result, the gas-kinetic numerical algorithm is established to study the three-dimensional complex flows from rarefied transition to continuum regimes. The parallel strategy adapted to the gas-kinetic numerical algorithm is investigated by analyzing the inner parallel degree of the algorithm, and then the HPF parallel processing program is developed. To test the reliability of the present gas-kinetic numerical method, the three-dimensional complex flows around sphere and spacecraft shape with various Knudsen numbers are simulated by HPF parallel computing. The computational results are found in high resolution of the flow fields and good agreement with the theoretical and experimental data. The computing practice has confirmed that the present gas-kinetic algorithm probably provides a promising approach to resolve the hypersonic aerothermodynamic problems with the complete spectrum of flow regimes from the gas-kinetic point of view of solving the Boltzmann model equation.
Ender, I A; Flegontova, E Yu; Gerasimenko, A B
2016-01-01
An algorithm for sequential calculation of non-isotropic matrix elements of the collision integral which are necessary for the solution of the non-linear Boltzmann equation by moment method is proposed. Isotropic matrix elements that we believe are known, are starting ones. The procedure is valid for any interaction law and any mass ratio of the colliding particles.
Ahrens, Cory D
2014-01-01
The classical $S_n$ equations of Carlson and Lee have been a mainstay in multi-dimensional radiation transport calculations. In this paper, an alternative to the $S_n$ equations, the "Lagrange Discrete Ordinate" (LDO) equations are derived. These equations are based on an interpolatory framework for functions on the unit sphere in three dimensions. While the LDO equations retain the formal structure of the classical $S_n$ equations, they have a number of important differences. The LDO equations naturally allow the angular flux to be evaluated in directions other than those found in the quadrature set. To calculate the scattering source in the LDO equations, no spherical harmonic moments are needed--only values of the angular flux. Moreover, the LDO scattering source preserves the eigenstructure of the continuous scattering operator. The formal similarity of the LDO equations with the $S_n$ equations should allow easy modification of mature 3D $S_n$ codes such as PARTISN or PENTRAN to solve the LDO equations. ...
Energy Technology Data Exchange (ETDEWEB)
Fidler, Christian
2011-12-16
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.
Singularity formation for one dimensional full Euler equations
Pan, Ronghua; Zhu, Yi
2016-12-01
We investigate the basic open question on the global existence v.s. finite time blow-up phenomena of classical solutions for the one-dimensional compressible Euler equations of adiabatic flow. For isentropic flows, it is well-known that the solutions develop singularity if and only if initial data contain any compression (the Riemann variables have negative spatial derivative). The situation for non-isentropic flow is not quite clear so far, due to the presence of non-constant entropy. In [4], it is shown that initial weak compressions do not necessarily develop singularity in finite time, unless the compression is strong enough for general data. In this paper, we identify a class of solutions of the full (non-isentropic) Euler equations, developing singularity in finite time even though their initial data do not contain any compression. This is in sharp contrast to the isentropic flow.
Generalized Poisson-Boltzmann Equation Taking into Account Ionic Interaction and Steric Effects
Institute of Scientific and Technical Information of China (English)
刘新敏; 李航; 李睿; 田锐; 许晨阳
2012-01-01
Generalized Poisson l3oltzmann equation which takes into account both ionic interaction in bulk solution and steric effects of adsorbed ions has been suggested. We found that, for inorganic cations adsorption on negatively charged surface, the steric effect is not significant for surface charge density 〈 0.0032 C/dm2, while the ionic interaction is an important effect for electrolyte concentration 〉 0.15 tool/1 in bulk solution. We conclude that for most actual cases the original PB equation can give reliable result in describing inorganic cation adsorption.
Hu, Kainan; Zhang, Hongwu; Geng, Shaojuan
2016-10-01
A decoupled scheme based on the Hermite expansion to construct lattice Boltzmann models for the compressible Navier-Stokes equations with arbitrary specific heat ratio is proposed. The local equilibrium distribution function including the rotational velocity of particle is decoupled into two parts, i.e., the local equilibrium distribution function of the translational velocity of particle and that of the rotational velocity of particle. From these two local equilibrium functions, two lattice Boltzmann models are derived via the Hermite expansion, namely one is in relation to the translational velocity and the other is connected with the rotational velocity. Accordingly, the distribution function is also decoupled. After this, the evolution equation is decoupled into the evolution equation of the translational velocity and that of the rotational velocity. The two evolution equations evolve separately. The lattice Boltzmann models used in the scheme proposed by this work are constructed via the Hermite expansion, so it is easy to construct new schemes of higher-order accuracy. To validate the proposed scheme, a one-dimensional shock tube simulation is performed. The numerical results agree with the analytical solutions very well.
Hu, Kainan; Zhang, Hongwu; Geng, Shaojuan
2016-10-01
A decoupled scheme based on the Hermite expansion to construct lattice Boltzmann models for the compressible Navier-Stokes equations with arbitrary specific heat ratio is proposed. The local equilibrium distribution function including the rotational velocity of particle is decoupled into two parts, i.e., the local equilibrium distribution function of the translational velocity of particle and that of the rotational velocity of particle. From these two local equilibrium functions, two lattice Boltzmann models are derived via the Hermite expansion, namely one is in relation to the translational velocity and the other is connected with the rotational velocity. Accordingly, the distribution function is also decoupled. After this, the evolution equation is decoupled into the evolution equation of the translational velocity and that of the rotational velocity. The two evolution equations evolve separately. The lattice Boltzmann models used in the scheme proposed by this work are constructed via the Hermite expansion, so it is easy to construct new schemes of higher-order accuracy. To validate the proposed scheme, a one-dimensional shock tube simulation is performed. The numerical results agree with the analytical solutions very well.
Energy Technology Data Exchange (ETDEWEB)
Bankovic, A., E-mail: ana.bankovic@gmail.com [Institute of Physics, University of Belgrade, Pregrevica 118, 11080 Belgrade (Serbia); Dujko, S. [Institute of Physics, University of Belgrade, Pregrevica 118, 11080 Belgrade (Serbia); Centrum Wiskunde and Informatica (CWI), P.O. Box 94079, 1090 GB Amsterdam (Netherlands); ARC Centre for Antimatter-Matter Studies, School of Engineering and Physical Sciences, James Cook University, Townsville, QLD 4810 (Australia); White, R.D. [ARC Centre for Antimatter-Matter Studies, School of Engineering and Physical Sciences, James Cook University, Townsville, QLD 4810 (Australia); Buckman, S.J. [ARC Centre for Antimatter-Matter Studies, Australian National University, Canberra, ACT 0200 (Australia); Petrovic, Z.Lj. [Institute of Physics, University of Belgrade, Pregrevica 118, 11080 Belgrade (Serbia)
2012-05-15
This work reports on a new series of calculations of positron transport properties in molecular hydrogen under the influence of spatially homogeneous electric field. Calculations are performed using a Monte Carlo simulation technique and multi term theory for solving the Boltzmann equation. Values and general trends of the mean energy, drift velocity and diffusion coefficients as a function of the reduced electric field E/n{sub 0} are reported here. Emphasis is placed on the explicit and implicit effects of positronium (Ps) formation on the drift velocity and diffusion coefficients. Two important phenomena arise; first, for certain regions of E/n{sub 0} the bulk and flux components of the drift velocity and longitudinal diffusion coefficient are markedly different, both qualitatively and quantitatively. Second, and contrary to previous experience in electron swarm physics, there is negative differential conductivity (NDC) effect in the bulk drift velocity component with no indication of any NDC for the flux component. In order to understand this atypical manifestation of the drift and diffusion of positrons in H{sub 2} under the influence of electric field, the spatially dependent positron transport properties such as number of positrons, average energy and velocity and spatially resolved rate for Ps formation are calculated using a Monte Carlo simulation technique. The spatial variation of the positron average energy and extreme skewing of the spatial profile of positron swarm are shown to play a central role in understanding the phenomena.
Illg, Christian; Haag, Michael; Teeny, Nicolas; Wirth, Jens; Fähnle, Manfred
2016-03-01
Scatterings of electrons at quasiparticles or photons are very important for many topics in solid-state physics, e.g., spintronics, magnonics or photonics, and therefore a correct numerical treatment of these scatterings is very important. For a quantum-mechanical description of these scatterings, Fermi's golden rule is used to calculate the transition rate from an initial state to a final state in a first-order time-dependent perturbation theory. One can calculate the total transition rate from all initial states to all final states with Boltzmann rate equations involving Brillouin zone integrations. The numerical treatment of these integrations on a finite grid is often done via a replacement of the Dirac delta distribution by a Gaussian. The Dirac delta distribution appears in Fermi's golden rule where it describes the energy conservation among the interacting particles. Since the Dirac delta distribution is a not a function it is not clear from a mathematical point of view that this procedure is justified. We show with physical and mathematical arguments that this numerical procedure is in general correct, and we comment on critical points.
Priimak, Dmitri
2014-01-01
We present finite differences numerical algorithm for solving 2D spatially homogeneous Boltzmann transport equation for semiconductor superlattices (SL) subject to time dependant electric field along SL axis and constant perpendicular magnetic field. Algorithm is implemented in C language targeted to CPU and in CUDA C language targeted to commodity NVidia GPUs. We compare performance and merits of one implementation versus another and discuss various methods of optimization.
Fluctuating multicomponent lattice Boltzmann model.
Belardinelli, D; Sbragaglia, M; Biferale, L; Gross, M; Varnik, F
2015-02-01
Current implementations of fluctuating lattice Boltzmann equations (FLBEs) describe single component fluids. In this paper, a model based on the continuum kinetic Boltzmann equation for describing multicomponent fluids is extended to incorporate the effects of thermal fluctuations. The thus obtained fluctuating Boltzmann equation is first linearized to apply the theory of linear fluctuations, and expressions for the noise covariances are determined by invoking the fluctuation-dissipation theorem directly at the kinetic level. Crucial for our analysis is the projection of the Boltzmann equation onto the orthonormal Hermite basis. By integrating in space and time the fluctuating Boltzmann equation with a discrete number of velocities, the FLBE is obtained for both ideal and nonideal multicomponent fluids. Numerical simulations are specialized to the case where mean-field interactions are introduced on the lattice, indicating a proper thermalization of the system.
Puglisi, Andrea
2015-01-01
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...
Energy Technology Data Exchange (ETDEWEB)
Mikell, Justin K. [Department of Radiation Physics, University of Texas MD Anderson Cancer Center, Houston, Texas (United States); University of Texas Graduate School of Biomedical Sciences at Houston, Houston, Texas (United States); Klopp, Ann H. [Department of Radiation Oncology, University of Texas MD Anderson Cancer Center, Houston, Texas (United States); Gonzalez, Graciela M.N. [Department of Biostatistics, University of Texas MD Anderson Cancer Center, Houston, Texas (United States); Kisling, Kelly D. [Department of Radiation Physics-Patient Care, University of Texas MD Anderson Cancer Center, Houston, Texas (United States); University of Texas Graduate School of Biomedical Sciences at Houston, Houston, Texas (United States); Price, Michael J. [Department of Physics and Astronomy, Louisiana State University and Agricultural and Mechanical College, Baton Rouge, Louisiana, and Mary Bird Perkins Cancer Center, Baton Rouge, Louisiana (United States); Berner, Paula A. [Department of Radiation Physics, University of Texas MD Anderson Cancer Center, Houston, Texas (United States); Eifel, Patricia J. [Department of Radiation Oncology, University of Texas MD Anderson Cancer Center, Houston, Texas (United States); Mourtada, Firas, E-mail: fmourtad@christianacare.org [Department of Radiation Physics-Patient Care, University of Texas MD Anderson Cancer Center, Houston, Texas (United States); Department of Experimental Diagnostic Imaging, University of Texas MD Anderson Cancer Center, Houston, Texas (United States); Department of Radiation Oncology, Helen F. Graham Cancer Center, Newark, Delaware (United States)
2012-07-01
Purpose: To investigate the dosimetric impact of the heterogeneity dose calculation Acuros (Transpire Inc., Gig Harbor, WA), a grid-based Boltzmann equation solver (GBBS), for brachytherapy in a cohort of cervical cancer patients. Methods and Materials: The impact of heterogeneities was retrospectively assessed in treatment plans for 26 patients who had previously received {sup 192}Ir intracavitary brachytherapy for cervical cancer with computed tomography (CT)/magnetic resonance-compatible tandems and unshielded colpostats. The GBBS models sources, patient boundaries, applicators, and tissue heterogeneities. Multiple GBBS calculations were performed with and without solid model applicator, with and without overriding the patient contour to 1 g/cm{sup 3} muscle, and with and without overriding contrast materials to muscle or 2.25 g/cm{sup 3} bone. Impact of source and boundary modeling, applicator, tissue heterogeneities, and sensitivity of CT-to-material mapping of contrast were derived from the multiple calculations. American Association of Physicists in Medicine Task Group 43 (TG-43) guidelines and the GBBS were compared for the following clinical dosimetric parameters: Manchester points A and B, International Commission on Radiation Units and Measurements (ICRU) report 38 rectal and bladder points, three and nine o'clock, and {sub D2cm3} to the bladder, rectum, and sigmoid. Results: Points A and B, D{sub 2} cm{sup 3} bladder, ICRU bladder, and three and nine o'clock were within 5% of TG-43 for all GBBS calculations. The source and boundary and applicator account for most of the differences between the GBBS and TG-43 guidelines. The D{sub 2cm3} rectum (n = 3), D{sub 2cm3} sigmoid (n = 1), and ICRU rectum (n = 6) had differences of >5% from TG-43 for the worst case incorrect mapping of contrast to bone. Clinical dosimetric parameters were within 5% of TG-43 when rectal and balloon contrast were mapped to bone and radiopaque packing was not overridden
Institute of Scientific and Technical Information of China (English)
段雅丽; 陈先进; 孔令华
2015-01-01
We develop a lattice Boltzmann model for compound Burgers-Korteweg-de Vries ( cBKdV) equation. By properly treating dispersive term uxxx and applying Chapman-Enskog expansion, the governing equation is recovered correctly from lattice Boltzmann equation and local equilibrium distribution functions are obtained. Numerical experiments show that our results agree well with exact solutions and have better numerical accuracy compared with previous numerical results. This hence indicates that the model is satisfactory and efficient.%针对Burgers-Korteweg-de Vries ( cBKdV)复合方程提出一种格子Boltzmann模型。通过恰当地处理色散项uxxx 并运用Chapman-Enskog展开从格子Boltzmann方程推导出宏观方程，从而得到联系微观量与宏观量的局部平衡分布函数。对不同微分方程进行数值实验，数值解与解析解非常吻合，相比于其它数值结果，该格子Boltzmann模型的数值结果更精确，说明该数值模型的高效性。
André, Raíla
2014-01-01
In this work we analyze the dynamics of collisionless self-gravitating systems described by the f(R)-gravity and Boltzmann equation in the weak field approximation, focusing on the Jeans instability for theses systems. The field equations in this approximation were obtained within the Palatini formalism. Through the solution of coupled equations we achieved the collapse criterion for infinite homogeneous fluid and stellar systems, which is given by a dispersion relation. This result is compared with the results of the standard case and the case for f(R)-gravity in metric formalism, in order to see the difference among them. The limit of instability varies according to which theory of gravity is adopted.
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Mendes, Albert C.R., E-mail: albert@fisica.ufjf.br [Departamento de Física, Universidade Federal de Juiz de Fora, 36036-330, Juiz de Fora - MG (Brazil); Takakura, Flavio I., E-mail: takakura@fisica.ufjf.br [Departamento de Física, Universidade Federal de Juiz de Fora, 36036-330, Juiz de Fora - MG (Brazil); Abreu, Everton M.C., E-mail: evertonabreu@ufrrj.br [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: jorge@fisica.ufjf.br [Departamento de Física, Universidade Federal de Juiz de Fora, 36036-330, Juiz de Fora - MG (Brazil)
2017-05-15
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.
Noronha, Jorge
2015-01-01
In this paper we obtain an analytical solution of the relativistic Boltzmann equation under the relaxation time approximation that describes the out-of-equilibrium dynamics of a radially expanding massless gas. This solution is found by mapping this expanding system in flat spacetime to a static flow in the curved spacetime $\\mathrm{AdS}_{2}\\otimes \\mathrm{S}_{2}$. We further derive explicit analytic expressions for the momentum dependence of the single particle distribution function as well as for the spatial dependence of its moments. We find that this dissipative system has the ability to flow as a perfect fluid even though its entropy density does not match the equilibrium form. The non-equilibrium contribution to the entropy density is shown to be due to higher order scalar moments (which possess no hydrodynamical interpretation) of the Boltzmann equation that can remain out of equilibrium but do not couple to the energy-momentum tensor of the system. Thus, in this system the slowly moving hydrodynamic d...
Fogolari, Federico; Corazza, Alessandra; Esposito, Gennaro
2015-04-05
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.
Ludwig Boltzmann: Atomic genius
Energy Technology Data Exchange (ETDEWEB)
Cercignani, C. [Department of Mathematics, Politecnico di Milano (Italy)]. E-mail: carcer@mate.polimi.it
2006-09-15
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.)
KdV方程带修正函数的格子Boltzmann模拟%Lattice Boltzmann model with amending function for KdV equation
Institute of Scientific and Technical Information of China (English)
何郁波; 董晓亮; 林晓艳
2012-01-01
A new lattice Boltzmann model with amending-function for KdV equation is presented. Monotonicity and stability of the scheme are analyzed. The conditions of monotonicity are obtained, under which the stability of the scheme is proved in the L1 norm. Applied the proposed scheme, the solutions of KdV equation are simulated, and numerical results agree with the analytical solutions quite well.%采用一种带修正函数的新格子Boltzmann模型模拟了KdV方程,分析了由此得出的迭代格式的单调性和稳定性,得到了格式的单调性条件.在单调性条件下,迭代格式是L1稳定的.数值模拟结果表明该格式是可行的.
Mendes, Albert C R; Abreu, Everton M C; Neto, Jorge Ananias
2016-01-01
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.
Energy Technology Data Exchange (ETDEWEB)
Horsten, N., E-mail: niels.horsten@kuleuven.be; Baelmans, M. [KU Leuven, Department of Mechanical Engineering, Celestijnenlaan 300A, 3001 Leuven (Belgium); Dekeyser, W. [ITER Organization, route de Vinon-sur-Verdon, 13067 St. Paul lez Durance Cedex (France); Samaey, G. [KU Leuven, Department of Computer Science, Celestijnenlaan 200A, 3001 Leuven (Belgium)
2016-01-15
We derive fluid neutral approximations for a simplified 1D edge plasma model, suitable to study the neutral behavior close to the target of a nuclear fusion divertor, and compare its solutions to the solution of the corresponding kinetic Boltzmann equation. The plasma is considered as a fixed background extracted from a detached 2D simulation. We show that the Maxwellian equilibrium distribution is already obtained very close to the target, justifying the use of a fluid approximation. We compare three fluid neutral models: (i) a diffusion model; (ii) a pressure-diffusion model (i.e., a combination of a continuity and momentum equation) assuming equal neutral and ion temperatures; and (iii) the pressure-diffusion model coupled to a neutral energy equation taking into account temperature differences between neutrals and ions. Partial reflection of neutrals reaching the boundaries is included in both the kinetic and fluid models. We propose two methods to obtain an incident neutral flux boundary condition for the fluid models: one based on a diffusion approximation and the other assuming a truncated Chapman-Enskog distribution. The pressure-diffusion model predicts the plasma sources very well. The diffusion boundary condition gives slightly better results overall. Although including an energy equation still improves the results, the assumption of equal ion and neutral temperature already gives a very good approximation.
Nonequilibrium phenomena in QCD and BEC. Boltzmann and beyond
Energy Technology Data Exchange (ETDEWEB)
Stockamp, T.
2006-12-22
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.)
Electrostatic forces in the Poisson-Boltzmann systems.
Xiao, Li; Cai, Qin; Ye, Xiang; Wang, Jun; Luo, Ray
2013-09-07
Continuum modeling of electrostatic interactions based upon numerical solutions of the Poisson-Boltzmann equation has been widely used in structural and functional analyses of biomolecules. A limitation of the numerical strategies is that it is conceptually difficult to incorporate these types of models into molecular mechanics simulations, mainly because of the issue in assigning atomic forces. In this theoretical study, we first derived the Maxwell stress tensor for molecular systems obeying the full nonlinear Poisson-Boltzmann equation. We further derived formulations of analytical electrostatic forces given the Maxwell stress tensor and discussed the relations of the formulations with those published in the literature. We showed that the formulations derived from the Maxwell stress tensor require a weaker condition for its validity, applicable to nonlinear Poisson-Boltzmann systems with a finite number of singularities such as atomic point charges and the existence of discontinuous dielectric as in the widely used classical piece-wise constant dielectric models.
Lattice-Boltzmann simulation of laser interaction with weakly ionized helium plasmas.
Li, Huayu; Ki, Hyungson
2010-07-01
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.
Reciprocal Symmetric Boltzmann Function and Unified Boson-Fermion Statistics
2007-01-01
The differential equation for Boltzmann's function is replaced by the corresponding discrete finite difference equation. The difference equation is, then, symmetrized so that the equation remains invariant when step d is replaced by -d. The solutions of this equation come in Boson-Fermion pairs. Reciprocal symmetric Boltzmann's function, thus, unifies both Bosonic and Fermionic distributions.
关于Boltzmann方程的空间均匀的ES模型%On the Spatially Homogeneous ES Model of the Boltzmann Equation
Institute of Scientific and Technical Information of China (English)
张显文; 胡适耕; 张诚坚
2006-01-01
We discuss the solution properties of the spatially homogeneous ellipsoidal statistical model (ES model) of the Boltzmann equation with Prandtl number Pr ∈ [2/3,∞). First, we establish the existence and uniqueness result. Second, the trend towards equilibrium and the Maxwellian lower bound estimates of the solution are established. Finally, we prove the entropy identity and obtain dissipation of the entropy of this equation.%在Prandtl数Pr∈[2/3,∞)的情况下,我们讨论了Boltzmann方程的空间均匀的椭圆统计模型.首先,我们建立了解的存在唯一性.其次,我们证明了该解收敛到平衡态并给出了其Maxwell分布型的下界估计.最后,我们给出了熵等式从而证明了该方程的熵是衰减的.
1989-12-01
and f,+ respectively from equation (39). 3.1.4 (I-hC) Matriz This section of the program simultaneously loads the C-matrix, mul- tiplies it by the time...and others. Numerical Analysis. Boston : Prindle, Weber and Schmidt, 1981. 14. Kieffer, L.J. "A Compilation of Electron Collision Cross Section Data
Full Symmetry Groups and Exact Solutions to BKP and GKP Equations
Directory of Open Access Journals (Sweden)
Bo Ren
2014-01-01
Full Text Available We investigate the (2+1-dimensional nonlinear BKP and GKP equations with the modified direct CK’s method. Then, we get its Lie point groups and the full symmetry group, and a relationship is constructed between the new solutions and the old one. Based on the relationship, the new solutions can be obtained by using a given solution of the equations.
Parametric lattice Boltzmann method
Shim, Jae Wan
2017-06-01
The discretized equilibrium distributions of the lattice Boltzmann method are presented by using the coefficients of the Lagrange interpolating polynomials that pass through the points related to discrete velocities and using moments of the Maxwell-Boltzmann distribution. The ranges of flow velocity and temperature providing positive valued distributions vary with regulating discrete velocities as parameters. New isothermal and thermal compressible models are proposed for flows of the level of the isothermal and thermal compressible Navier-Stokes equations. Thermal compressible shock tube flows are simulated by only five on-lattice discrete velocities. Two-dimensional isothermal and thermal vortices provoked by the Kelvin-Helmholtz instability are simulated by the parametric models.
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Kawakami, H.; Urabe, J.; Yukimura, K. (Doshisha Univ., Kyoto (Japan))
1991-03-20
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.
Wang, Peng; Wang, Lian-Ping; Guo, Zhaoli
2016-10-01
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.
Wang, Peng; Wang, Lian-Ping; Guo, Zhaoli
2016-10-01
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 kmaxη >3 , where kmax is the maximum resolved wave number and η is the flow Kolmogorov length. This resolution requirement can be contrasted with the requirements of kmaxη >1 for the pseudospectral method and kmaxη >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.
Institute of Scientific and Technical Information of China (English)
Deng Yun-Kun; Xiao Deng-Ming
2013-01-01
The electron swarm parameters including the density-normalized effective ionization coefficients (α-η)/N and the electron drift velocities Ve are calculated for a gas mixture of CF3I with N2 and CO2 by solving the Boltzmann equation in the condition of a steady-state Townsend (SST) experiment.The overall density-reduced electric field strength is from 100 Td to 1000 Td (1 Td =10-17 V·cm2),while the CF3I content k in the gas mixture can be varied over the range from 0％ to 100％.From the variation of (α-η)/N with the CF3I mixture ratio k,the limiting field strength (E/N)lim for each CF3I concentration is derived.It is found that for the mixtures with 70％ CF3I,the values of (E/N)lim are essentially the same as that for pure SF6.Additionally,the global warming potential (GWP) and the liquefaction temperature of the gas mixtures are also taken into account to evaluate the possibility of application in the gas insulation of power equipment.
A lattice Boltzmann coupled to finite volumes method for solving phase change problems
Directory of Open Access Journals (Sweden)
El Ganaoui Mohammed
2009-01-01
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.
Franz, Delbert D.; Melching, Charles S.
1997-01-01
The Full EQuations (FEQ) model is a computer program for solution of the full, dynamic equations of motion for one-dimensional unsteady flow in open channels and through control structures. A stream system that is simulated by application of FEQ is subdivided into stream reaches (branches), parts of the stream system for which complete information on flow and depth are not required (dummy branches), and level-pool reservoirs. These components are connected by special features; that is, hydraulic control structures, including junctions, bridges, culverts, dams, waterfalls, spillways, weirs, side weirs, and pumps. The principles of conservation of mass and conservation of momentum are used to calculate the flow and depth throughout the stream system resulting from known initial and boundary conditions by means of an implicit finite-difference approximation at fixed points (computational nodes). The hydraulic characteristics of (1) branches including top width, area, first moment of area with respect to the water surface, conveyance, and flux coefficients and (2) special features (relations between flow and headwater and (or) tail-water elevations, including the operation of variable-geometry structures) are stored in function tables calculated in the companion program, Full EQuations UTiLities (FEQUTL). Function tables containing other information used in unsteady-flow simulation (boundary conditions, tributary inflows or outflows, gate settings, correction factors, characteristics of dummy branches and level-pool reservoirs, and wind speed and direction) are prepared by the user as detailed in this report. In the iterative solution scheme for flow and depth throughout the stream system, an interpolation of the function tables corresponding to the computational nodes throughout the stream system is done in the model. FEQ can be applied in the simulation of a wide range of stream configurations (including loops), lateral-inflow conditions, and special features. The
Boltzmann-Langevin one-body dynamics for fermionic systems
Directory of Open Access Journals (Sweden)
Napolitani P.
2012-07-01
Full Text Available A full implementation of the Boltzmann-Langevin equation for fermionic systems is introduced in a transport model for dissipative collisions among heavy nuclei. Fluctuations are injected in phase space and not, like in more conventional approaches, as a projection on suitable subspaces. The advantage of this model is to be specifically adapted to describe processes characterised by instabilities, like the formation of fragments from a hot nuclear system, and by dissipation, like the transparency in nucleus-nucleus collisions.
Lou, James; Shih, Chun-Yu; Lee, Eric
2010-01-05
Diffusiophoresis in concentrated suspensions of spherical colloids with charge-regulated surface is investigated theoretically. The charge-regulated surface considered here is the generalization of conventional constant surface potential and constant surface charge density situations. Kuwabara's unit cell model is adopted to describe the system and a pseudospectral method based on Chebyshev polynomial is employed to solve the governing general electrokinetic equations. Excellent agreements with experimental data available in literature were obtained for the limiting case of constant surface potential and very dilute suspension. It is found, among other things, that in general the larger the number of dissociated functional groups on particle surface is, the higher the particle surface potential, hence the larger the magnitude of the particle mobility. The electric potential on particle surface depends on both the concentration of dissociated hydrogen ions and the concentration of electrolyte in the solution. The electric potential on particle surface turns out to be the dominant factor in the determination of the eventual particle diffusiophoretic mobility. Local maximum of diffusiophoretic mobility as a function of double layer thickness is observed. Its reason and influence is discussed. Corresponding behavior for the constant potential situation, however, may yield a monotonously increasing profile.
Directory of Open Access Journals (Sweden)
Xinzhi Liu
1998-01-01
Full Text Available This paper studies a class of high order delay partial differential equations. Employing high order delay differential inequalities, several oscillation criteria are established for such equations subject to two different boundary conditions. Two examples are also given.
Mathematical analysis of plasmonic resonances for nanoparticles: The full Maxwell equations
Ammari, Habib; Ruiz, Matias; Yu, Sanghyeon; Zhang, Hai
2016-09-01
In this paper we use the full Maxwell equations for light propagation in order to analyze plasmonic resonances for nanoparticles. We mathematically define the notion of plasmonic resonance and analyze its shift and broadening with respect to changes in size, shape, and arrangement of the nanoparticles, using the layer potential techniques associated with the full Maxwell equations. We present an effective medium theory for resonant plasmonic systems and derive a condition on the volume fraction under which the Maxwell-Garnett theory is valid at plasmonic resonances.
The Einstein-Boltzmann system and positivity
Lee, Ho
2012-01-01
The Einstein-Boltzmann system is studied, with particular attention to the non-negativity of the solution of the Boltzmann equation. A new parametrization of post-collisional momenta in general relativity is introduced and then used to simplify the conditions on the collision cross-section given by Bancel and Choquet-Bruhat. The non-negativity of solutions of the Boltzmann equation on a given curved spacetime has been studied by Bichteler and by Tadmon. By examining to what extent the results of these authors apply in the framework of Bancel and Choquet-Bruhat, the non-negativity problem for the Einstein-Boltzmann system is resolved for a certain class of scattering kernels. It is emphasized that it is a challenge to extend the existing theory of the Cauchy problem for the Einstein-Boltzmann system so as to include scattering kernels which are physically well-motivated.
Energy Technology Data Exchange (ETDEWEB)
Uchida, S.; Sugawara, H.; Ventzek, P.; Sakai, Y. [Hokkaido University, Sapporo (Japan)
1998-06-01
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.
Newton-Krylov-Schwarz algorithms for the 2D full potential equation
Energy Technology Data Exchange (ETDEWEB)
Cai, Xiao-Chuan [Univ. of Colorado, Boulder, CO (United States); Gropp, W.D. [Argonne National Lab., IL (United States); Keyes, D.E. [Old Dominion Univ. Norfolk, VA (United States)] [and others
1996-12-31
We study parallel two-level overlapping Schwarz algorithms for solving nonlinear finite element problems, in particular, for the full potential equation of aerodynamics discretized in two dimensions with bilinear elements. The main algorithm, Newton-Krylov-Schwarz (NKS), employs an inexact finite-difference Newton method and a Krylov space iterative method, with a two-level overlapping Schwarz method as a preconditioner. We demonstrate that NKS, combined with a density upwinding continuation strategy for problems with weak shocks, can be made robust for this class of mixed elliptic-hyperbolic nonlinear partial differential equations, with proper specification of several parameters. We study upwinding parameters, inner convergence tolerance, coarse grid density, subdomain overlap, and the level of fill-in in the incomplete factorization, and report favorable choices for numerical convergence rate and overall execution time on a distributed-memory parallel computer.
Unsteady interaction of a viscous fluid with an elastic shell modeled by full von Karman equations
Chueshov, Igor; Ryzhkova, Iryna
We study well-posedness and asymptotic dynamics of a coupled system consisting of linearized 3D Navier-Stokes equations in a bounded domain and a classical (nonlinear) full von Karman shallow shell equation that accounts for both transversal and lateral displacements on a flexible part of the boundary. We also take into account rotational inertia of filaments of the shell. Our main result shows that the problem generates a semiflow in an appropriate phase space. The regularity provided by viscous dissipation in the fluid allows us to consider simultaneously both cases of presence inertia in the lateral displacements and its absence. Our second result states the existence of a compact global attractor for this semiflow in the case of presence of (rotational) damping in the transversal component and a particular structure of external forces.
Unsteady interaction of a viscous fluid with an elastic shell modeled by full von Karman equations
Chueshov, Igor
2011-01-01
We study well-posedness and asymptotic dynamics of a coupled system consisting of linearized 3D Navier--Stokes equations in a bounded domain and a classical (nonlinear) full von Karman shallow shell equations that accounts for both transversal and lateral displacements on a flexible part of the boundary. We also take into account rotational inertia of filaments of the shell. Out main result shows that the problem generates a semiflow in an appropriate phase space. The regularity provided by viscous dissipation in the fluid allows us to consider simultaneously both cases of presence inertia in the lateral displacements and its absence. Our second result states the existence of a compact global attractor for this semiflow in the case of presence of (rotational) damping in the transversal component and a particular structure of external forces.
Energy Technology Data Exchange (ETDEWEB)
Haverkort, J.W. [Centrum Wiskunde & Informatica, P.O. Box 94079, 1090 GB Amsterdam (Netherlands); Dutch Institute for Fundamental Energy Research, P.O. Box 6336, 5600 HH Eindhoven (Netherlands); Blank, H.J. de [Dutch Institute for Fundamental Energy Research, P.O. Box 6336, 5600 HH Eindhoven (Netherlands); Huysmans, G.T.A. [ITER Organization, Route de Vinon sur Verdon, 13115 Saint Paul Lez Durance (France); Pratt, J. [Dutch Institute for Fundamental Energy Research, P.O. Box 6336, 5600 HH Eindhoven (Netherlands); Koren, B., E-mail: b.koren@tue.nl [Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven (Netherlands)
2016-07-01
Numerical simulations form an indispensable tool to understand the behavior of a hot plasma that is created inside a tokamak for providing nuclear fusion energy. Various aspects of tokamak plasmas have been successfully studied through the reduced magnetohydrodynamic (MHD) model. The need for more complete modeling through the full MHD equations is addressed here. Our computational method is presented along with measures against possible problems regarding pollution, stability, and regularity. The problem of ensuring continuity of solutions in the center of a polar grid is addressed in the context of a finite element discretization of the full MHD equations. A rigorous and generally applicable solution is proposed here. Useful analytical test cases are devised to verify the correct implementation of the momentum and induction equation, the hyperdiffusive terms, and the accuracy with which highly anisotropic diffusion can be simulated. A striking observation is that highly anisotropic diffusion can be treated with the same order of accuracy as isotropic diffusion, even on non-aligned grids, as long as these grids are generated with sufficient care. This property is shown to be associated with our use of a magnetic vector potential to describe the magnetic field. Several well-known instabilities are simulated to demonstrate the capabilities of the new method. The linear growth rate of an internal kink mode and a tearing mode are benchmarked against the results of a linear MHD code. The evolution of a tearing mode and the resulting magnetic islands are simulated well into the nonlinear regime. The results are compared with predictions from the reduced MHD model. Finally, a simulation of a ballooning mode illustrates the possibility to use our method as an ideal MHD method without the need to add any physical dissipation.
Revisiting the radio interferometer measurement equation. I. A full-sky Jones formalism
Smirnov, Oleg M
2011-01-01
Since its formulation by Hamaker et al., the radio interferometer measurement equation (RIME) has provided a rigorous mathematical basis for the development of novel calibration methods and techniques, including various approaches to the problem of direction-dependent effects (DDEs). This series of papers aims to place recent developments in the treatment of DDEs into one RIME-based mathematical framework, and to demonstrate the ease with which the various effects can be described and understood. It also aims to show the benefits of a RIME-based approach to calibration. Paper I re-derives the RIME from first principles, extends the formalism to the full-sky case, and incorporates DDEs. Paper II then uses the formalism to describe self-calibration, both with a full RIME, and with the approximate equations of older software packages, and shows how this is affected by DDEs. It also gives an overview of real-life DDEs and proposed methods of dealing with them. Applying this to WSRT data (Paper III) results in a n...
Schminkey, Donna L; von Oertzen, Timo; Bullock, Linda
2016-08-01
With increasing access to population-based data and electronic health records for secondary analysis, missing data are common. In the social and behavioral sciences, missing data frequently are handled with multiple imputation methods or full information maximum likelihood (FIML) techniques, but healthcare researchers have not embraced these methodologies to the same extent and more often use either traditional imputation techniques or complete case analysis, which can compromise power and introduce unintended bias. This article is a review of options for handling missing data, concluding with a case study demonstrating the utility of multilevel structural equation modeling using full information maximum likelihood (MSEM with FIML) to handle large amounts of missing data. MSEM with FIML is a parsimonious and hypothesis-driven strategy to cope with large amounts of missing data without compromising power or introducing bias. This technique is relevant for nurse researchers faced with ever-increasing amounts of electronic data and decreasing research budgets. © 2016 Wiley Periodicals, Inc.
Schaap, M. G.
2013-12-01
This DOE-funded study is a collaboration between Oregon State University (led by Dr. Dorthe Wildenschild) and the University of Arizona to investigate pore-scale aspects of capillary trapping to enhance the efficiency of geological CO2 sequestration in deep saline aquifers where super-critical conditions prevail. Compared to most current reservoir-scale studies, our research takes several steps back in scale to observe and model trapping at the pore-scale using a combination of computed micro-tomography imaging (performed by OSU) and multi-phase/multi-component lattice Boltzmann (LB) simulations (carried out by UA). The main objective is to quantify how pore-scale mechanisms translate into continuum scale properties that can subsequently support improved modelling of sequestration at large spatio-temporal scales. For the purposes of this project it is important to correctly simulate the physical conditions under which super-critical CO2 will be present after injection into the host rock. In practice this means that the LB model should be able to handle the pressures (P), densities (ρ), temperatures (T) that prevail in deep geological media. A logical way of dealing with is is to combine a single-component LB model with and Equation of State (EOS) that describes the physical interrelations among P, ρ and T (Yuan and Scheafer, 2006). Previously, we showed that the Peng-Robinson (PR) EOS provides an excellent fit to super-critical conditions for the pure CO2 system. However, simulating pure-CO2 systems is not sufficient as the super-critical CO2 will co-exist (and interact) with brine present in the saline aquifers. In effect this means that we need to simulate multi-component systems: one phase being the super-critical CO2, the other phase being a brine of varying salinity. Previously, we have used used a Shan-Chen-type model (Shan Chen, 1993, 1994) as modified by Martys and Chen (1996) for simplified capillari pressure dominated air-water systems in porous media
A new full discrete stabilized viscosity method for transient Navier-Stokes equations
Institute of Scientific and Technical Information of China (English)
Yan-mei QIN; Min-fu FENG; Tian-xiao ZHOU
2009-01-01
A new full discrete stabilized viscosity method for the transient Navier-Stokes equations with the high Reynolds number (small viscosity coefficient) is proposed based on the pressure projection and the extrapolated trapezoidal rule.The transient NavierStokes equations are fully discretized by the continuous equal-order finite elements in space and the reduced Crank-Nicolson scheme in time.The new stabilized method is stable and has many attractive properties.First,the system is stable for the equal-order combination of discrete continuous velocity and pressure spaces because of adding a pressure projection term.Second,the artifical viscosity parameter is added to the viscosity coefficient as a stability factor,so the system is antidiffusive.Finally,the method requires only the solution to a linear system at every time step.Stability and convergence of the method is proved.The error estimation results show that the method has a second-order accuracy,and the constant in the estimation is independent of the viscosity coefficient.The numerical results are given,which demonstrate the advantages of the method presented.
Optimal decay rates of classical solutions for the full compressible MHD equations
Gao, Jincheng; Tao, Qiang; Yao, Zheng-an
2016-04-01
In this paper, we are concerned with optimal decay rates for higher-order spatial derivatives of classical solutions to the full compressible MHD equations in three-dimensional whole space. If the initial perturbation is small in {H^3}-norm and bounded in {L^q(qin [1, 6/5 ))}-norm, we apply the Fourier splitting method by Schonbek (Arch Ration Mech Anal 88:209-222, 1985) to establish optimal decay rates for the second-order spatial derivatives of solutions and the third-order spatial derivatives of magnetic field in {L^2}-norm. These results improve the work of Pu and Guo (Z Angew Math Phys 64:519-538, 2013).
Accurate deterministic solutions for the classic Boltzmann shock profile
Yue, Yubei
The Boltzmann equation or Boltzmann transport equation is a classical kinetic equation devised by Ludwig Boltzmann in 1872. It is regarded as a fundamental law in rarefied gas dynamics. Rather than using macroscopic quantities such as density, temperature, and pressure to describe the underlying physics, the Boltzmann equation uses a distribution function in phase space to describe the physical system, and all the macroscopic quantities are weighted averages of the distribution function. The information contained in the Boltzmann equation is surprisingly rich, and the Euler and Navier-Stokes equations of fluid dynamics can be derived from it using series expansions. Moreover, the Boltzmann equation can reach regimes far from the capabilities of fluid dynamical equations, such as the realm of rarefied gases---the topic of this thesis. Although the Boltzmann equation is very powerful, it is extremely difficult to solve in most situations. Thus the only hope is to solve it numerically. But soon one finds that even a numerical simulation of the equation is extremely difficult, due to both the complex and high-dimensional integral in the collision operator, and the hyperbolic phase-space advection terms. For this reason, until few years ago most numerical simulations had to rely on Monte Carlo techniques. In this thesis I will present a new and robust numerical scheme to compute direct deterministic solutions of the Boltzmann equation, and I will use it to explore some classical gas-dynamical problems. In particular, I will study in detail one of the most famous and intrinsically nonlinear problems in rarefied gas dynamics, namely the accurate determination of the Boltzmann shock profile for a gas of hard spheres.
Solution behaviors in coupled Schrödinger equations with full-modulated nonlinearities
Pınar, Zehra; Deliktaş, Ekin
2017-02-01
The nonlinear partial differential equations have an important role in real life problems. To obtain the exact solutions of the nonlinear partial differential equations, a number of approximate methods are known in the literature. In this work, a time- space modulated nonlinearities of coupled Schrödinger equations are considered. We provide a large class of Jacobi-elliptic solutions via the auxiliary equation method with sixth order nonlinearity and the Chebyshev approximation.
Hierarchical Boltzmann simulations and model error estimation
Torrilhon, Manuel; Sarna, Neeraj
2017-08-01
A hierarchical simulation approach for Boltzmann's equation should provide a single numerical framework in which a coarse representation can be used to compute gas flows as accurately and efficiently as in computational fluid dynamics, but a subsequent refinement allows to successively improve the result to the complete Boltzmann result. We use Hermite discretization, or moment equations, for the steady linearized Boltzmann equation for a proof-of-concept of such a framework. All representations of the hierarchy are rotationally invariant and the numerical method is formulated on fully unstructured triangular and quadrilateral meshes using a implicit discontinuous Galerkin formulation. We demonstrate the performance of the numerical method on model problems which in particular highlights the relevance of stability of boundary conditions on curved domains. The hierarchical nature of the method allows also to provide model error estimates by comparing subsequent representations. We present various model errors for a flow through a curved channel with obstacles.
Suzuki, Hideyuki; Imura, Jun-ichi; Horio, Yoshihiko; Aihara, Kazuyuki
2013-01-01
The chaotic Boltzmann machine proposed in this paper is a chaotic pseudo-billiard system that works as a Boltzmann machine. Chaotic Boltzmann machines are shown numerically to have computing abilities comparable to conventional (stochastic) Boltzmann machines. Since no randomness is required, efficient hardware implementation is expected. Moreover, the ferromagnetic phase transition of the Ising model is shown to be characterised by the largest Lyapunov exponent of the proposed system. In general, a method to relate probabilistic models to nonlinear dynamics by derandomising Gibbs sampling is presented.
Cartesian Mesh Linearized Euler Equations Solver for Aeroacoustic Problems around Full Aircraft
Directory of Open Access Journals (Sweden)
Yuma Fukushima
2015-01-01
Full Text Available The linearized Euler equations (LEEs solver for aeroacoustic problems has been developed on block-structured Cartesian mesh to address complex geometry. Taking advantage of the benefits of Cartesian mesh, we employ high-order schemes for spatial derivatives and for time integration. On the other hand, the difficulty of accommodating curved wall boundaries is addressed by the immersed boundary method. The resulting LEEs solver is robust to complex geometry and numerically efficient in a parallel environment. The accuracy and effectiveness of the present solver are validated by one-dimensional and three-dimensional test cases. Acoustic scattering around a sphere and noise propagation from the JT15D nacelle are computed. The results show good agreement with analytical, computational, and experimental results. Finally, noise propagation around fuselage-wing-nacelle configurations is computed as a practical example. The results show that the sound pressure level below the over-the-wing nacelle (OWN configuration is much lower than that of the conventional DLR-F6 aircraft configuration due to the shielding effect of the OWN configuration.
Boltzmann's "H"-Theorem and the Assumption of Molecular Chaos
Boozer, A. D.
2011-01-01
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.…
Li, Jingzhi; Liu, Hongyu; Rondi, Luca; Uhlmann, Gunther
2015-04-01
We develop a very general theory on the regularized approximate invisibility cloaking for the wave scattering governed by the Helmholtz equation in any space dimensions via the approach of transformation optics. There are four major ingredients in our proposed theory: (1) The non-singular cloaking medium is obtained by the push-forwarding construction through a transformation that blows up a subset in the virtual space, where is an asymptotic regularization parameter. will degenerate to K 0 as , and in our theory K 0 could be any convex compact set in , or any set whose boundary consists of Lipschitz hypersurfaces, or a finite combination of those sets. (2) A general lossy layer with the material parameters satisfying certain compatibility integral conditions is employed right between the cloaked and cloaking regions. (3) The contents being cloaked could also be extremely general, possibly including, at the same time, generic mediums and, sound-soft, sound-hard and impedance-type obstacles, as well as some sources or sinks. (4) In order to achieve a cloaking device of compact size, particularly for the case when is not "uniformly small", an assembly-by-components, the (ABC) geometry is developed for both the virtual and physical spaces and the blow-up construction is based on concatenating different components. Within the proposed framework, we show that the scattered wave field corresponding to a cloaking problem will converge to u 0 as , with u 0 being the scattered wave field corresponding to a sound-hard K 0. The convergence result is used to theoretically justify the approximate full and partial invisibility cloaks, depending on the geometry of K 0. On the other hand, the convergence results are conducted in a much more general setting than what is needed for the invisibility cloaking, so they are of significant mathematical interest for their own sake. As for applications, we construct three types of full and partial cloaks. Some numerical experiments are
Directory of Open Access Journals (Sweden)
Thereza A. Soares
2004-08-01
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.
Single-site Green function of the Dirac equation for full-potential electron scattering
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Kordt, Pascal
2012-05-30
I present an elaborated analytical examination of the Green function of an electron scattered at a single-site potential, for both the Schroedinger and the Dirac equation, followed by an efficient numerical solution, in both cases for potentials of arbitrary shape without an atomic sphere approximation. A numerically stable way to calculate the corresponding regular and irregular wave functions and the Green function is via the angular Lippmann-Schwinger integral equations. These are solved based on an expansion in Chebyshev polynomials and their recursion relations, allowing to rewrite the Lippmann-Schwinger equations into a system of algebraic linear equations. Gonzales et al. developed this method for the Schroedinger equation, where it gives a much higher accuracy compared to previous perturbation methods, with only modest increase in computational effort. In order to apply it to the Dirac equation, I developed relativistic Lippmann-Schwinger equations, based on a decomposition of the potential matrix into spin spherical harmonics, exploiting certain properties of this matrix. The resulting method was embedded into a Korringa-Kohn-Rostoker code for density functional calculations. As an example, the method is applied by calculating phase shifts and the Mott scattering of a tungsten impurity. (orig.)
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Chatterjee, Kausik, E-mail: kausik.chatterjee@aggiemail.usu.edu [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: john.roadcap@us.af.mil [Air Force Research Laboratory, Kirtland AFB, NM 87117 (United States); Singh, Surendra, E-mail: surendra-singh@utulsa.edu [Department of Electrical Engineering, The University of Tulsa, Tulsa, OK 74104 (United States)
2014-11-01
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.
Adaptive Lattice Boltzmann Model for Compressible Flows
Institute of Scientific and Technical Information of China (English)
无
2000-01-01
A new lattice Boltzmann model for compressible flows is presented. The main difference from the standard lattice Boltzmann model is that the particle velocities are no longer constant, but vary with the mean velocity and internal energy. The adaptive nature of the particle velocities permits the mean flow to have a high Mach number. The introduction of a particle potential energy makes the model suitable for a perfect gas with arbitrary specific heat ratio. The Navier-Stokes (N-S) equations are derived by the Chapman-Enskog method from the BGK Boltzmann equation. Two kinds of simulations have been carried out on the hexagonal lattice to test the proposed model. One is the Sod shock-tube simulation. The other is a strong shock of Mach number 5.09 diffracting around a corner.
Linearized lattice Boltzmann method for micro- and nanoscale flow and heat transfer.
Shi, Yong; Yap, Ying Wan; Sader, John E
2015-07-01
Ability to characterize the heat transfer in flowing gases is important for a wide range of applications involving micro- and nanoscale devices. Gas flows away from the continuum limit can be captured using the Boltzmann equation, whose analytical solution poses a formidable challenge. An efficient and accurate numerical simulation of the Boltzmann equation is thus highly desirable. In this article, the linearized Boltzmann Bhatnagar-Gross-Krook equation is used to develop a hierarchy of thermal lattice Boltzmann (LB) models based on half-space Gaussian-Hermite (GH) quadrature ranging from low to high algebraic precision, using double distribution functions. Simplified versions of the LB models in the continuum limit are also derived, and are shown to be consistent with existing thermal LB models for noncontinuum heat transfer reported in the literature. Accuracy of the proposed LB hierarchy is assessed by simulating thermal Couette flows for a wide range of Knudsen numbers. Effects of the underlying quadrature schemes (half-space GH vs full-space GH) and continuum-limit simplifications on computational accuracy are also elaborated. The numerical findings in this article provide direct evidence of improved computational capability of the proposed LB models for modeling noncontinuum flows and heat transfer at small length scales.
Sman, van der R.G.M.
2006-01-01
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
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Uesaka, S. [Kyoto University, Kyoto (Japan). Faculty of Engineering; Watanabe, T.; Sassa, K. [Kyoto University, Kyoto (Japan)
1997-05-27
Algorithm is constructed and a program developed for a full-wave inversion (FWI) method utilizing the elastic wave equation in seismic exploration. The FWI method is a method for obtaining a physical property distribution using the whole observed waveforms as the data. It is capable of high resolution which is several times smaller than the wavelength since it can handle such phenomena as wave reflection and dispersion. The method for determining the P-wave velocity structure by use of the acoustic wave equation does not provide information about the S-wave velocity since it does not consider S-waves or converted waves. In an analysis using the elastic wave equation, on the other hand, not only P-wave data but also S-wave data can be utilized. In this report, under such circumstances, an inverse analysis algorithm is constructed on the basis of the elastic wave equation, and a basic program is developed. On the basis of the methods of Mora and of Luo and Schuster, the correction factors for P-wave and S-wave velocities are formulated directly from the elastic wave equation. Computations are performed and the effects of the hypocenter frequency and vibration transmission direction are examined. 6 refs., 8 figs.
Crystallographic Lattice Boltzmann Method
Namburi, Manjusha; Krithivasan, Siddharth; Ansumali, Santosh
2016-01-01
Current approaches to Direct Numerical Simulation (DNS) are computationally quite expensive for most realistic scientific and engineering applications of Fluid Dynamics such as automobiles or atmospheric flows. The Lattice Boltzmann Method (LBM), with its simplified kinetic descriptions, has emerged as an important tool for simulating hydrodynamics. In a heterogeneous computing environment, it is often preferred due to its flexibility and better parallel scaling. However, direct simulation of realistic applications, without the use of turbulence models, remains a distant dream even with highly efficient methods such as LBM. In LBM, a fictitious lattice with suitable isotropy in the velocity space is considered to recover Navier-Stokes hydrodynamics in macroscopic limit. The same lattice is mapped onto a cartesian grid for spatial discretization of the kinetic equation. In this paper, we present an inverted argument of the LBM, by making spatial discretization as the central theme. We argue that the optimal spatial discretization for LBM is a Body Centered Cubic (BCC) arrangement of grid points. We illustrate an order-of-magnitude gain in efficiency for LBM and thus a significant progress towards feasibility of DNS for realistic flows. PMID:27251098
Integrated inversion using combined wave-equation tomography and full waveform inversion
Wang, Haiyang; Singh, Satish C.; Calandra, Henri
2014-07-01
Wave-equation tomography (WT) and full waveform inversion (FWI) are combined through a hybrid misfit function to estimate high-resolution subsurface structures starting from a poorly constrained initial velocity model. Both methods share the same wavefield forward modelling and inversion schemes, while they differ only on the ways to calculate misfit functions and hence the ways to sample in the model space. Aiming at minimizing the cross-correlation phase delay between synthetic and real data, WT can be used to retrieve the long- and middle-wavelength model components, which are essential to FWI. Compared to ray-based traveltime tomography that is based on asymptotic high-frequency approximation, WT provides a better resolution by exploring the band-limited feature of seismic wavefield. On the other hand, FWI is capable of resolving the short-wavelength model component, complementing the WT. In this study, we apply WT to surface first-arrival refraction data, and apply FWI to both refraction and reflection data. We assign adaptive weights to the two different misfit measurements and build a progressive inversion strategy. To illustrate the advantage of our strategy over conventional `ray tomography + FWI' approach, we show in a synthetic lens test that WT can provide extra subsurface information that is critical for a successful FWI application. To further show the efficiency, we test our strategy on the 2-D Marmousi model where satisfactory inversion results are achieved without much manual intervention. Finally, we apply the inversion strategy to a deep-water seismic data set acquired offshore Sumatra with a 12-km-long streamer. In order to alleviate several practical problems posed by the deep-water setting, we apply downward continuation (DC) to generate a virtual ocean bottom experiment data set prior to inversion. The new geometry after DC boosts up the shallow refractions, as well as avoiding cumbersome modelling through the thick water column, thus
Dynamically coupling the full Stokes equations and the SIA - the ISCAL method
Ahlkrona, Josefin; Lötstedt, Per; Zwinger, Thomas
2016-04-01
The ISCAL (Ice Sheet Coupled Approximation Level) is a new method for computing ice sheet flow, implemented in the community ice-sheet model Elmer/Ice. ISCAL combines the exact Stokes equations with the set of equations obained applying the Shallow Ice Approximation (SIA), confining the computationally cheap SIA to only those areas where it is accurate enough. The coupling is done dynamically and is based on SIA error estimates. In this way, the simulation times are significantly reduced, while the user has control over the error introduced by the approximations applied. Accuracy and efficiency is demonstrated using synthetic set-ups as well as for the Greenland Ice Sheet.
Boltzmann-Electron Model in Aleph.
Energy Technology Data Exchange (ETDEWEB)
Hughes, Thomas Patrick; Hooper, Russell
2014-11-01
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
How good is the Lattice Boltzmann method?
Kocheemoolayil, Joseph; Barad, Michael; Kiris, Cetin
2016-11-01
Conflicting opinions exist in literature regarding how efficient the lattice Boltzmann method is relative to high-order finite difference approximations of the Navier-Stokes equations on Cartesian meshes, especially at high Mach numbers. We address the question from the pragmatic viewpoint of a practitioner. Dispersion, dissipation and aliasing errors of various lattice Boltzmann models are systematically quantified. The number of floating point operations and memory required for a desired accuracy level are carefully compared for the two numerical methods. Turbulent kinetic energy budgets for several standard test cases such as the decaying Taylor-Green vortex problem are used to evaluate how effective the stabilization mechanisms necessary for lattice Boltzmann method at high Reynolds numbers are. Detailed comments regarding the cyclomatic complexity of the underlying software, scalability of the underlying algorithm on state-of-the-art high-performance computing platforms and wall clock times and relative accuracy for selected simulations conducted using the two approaches are also made.
Energy Technology Data Exchange (ETDEWEB)
Watanabe, T.; Sassa, K. [Kyoto University, Kyoto (Japan); Uesaka, S. [Kyoto University, Kyoto (Japan). Faculty of Engineering
1996-10-01
The effect of initial models on full-wave inversion (FWI) analysis based on acoustic wave-equation was studied for elastic wave tomography of underground structures. At present, travel time inversion using initial motion travel time is generally used, and inverse analysis is conducted using the concept `ray,` assuming very high wave frequency. Although this method can derive stable solutions relatively unaffected by initial model, it uses only the data of initial motion travel time. FWI calculates theoretical waveform at each receiver using all of observed waveforms as data by wave equation modeling where 2-D underground structure is calculated by difference calculus under the assumption that wave propagation is described by wave equation of P wave. Although it is a weak point that FWI is easily affected by noises in an initial model and data, it is featured by high resolution of solutions. This method offers very excellent convergence as a proper initial model is used, resulting in sufficient performance, however, it is strongly affected by initial model. 2 refs., 7 figs., 1 tab.
Meshless lattice Boltzmann method for the simulation of fluid flows.
Musavi, S Hossein; Ashrafizaadeh, Mahmud
2015-02-01
A meshless lattice Boltzmann numerical method is proposed. The collision and streaming operators of the lattice Boltzmann equation are separated, as in the usual lattice Boltzmann models. While the purely local collision equation remains the same, we rewrite the streaming equation as a pure advection equation and discretize the resulting partial differential equation using the Lax-Wendroff scheme in time and the meshless local Petrov-Galerkin scheme based on augmented radial basis functions in space. The meshless feature of the proposed method makes it a more powerful lattice Boltzmann solver, especially for cases in which using meshes introduces significant numerical errors into the solution, or when improving the mesh quality is a complex and time-consuming process. Three well-known benchmark fluid flow problems, namely the plane Couette flow, the circular Couette flow, and the impulsively started cylinder flow, are simulated for the validation of the proposed method. Excellent agreement with analytical solutions or with previous experimental and numerical results in the literature is observed in all the simulations. Although the computational resources required for the meshless method per node are higher compared to that of the standard lattice Boltzmann method, it is shown that for cases in which the total number of nodes is significantly reduced, the present method actually outperforms the standard lattice Boltzmann method.
Isotropy conditions for lattice Boltzmann schemes. Application to D2Q9*
Directory of Open Access Journals (Sweden)
Graille Benjamin
2012-04-01
Full Text Available In this paper, we recall the linear version of the lattice Boltzmann schemes in the framework proposed by d’Humières. According to the equivalent equations we introduce a definition for a scheme to be isotropic at some order. This definition is chosen such that the equivalent equations are preserved by orthogonal transformations of the frame. The property of isotropy can be read through a group operation and then implies a sequence of relations on relaxation times and equilibrium states that characterizes a lattice Boltzmann scheme. We propose a method to select the parameters of the scheme according to the desired order of isotropy. Applying it to the D2Q9 scheme yields the classical constraints for the first and second orders and some non classical for the third and fourth orders.
Wang, H.; Singh, S. C.; Ghosal, D.
2012-12-01
Seismic full waveform inversion is an emerging technique to determine fine-scale subsurface velocity structure. However, it requires a good starting velocity model, which is generally obtained using travel time tomography, to converge to a global minimum. Furthermore, the computing cost of full waveform inversion could be very high. In order to converge to a global solution, we have developed a combined full wave equation tomography (WET) and full waveform inversion (FWI) where the large and medium scale velocity is determined using full wave equation tomography first and then the fine-scale elastic parameters are inverted using full waveform inversion. WET and FWI both utilize full wavefield modeling, but differ on the definition of objective functions: WET aims to minimize L2-norm of cross-correlation synthetic and observed data, which is mainly sensitive to travel times, while FWI aims to optimize the L2-norm full waveform misfit, which is sensitive to both amplitudes and travel times of arrivals. Adjoint method is used to calculate the gradient for both methods efficiently. To compensate the energy loss due to wave propagation in the adjoint calculation and geometric limitation of survey, we apply an approximate Hessian preconditioning to the gradient. Further more, to stabilize WET, we precondition the time delay measures observed from cross-relation with maximum cross-correlation coefficients and perform tomographic model regularization to avoid local minimum. By exploring the band-limited feature of seismic wavefield, WET can provide better resolution than ray-based travel time tomography, and hence better suited for the FWI to converge to the true model, which provides very fine detail P and S-wave velocity. Both WET and WFI are based on the solution of full elastic wave equation and hence can model all types of wave present data. In order to reduce the computation cost and to invert seismic refraction arrivals first, we downward continue the streamer data
Sugimoto, Yu; Kitazumi, Yuki; Shirai, Osamu; Yamamoto, Masahiro; Kano, Kenji
2016-03-31
To understand electrostatic interactions in biomolecules, the bimolecular rate constants (k) between redox enzymes and charged substrates (in this study, redox mediators in the electrode reaction) were evaluated at various ionic strengths (I) for the mediated bioelectrocatalytic reaction. The k value between bilirubin oxidase (BOD) and positively charged mediators increased with I, while that between BOD and negatively charged mediators decreased with I. The opposite trend was observed for the reaction of glucose oxidase (GOD). In the case of noncharged mediators, the k value was independent of I for both BOD and GOD. These results reflect the electrostatic interactions between the enzymes and the mediators. Furthermore, we estimated k/k° (k° being the thermodynamic rate constant) by numerical simulation (finite element method) based on the Poisson-Boltzmann (PB) equation. By considering the charges of individual atoms involved in the amino acids around the substrate binding sites in the enzymes, the simulated k/k° values well reproduced the experimental data. In conclusion, k/k° can be predicted by PB-based simulation as long as the crystal structure of the enzyme and the substrate binding site are known.
Full gradient stabilized cut finite element methods for surface partial differential equations
Burman, Erik; Hansbo, Peter; Larson, Mats G.; Massing, André; Zahedi, Sara
2016-10-01
We propose and analyze a new stabilized cut finite element method for the Laplace-Beltrami operator on a closed surface. The new stabilization term provides control of the full $\\mathbb{R}^3$ gradient on the active mesh consisting of the elements that intersect the surface. Compared to face stabilization, based on controlling the jumps in the normal gradient across faces between elements in the active mesh, the full gradient stabilization is easier to implement and does not significantly increase the number of nonzero elements in the mass and stiffness matrices. The full gradient stabilization term may be combined with a variational formulation of the Laplace-Beltrami operator based on tangential or full gradients and we present a simple and unified analysis that covers both cases. The full gradient stabilization term gives rise to a consistency error which, however, is of optimal order for piecewise linear elements, and we obtain optimal order a priori error estimates in the energy and $L^2$ norms as well as an optimal bound of the condition number. Finally, we present detailed numerical examples where we in particular study the sensitivity of the condition number and error on the stabilization parameter.
Zhang, Wei; Su, Housheng; Wang, Hongwei; Han, Zhengzhi
2012-12-01
This paper aims to design full-order and reduced-order observers for one-sided Lipschitz nonlinear systems. The system under consideration is an extension of its known Lipschitz counterpart and possesses inherent advantages with respect to conservativeness. For such system, we first develop a novel Riccati equation approach to design a full-order observer, for which rigorous mathematical analysis is performed. Consequently, we show that the conditions under which a full-order observer exists also guarantee the existence of a reduced-order observer. A design method for the reduced-order observer that is dependent on the solution of the Riccati equation is then presented. The proposed conditions are easily and numerically tractable via standard numerical software. Furthermore, it is theoretically proven that the obtained conditions are less conservative than some existing ones in recent literature. The effectiveness of the proposed observers is illustrated via a simulative example.
Equation of state for hot quark-gluon plasma transitions to hadrons with full QCD potential
Sheikholeslami-Sabzevari, Bijan
2002-05-01
A practical method based on Mayer's cluster expansion to calculate critical values for a quark-gluon plasma (QGP) phase transition to hadrons is represented. It can be applied to a high-temperature QGP for clustering of quarks to mesons and baryons. The potential used is the Cornell potential, i.e., a potential containing both confining and gluon exchange terms. Debye screening effects are included. An equation of state (EOS) for hadron production is found by analytical methods, which is valid near the critical point. The example of the formation of J/ψ and Υ is recalculated. It is shown that in the range of temperatures available by today's accelerators, the latter particles are suppressed. This is further confirmation for heavy quarkonia suppression and, hence, for a signature of a QGP. The EOS presented here also shows that in future colliders there will be no heavy quarkonia production by the mechanism of phase transition. Hence, if there will be heavy quarkonia production, it must be based on some other mechanisms, perhaps on the basis of some recently suggested possibilities.
Supernova equations of state including full nuclear ensemble with in-medium effects
Furusawa, Shun; Sumiyoshi, Kohsuke; Yamada, Shoichi; Suzuki, Hideyuki
2017-01-01
We construct new equations of state for baryons at sub-nuclear densities for the use in core-collapse supernova simulations. The abundance of various nuclei is obtained together with thermodynamic quantities. The formulation is an extension of the previous model, in which we adopted the relativistic mean field theory with the TM1 parameter set for nucleons, the quantum approach for d, t, h and α as well as the liquid drop model for the other nuclei under the nuclear statistical equilibrium. We reformulate the model of the light nuclei other than d, t, h and α based on the quasi-particle description. Furthermore, we modify the model so that the temperature dependences of surface and shell energies of heavy nuclei could be taken into account. The pasta phases for heavy nuclei and the Pauli- and self-energy shifts for d, t, h and α are taken into account in the same way as in the previous model. We find that nuclear composition is considerably affected by the modifications in this work, whereas thermodynamical quantities are not changed much. In particular, the washout of shell effect has a great impact on the mass distribution above T ∼ 1 MeV. This improvement may have an important effect on the rates of electron captures and coherent neutrino scatterings on nuclei in supernova cores.
Directory of Open Access Journals (Sweden)
Hose Rod
2009-10-01
Full Text Available Abstract Background Systolic blood flow has been simulated in the abdominal aorta and the superior mesenteric artery. The simulations were carried out using two different computational hemodynamic methods: the finite element method to solve the Navier Stokes equations and the lattice Boltzmann method. Results We have validated the lattice Boltzmann method for systolic flows by comparing the velocity and pressure profiles of simulated blood flow between methods. We have also analyzed flow-specific characteristics such as the formation of a vortex at curvatures and traces of flow. Conclusion The lattice Boltzmann Method is as accurate as a Navier Stokes solver for computing complex blood flows. As such it is a good alternative for computational hemodynamics, certainly in situation where coupling to other models is required.
Navier-Stokes Dynamics by a Discrete Boltzmann Model
Rubinstein, Robet
2010-01-01
This work investigates the possibility of particle-based algorithms for the Navier-Stokes equations and higher order continuum approximations of the Boltzmann equation; such algorithms would generalize the well-known Pullin scheme for the Euler equations. One such method is proposed in the context of a discrete velocity model of the Boltzmann equation. Preliminary results on shock structure are consistent with the expectation that the shock should be much broader than the near discontinuity predicted by the Pullin scheme, yet narrower than the prediction of the Boltzmann equation. We discuss the extension of this essentially deterministic method to a stochastic particle method that, like DSMC, samples the distribution function rather than resolving it completely.
Lattice Boltzmann Stokesian dynamics.
Ding, E J
2015-11-01
Lattice Boltzmann Stokesian dynamics (LBSD) is presented for simulation of particle suspension in Stokes flows. This method is developed from Stokesian dynamics (SD) with resistance and mobility matrices calculated using the time-independent lattice Boltzmann algorithm (TILBA). TILBA is distinguished from the traditional lattice Boltzmann method (LBM) in that a background matrix is generated prior to the calculation. The background matrix, once generated, can be reused for calculations for different scenarios, thus the computational cost for each such subsequent calculation is significantly reduced. The LBSD inherits the merits of the SD where both near- and far-field interactions are considered. It also inherits the merits of the LBM that the computational cost is almost independent of the particle shape.
Lindley, David
2002-01-01
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.
Maxwell iteration for the lattice Boltzmann method with diffusive scaling.
Zhao, Weifeng; Yong, Wen-An
2017-03-01
In this work, we present an alternative derivation of the Navier-Stokes equations from Bhatnagar-Gross-Krook models of the lattice Boltzmann method with diffusive scaling. This derivation is based on the Maxwell iteration and can expose certain important features of the lattice Boltzmann solutions. Moreover, it will be seen to be much more straightforward and logically clearer than the existing approaches including the Chapman-Enskog expansion.
Verrière, Marc; Dubray, Noël; Schunck, Nicolas; Regnier, David; Dossantos-Uzarralde, Pierre
2017-09-01
Dynamical description of low energy fission is, in our full microscopic approach, decomposed in two steps. In the first step we generate the Potential Energy Surface (PES) of the compound system we want to describe with the Hartree-Fock-Bogoliubov (HFB) method with a Gogny interaction. The second step uses the Time Dependent Generator Coordinate Method (TDGCM) with the Gaussian Overlap Approximation (GOA). The GOA holds in two assumptions: the overlap matrix between HFB states has a gaussian shape (with respect to the difference between coordinates of states in deformation space); and the expectation value of the collective hamiltonian between these states can be expanded up to order two, leading in this case to a Schrödinger-like equation. In this work we replace TDGCM+GOA in the second step of our approach by an exact treatment of the TDGCM. The main equation of this method is the time-dependent Hill-Wheeler equation and involves two objects: the overlap matrix and the collective hamiltonian. We first calculate these matrices on a PES. Then, we build an "exact TDGCM" solver using a finite element method and a Crank-Nicolson scheme. In this talk, we will present the time-dependent Hill-Wheeler equation and discretization schemes (in time and deformation space). The analytic calculation of overlap matrix and collective hamiltonian will be detailed. Finally, first results with an exact treatment of the TDGCM will be discussed.
Lindner, Manfred
2007-01-01
Boltzmann equations are often used to describe the non-equilibrium time-evolution of many-body systems in particle physics. Prominent examples are the computation of the baryon asymmetry of the universe and the evolution of the quark-gluon plasma after a relativistic heavy ion collision. However, Boltzmann equations are only a classical approximation of the quantum thermalization process, which is described by so-called Kadanoff-Baym equations. This raises the question how reliable Boltzmann equations are as approximations to the complete Kadanoff-Baym equations. Therefore, we present in this article a detailed comparison of Boltzmann and Kadanoff-Baym equations in the framework of a chirally invariant Yukawa-type quantum field theory including fermions and scalars. The obtained numerical results reveal significant differences between both types of equations. Apart from quantitative differences, on a qualitative level the late-time universality respected by Kadanoff-Baym equations is severely restricted in th...
A Unified Theory of Non-Ideal Gas Lattice Boltzmann Models
Luo, Li-Shi
1998-01-01
A non-ideal gas lattice Boltzmann model is directly derived, in an a priori fashion, from the Enskog equation for dense gases. The model is rigorously obtained by a systematic procedure to discretize the Enskog equation (in the presence of an external force) in both phase space and time. The lattice Boltzmann model derived here is thermodynamically consistent and is free of the defects which exist in previous lattice Boltzmann models for non-ideal gases. The existing lattice Boltzmann models for non-ideal gases are analyzed and compared with the model derived here.
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...
Lattice Boltzmann Large Eddy Simulation Model of MHD
Flint, Christopher
2016-01-01
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...
On full-tensor permeabilities of porous media from numerical solutions of the Navier-Stokes equation
Wang, Y.
2013-01-01
A numerical method is proposed to compute full-tensor permeability of porous media without artificial simplification. Navier-Stokes (N-S) equation and Darcy\\'s law are combined to design these numerical experiments. This method can successfully detect the permeability values in principle directions of the porous media and the anisotropic degrees. It is found that the same configuration of porous media may possess isotropic features at lower Reynolds numbers while manifesting anisotropic features at higher Reynolds numbers due to the nonlinearity from convection. Anisotropy becomes pronounced especially when convection is dominant. 2013 Yi Wang et al.
Energy Technology Data Exchange (ETDEWEB)
Koning, J; Rieben, R; Rodrigue, G
2004-12-09
We measure the loss of power incurred by the bending of a single mode step-indexed optical fiber using vector finite element modeling of the full-wave Maxwell equations in the optical regime. We demonstrate fewer grid elements can be used to model light transmission in longer fiber lengths by using high-order basis functions in conjunction with a high order energy conserving time integration method. The power in the core is measured at several points to determine the percentage loss. We also demonstrate the effect of bending on the light polarization.
Levashov, P R; Sin'ko, G V; Smirnov, N A; Minakov, D V; Shemyakin, O P; Khishchenko, K V
2010-12-22
In the present work, we compare the thermal contribution of electrons to thermodynamic functions of metals in different models at high densities and electron temperatures. One of the theoretical approaches, the full-potential linear-muffin-tin-orbital method, treats all electrons in the framework of density functional theory (DFT). The other approach, VASP, uses projector-augmented-wave pseudopotentials for the core electrons and considers the valent electrons also in the context of DFT. We analyze the limitations of the pseudopotential approach and compare the DFT results with a finite-temperature Thomas-Fermi model and two semiempirical equations of state.
Madsen, Niels K; Godtliebsen, Ian H; Christiansen, Ove
2017-04-07
Vibrational coupled-cluster (VCC) theory provides an accurate method for calculating vibrational spectra and properties of small to medium-sized molecules. Obtaining these properties requires the solution of the non-linear VCC equations which can in some cases be hard to converge depending on the molecule, the basis set, and the vibrational state in question. We present and compare a range of different algorithms for solving the VCC equations ranging from a full Newton-Raphson method to approximate quasi-Newton models using an array of different convergence-acceleration schemes. The convergence properties and computational cost of the algorithms are compared for the optimization of VCC states. This includes both simple ground-state problems and difficult excited states with strong non-linearities. Furthermore, the effects of using tensor-decomposed solution vectors and residuals are investigated and discussed. The results show that for standard ground-state calculations, the conjugate residual with optimal trial vectors algorithm has the shortest time-to-solution although the full Newton-Raphson method converges in fewer macro-iterations. Using decomposed tensors does not affect the observed convergence rates in our test calculations as long as the tensors are decomposed to sufficient accuracy.
Noise source identification with the lattice Boltzmann method.
Vergnault, Etienne; Malaspinas, Orestis; Sagaut, Pierre
2013-03-01
In this paper the sound source identification problem is addressed with the use of the lattice Boltzmann method. To this aim, a time-reversed problem coupled to a complex differentiation method is used. In order to circumvent the inherent instability of the time-reversed lattice Boltzmann scheme, a method based on a split of the lattice Boltzmann equation into a mean and a perturbation component is used. Lattice Boltzmann method formulation around an arbitrary base flow is recalled and specific applications to acoustics are presented. The implementation of the noise source detection method for two-dimensional weakly compressible (low Mach number) flows is discussed, and the applicability of the method is demonstrated.
Explicit finite-difference lattice Boltzmann method for curvilinear coordinates.
Guo, Zhaoli; Zhao, T S
2003-06-01
In this paper a finite-difference-based lattice Boltzmann method for curvilinear coordinates is proposed in order to improve the computational efficiency and numerical stability of a recent method [R. Mei and W. Shyy, J. Comput. Phys. 143, 426 (1998)] in which the collision term of the Boltzmann Bhatnagar-Gross-Krook equation for discrete velocities is treated implicitly. In the present method, the implicitness of the numerical scheme is removed by introducing a distribution function different from that being used currently. As a result, an explicit finite-difference lattice Boltzmann method for curvilinear coordinates is obtained. The scheme is applied to a two-dimensional Poiseuille flow, an unsteady Couette flow, a lid-driven cavity flow, and a steady flow around a circular cylinder. The numerical results are in good agreement with the results of previous studies. Extensions to other lattice Boltzmann models based on nonuniform meshes are also discussed.
SIMULATION OF MIXED CONVECTIVE HEAT TRANSFER USING LATTICE BOLTZMANN METHOD
Directory of Open Access Journals (Sweden)
A. R. M. Rosdzimin
2010-12-01
Full Text Available In this paper, mixed (forced–natural convective heat transfer around a heated square cylinder located inside a lid driven cavity has been studied numerically using the lattice Boltzmann method in the range of 100≤ Re ≤ 1000 with the corresponding Richardson number 0.01≤Ri≤10. The double-population lattice Boltzmann formulation is used as the governing equation. Two dimensional nine-velocity models are used for the computation of the velocity field while a four-velocity model is used for the computation of the temperature field. We found that the combination of nine- and four-velocity models can be applied to the calculation without losing its accuracy. The results are presented in the form of streamline and isotherm plots as well as the variation of local Nusselt number at the top surface of the heated square. The computational results demonstrate that the flow pattern, formation of vortex and also the Nusselt number are influence by the Reynolds number and Richardson number.
A Boltzmann Transport Simulation Using Open Source Physics
Hasbun, Javier
2004-03-01
The speed of a charged particle, under an applied electric field, in a conducting media, is, usually, simply modelled by writing Newton's 2nd law in the form mfrac ddtv=qE-mfrac vτ ; (1), where v is the speed, E is the applied electric field, q is the charge, m is the mass, and τ is the scattering time between collisions. Here, we simulate a numerical solution of the Boltzmann transport equation,frac partial partial tf+ vot nabla _rf+Fot nabla _pf=frac partial partial tf|_coll (2), where in general the Boltzmann distribution function f=f(r,p,t) depends on position, momentum, and time. Our numerical solution is made possible by neglecting the 2nd term on the LHS, and by modelling the RHS collision term as fracpartial partial tf|_coll=-frac 1τ . With these approximations, in addition to considering only one dimension, we find, our numerical solution of (2). The average velocity numerically obtained through the resulting distribution is compared to that obtained by the analytic solution of (1). An efficient method of carrying out the numerical solution of (2) due to P. Drallos and M. Wadehra [Journal of Applied Physics 63, 5601(1988)] is incorporated here. A final version of an applet that performs the full Java simulation will be located at http://www.westga.edu/ jhasbun/osp/osp.htm.
Energy Technology Data Exchange (ETDEWEB)
Morgansen, K.A.; Pin, F.G.
1995-03-01
This paper describes an enhanced version of the code for the Full Space Parameterization (FSP) method that has recently been presented for determining optimized (and possibly constrained) solutions, x, to underspecified system`s of algebraic equations b = Ax. The enhanced code uses the conditions necessary for linear independence of the m {minus} n + 1 vectors forming the solution as a basis for an efficient search pattern to quickly find the full set of solution vectors. A discussion is made of the complications which may be present due to the particular combination of the matrix A and the vector b. The first part of the code implements the various methods needed to handle these particular cases before the solution vectors are calculated so that computation time may be decreased. The second portion of the code implements methods which can be used to calculate the necessary solution vectors. The respective expressions of the full solution space, S, for the cases of the matrix A being full rank and rank deficient are given. Finally, examples of the resolution of particular cases are provided, and a sample application to the joint motion of a mobile manipulator for a given end-effector trajectory is presented.
Lattice Boltzmann Numerical Simulation of a Circular Cylinder
Institute of Scientific and Technical Information of China (English)
冯士德; 赵颖; 郜宪林; 季仲贞
2002-01-01
The lattice Boltzmann equation (LBE) model based on the Boltzmann equation is suitable for the numerical simulation of various flow fields. The fluid dynamics equation can be recovered from the LBE model. However,compared to the Navier-Stokes transport equation, the fluid dynamics equation derived from the LBE model is somewhat different in the viscosity transport term, which contains not only the Navier-Stokes transport equation but also nonsteady pressure and momentum flux terms. The two nonsteady terms can produce the same function as the random stirring force term introduced in the direct numerical or large-eddy vortex simulation of turbulence.Through computation of a circular cylinder, it is verified that the influence of the two nonsteady terms on flow field stability cannot be ignored, which is helpful for the study of turbulence.
Yu, P. X.; Tian, Z. F.; Ying, A. Y.; Abdou, M. A.
2017-10-01
In this paper, an effective and accurate numerical model that involves a suggested mathematical formulation, viz., the stream functions (ψ and A)-velocity-magnetic induction formulation and a fourth-order compact difference algorithm is proposed for solving the two-dimensional (2D) steady incompressible full magnetohydrodynamic (MHD) flow equations. The stream functions-velocity-magnetic induction formulation of the 2D incompressible full MHD equations is able to circumvent the difficulty of handling the pressure variable in the primitive variable formulation or determining the vorticity values on the boundary in the stream function-vorticity formulation, and also ensure the divergence-free constraint condition of the magnetic field inherently. A test problem with the analytical solution, the well-studied lid-driven cavity problem in viscous fluid flow and the lid-driven MHD flow in a square cavity are performed to assess and verify the accuracy and the behavior of the method proposed currently. Numerical results for the present method are compared with the analytical solution and the other high-order accurate results. It is shown that the proposed stream function-velocity-magnetic induction compact difference method not only has the excellent performances in computational accuracy and efficiency, but also matches well with the divergence-free constraint of the magnetic field. Moreover, the benchmark solutions for the lid-driven cavity MHD flow in the presence of the aligned and transverse magnetic field for Reynolds number (Re) up to 5000 are provided for the wide range of magnetic Reynolds number (Rem) from 0.01 to 100 and Hartmann number (Ha) up to 4000.
Lattice Boltzmann method and its applications in engineering thermophysics
Institute of Scientific and Technical Information of China (English)
HE YaLing; LI Qing; WANG Yong; TANG GuiHua
2009-01-01
The lattice Boltzmann method (LBM),a mesoscopic method between the molecular dynamics method and the conventional numerical methods,has been developed into a very efficient numerical alternative in the past two decades.Unlike conventional numerical methods,the kinetic theory based LBM simulates fluid flows by tracking the evolution of the particle distribution function,and then accumulates the distribution to obtain macroscopic averaged properties.In this article we review some work on LBM applications in engineering thermophysics:(1) brief introduction to the development of the LBM; (2)fundamental theory of LBM including the Boltzmann equation,Maxwell distribution function,Boltzmann-BGK equation,and the lattice Boltzmann-BGK equation; (3) lattice Boltzmann models for compressible flows and non-equilibrium gas flows,bounce back-specular-reflection boundary scheme for microscale gaseous flows,the mass modified outlet boundary scheme for fully developed flows,and an implicit-explicit finite-difference-based LBM; and (4) applications of the LBM to oscillating flow,compressible flow,porous media flow,non-equilibrium flow,and gas resonant oscillating flow.
Poisson-Boltzmann versus Size-Modified Poisson-Boltzmann Electrostatics Applied to Lipid Bilayers.
Wang, Nuo; Zhou, Shenggao; Kekenes-Huskey, Peter M; Li, Bo; McCammon, J Andrew
2014-12-26
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.
Poisson–Boltzmann versus Size-Modified Poisson–Boltzmann Electrostatics Applied to Lipid Bilayers
2015-01-01
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. PMID:25426875
Two Dimensional Lattice Boltzmann Method for Cavity Flow Simulation
Directory of Open Access Journals (Sweden)
Panjit MUSIK
2004-01-01
Full Text Available 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 specific two-dimensional nine-velocity square Lattice model (D2Q9 Model is used in the simulation with the velocity at the top of the cavity kept fixed. LBM is an efficient method for reproducing the dynamics of cavity flow and the results which are comparable to those of previous work.
Lattice Boltzmann Model for Compressible Fluid on a Square Lattice
Institute of Scientific and Technical Information of China (English)
SUN Cheng-Hai
2000-01-01
A two-level four-direction lattice Boltzmann model is formulated on a square lattice to simulate compressible flows with a high Mach number. The particle velocities are adaptive to the mean velocity and internal energy. Therefore, the mean flow can have a high Mach number. Due to the simple form of the equilibrium distribution, the 4th order velocity tensors are not involved in the calculations. Unlike the standard lattice Boltzmann model, o special treatment is need for the homogeneity of 4th order velocity tensors on square lattices. The Navier-Stokes equations were derived by the Chapman-Enskog method from the BGK Boltzmann equation. The model can be easily extended to three-dimensional cubic lattices. Two-dimensional shock-wave propagation was simulated
An integrable 3D lattice model with positive Boltzmann weights
Mangazeev, Vladimir V; Sergeev, Sergey M
2013-01-01
In this paper we construct a three-dimensional (3D) solvable lattice model with non-negative Boltzmann weights. The spin variables in the model are assigned to edges of the 3D cubic lattice and run over an infinite number of discrete states. The Boltzmann weights satisfy the tetrahedron equation, which is a 3D generalisation of the Yang-Baxter equation. The weights depend on a free parameter 0Boltzmann weights.
Learning thermodynamics with Boltzmann machines
Torlai, Giacomo; Melko, Roger G.
2016-10-01
A Boltzmann machine is a stochastic neural network that has been extensively used in the layers of deep architectures for modern machine learning applications. In this paper, we develop a Boltzmann machine that is capable of modeling thermodynamic observables for physical systems in thermal equilibrium. Through unsupervised learning, we train the Boltzmann machine on data sets constructed with spin configurations importance sampled from the partition function of an Ising Hamiltonian at different temperatures using Monte Carlo (MC) methods. The trained Boltzmann machine is then used to generate spin states, for which we compare thermodynamic observables to those computed by direct MC sampling. We demonstrate that the Boltzmann machine can faithfully reproduce the observables of the physical system. Further, we observe that the number of neurons required to obtain accurate results increases as the system is brought close to criticality.
Directory of Open Access Journals (Sweden)
Haiqing Si
2015-03-01
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.
Fecher, T.; Pail, R.; Gruber, T.
2017-05-01
GOCO05c is a gravity field model computed as a combined solution of a satellite-only model and a global data set of gravity anomalies. It is resolved up to degree and order 720. It is the first model applying regionally varying weighting. Since this causes strong correlations among all gravity field parameters, the resulting full normal equation system with a size of 2 TB had to be solved rigorously by applying high-performance computing. GOCO05c is the first combined gravity field model independent of EGM2008 that contains GOCE data of the whole mission period. The performance of GOCO05c is externally validated by GNSS-levelling comparisons, orbit tests, and computation of the mean dynamic topography, achieving at least the quality of existing high-resolution models. Results show that the additional GOCE information is highly beneficial in insufficiently observed areas, and that due to the weighting scheme of individual data the spectral and spatial consistency of the model is significantly improved. Due to usage of fill-in data in specific regions, the model cannot be used for physical interpretations in these regions.
Xin, Bo; Sun, Dakun; Jing, Xiaodong; Sun, Xiaofeng
2016-07-01
Lined ducts are extensively applied to suppress noise emission from aero-engines and other turbomachines. The complex noise/flow interaction in a lined duct possibly leads to acoustic instability in certain conditions. To investigate the instability, the full linearized Navier-Stokes equations with eddy viscosity considered are solved in frequency domain using a Galerkin finite element method to compute the sound transmission in shear flow in the lined duct as well as the flow perturbation over the impedance wall. A good agreement between the numerical predictions and the published experimental results is obtained for the sound transmission, showing that a transmission peak occurs around the resonant frequency of the acoustic liner in the presence of shear flow. The eddy viscosity is an important influential factor that plays the roles of both providing destabilizing and making coupling between the acoustic and flow motions over the acoustic liner. Moreover, it is shown from the numerical investigation that the occurrence of the sound amplification and the magnitude of transmission coefficient are closely related to the realistic velocity profile, and we find it essential that the actual variation of the velocity profile in the axial direction over the liner surface be included in the computation. The simulation results of the periodic flow patterns possess the proper features of the convective instability over the liner, as observed in Marx et al.'s experiment. A quantitative comparison between numerical and experimental results of amplitude and phase of the instability is performed. The corresponding eigenvalues achieve great agreement.
Hu, Kainan; Geng, Shaojuan
2016-01-01
A new lattice Boltzmann scheme associated with flexible specific heat ratio is proposed. The new free degree is introduced via the internal energy associated with the internal structure. The evolution equation of the distribution function is reduced to two evolution equations. One is connected to the density and velocity, the other is of the energy. A two-dimensional lattice Boltzmann model and a three-dimensional lattice Boltzmann model are derived via the Hermite expansion. The two lattice Boltzmann models are applied to simulating the shock tube of one dimension. Good agreement between the numerical results and the analytical solutions are obtained.
Lattice Boltzmann Model for Numerical Relativity
Ilseven, E
2015-01-01
In the Bona-Masso 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. Our model is validated with well-established tests, showing good agreement with analytical solutions. Furthermore, we show 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 improves. Finally, in order to show the potential of our approach a linear scaling law for parallelisation with respect to number of CPU cores is demonstrated. Our model represents the first step in using lattice kinetic theory to solve gravitational problems.
Lattice Boltzmann model for numerical relativity.
Ilseven, E; Mendoza, M
2016-02-01
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.
EXTERNAL BODY FORCE IN FINITE DIFFERENCE LATTICE BOLTZMANN METHOD
Institute of Scientific and Technical Information of China (English)
CHEN Sheng; LIU Zhao-hui; SHI Bao-chang; ZHENG Chu-guang
2005-01-01
A new finite difference lattice Boltzmann scheme is developed. Because of analyzing the influence of external body force roundly, the correct Navier-Stokes equations with the external body force are recovered, without any additional unphysical terms. And some numerical results are presented. The result which close agreement with analytical data shows the good performance of the model.
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...... 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...... 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....
Lattice Boltzmann model for nanofluids
Energy Technology Data Exchange (ETDEWEB)
Xuan Yimin; Yao Zhengping [Nanjing University of Science and Technology, School of Power Engineering, Nanjing (China)
2005-01-01
A nanofluid is a particle suspension that consists of base liquids and nanoparticles and has great potential for heat transfer enhancement. By accounting for the external and internal forces acting on the suspended nanoparticles and interactions among the nanoparticles and fluid particles, a lattice Boltzmann model is proposed for simulating flow and energy transport processes inside the nanofluids. First, we briefly introduce the conventional lattice Boltzmann model for multicomponent systems. Then, we discuss the irregular motion of the nanoparticles and inherent dynamic behavior of nanofluids and describe a lattice Boltzmann model for simulating nanofluids. Finally, we conduct some calculations for the distribution of the suspended nanoparticles. (orig.)
Directory of Open Access Journals (Sweden)
Khoie R.
1996-01-01
Full Text Available A self-consistent Boltzmann-Poisson-Schrödinger solver for High Electron Mobility Transistor is presented. The quantization of electrons in the quantum well normal to the heterojunction is taken into account by solving the two higher moments of Boltzmann equation along with the Schrödinger and Poisson equations, self-consistently. The Boltzmann transport equation in the form of a current continuity equation and an energy balance equation are solved to obtain the transient and steady-state transport behavior. The numerical instability problems associated with the simulator are presented, and the criteria for smooth convergence of the solutions are discussed. The current-voltage characteristics, transconductance, gate capacitance, and unity-gain frequency of a single quantum well HEMT is discussed. It has been found that a HEMT device with a gate length of 0.7 μ m , and with a gate bias voltage of 0.625 V, has a transconductance of 579.2 mS/mm, which together with the gate capacitance of 19.28 pF/cm, can operate at a maximum unity-gain frequency of 47.8 GHz.
Directory of Open Access Journals (Sweden)
R. Khoie
1996-01-01
Full Text Available A self-consistent Boltzmann-Poisson-Schrödinger solver for High Electron Mobility Transistor is presented. The quantization of electrons in the quantum well normal to the heterojunction is taken into account by solving the two higher moments of Boltzmann equation along with the Schrödinger and Poisson equations, self-consistently. The Boltzmann transport equation in the form of a current continuity equation and an energy balance equation are solved to obtain the transient and steady-state transport behavior. The numerical instability problems associated with the simulator are presented, and the criteria for smooth convergence of the solutions are discussed. The current-voltage characteristics, transconductance, gate capacitance, and unity-gain frequency of a single quantum well HEMT is discussed. It has been found that a HEMT device with a gate length of 0.7 μm, and with a gate bias voltage of 0.625 V, has a transconductance of 579.2 mS/mm, which together with the gate capacitance of 19.28 pF/cm, can operate at a maximum unity-gain frequency of 47.8 GHz.
Contact line dynamics in binary lattice Boltzmann simulations
Pooley, C M; Yeomans, J M; 10.1103/PhysRevE.78.056709
2008-01-01
We show that, when a single relaxation time lattice Boltzmann algorithm is used to solve the hydrodynamic equations of a binary fluid for which the two components have different viscosities, strong spurious velocities in the steady state lead to incorrect results for the equilibrium contact angle. We identify the origins of these spurious currents, and demonstrate how the results can be greatly improved by using a lattice Boltzmann method based on a multiple-relaxation-time algorithm. By considering capillary filling we describe the dependence of the advancing contact angle on the interface velocity.
Lattice Boltzmann method for linear oscillatory noncontinuum flows.
Shi, Yong; Yap, Ying Wan; Sader, John E
2014-03-01
Oscillatory gas flows are commonly generated by micro- and nanoelectromechanical systems. Due to their small size and high operating frequencies, these devices often produce noncontinuum gas flows. Theoretical analysis of such flows requires solution of the unsteady Boltzmann equation, which can present a formidable challenge. In this article, we explore the applicability of the lattice Boltzmann (LB) method to such linear oscillatory noncontinuum flows; this method is derived from the linearized Boltzmann Bhatnagar-Gross-Krook (BGK) equation. We formulate four linearized LB models in the frequency domain, based on Gaussian-Hermite quadratures of different algebraic precision (AP). The performance of each model is assessed by comparison to high-accuracy numerical solutions to the linearized Boltzmann-BGK equation for oscillatory Couette flow. The numerical results demonstrate that high even-order LB models provide superior performance over the greatest noncontinuum range. Our results also highlight intrinsic deficiencies in the current LB framework, which is incapable of capturing noncontinuum behavior at high oscillation frequencies, regardless of quadrature AP and the Knudsen number.
Lattice Boltzmann method for linear oscillatory noncontinuum flows
Shi, Yong; Yap, Ying Wan; Sader, John E.
2014-03-01
Oscillatory gas flows are commonly generated by micro- and nanoelectromechanical systems. Due to their small size and high operating frequencies, these devices often produce noncontinuum gas flows. Theoretical analysis of such flows requires solution of the unsteady Boltzmann equation, which can present a formidable challenge. In this article, we explore the applicability of the lattice Boltzmann (LB) method to such linear oscillatory noncontinuum flows; this method is derived from the linearized Boltzmann Bhatnagar-Gross-Krook (BGK) equation. We formulate four linearized LB models in the frequency domain, based on Gaussian-Hermite quadratures of different algebraic precision (AP). The performance of each model is assessed by comparison to high-accuracy numerical solutions to the linearized Boltzmann-BGK equation for oscillatory Couette flow. The numerical results demonstrate that high even-order LB models provide superior performance over the greatest noncontinuum range. Our results also highlight intrinsic deficiencies in the current LB framework, which is incapable of capturing noncontinuum behavior at high oscillation frequencies, regardless of quadrature AP and the Knudsen number.
Energy Technology Data Exchange (ETDEWEB)
Yomba, Emmanuel, E-mail: emmanuel.yomba@csun.edu; Zakeri, Gholam-Ali, E-mail: ali.zakeri@csun.edu
2016-02-05
We investigate the existence of various solitary wave solutions in coupled Schrödinger equations with specific cubic and quintic nonlinearities. This system arises in wave propagation in fiber optics with focusing and defocusing with modulated nonlinearities. We obtain front–front, bright–bright, dark–dark, and dark–bright like solitons using a direct approach, and then, by reducing the system of equations to a single auxiliary equation of a Duffing-type ordinary differential equation, we provide a large class of Jacobi-elliptic solutions. These solutions are presented in the exact form and analyzed. We find a class of wide localized and snake-like (in both space and time) vector solitons. One of the novel aspects of this study is that we have shown that the qualitative behavior of the solutions is independent of the choice of similarity variables. Numerical results show that the solutions of the above system are stable with up to 10% white noises. - Highlights: • Dynamics of wide and snake-like pulses is analyzed for coupled Schrödinger equations. • Qualitative appearance of solutions is analyzed using various similarity variables. • Effect of change in parameter-values on dynamics of the solutions is investigated. • Vectors front–front, bright–bright, dark–dark and dark–bright solitons are obtained.
Caveat on the Boltzmann distribution function use in biology.
Sevcik, Carlos
2017-08-01
Sigmoid semilogarithmic functions with shape of Boltzmann equations, have become extremely popular to describe diverse biological situations. Part of the popularity is due to the easy availability of software which fits Boltzmann functions to data, without much knowledge of the fitting procedure or the statistical properties of the parameters derived from the procedure. The purpose of this paper is to explore the plasticity of the Boltzmann function to fit data, some aspects of the optimization procedure to fit the function to data and how to use this plastic function to differentiate the effect of treatment on data and to attest the statistical significance of treatment effect on the data. Copyright © 2017. Published by Elsevier Ltd.
Directory of Open Access Journals (Sweden)
Kuznetsov Vladimir V.
2016-01-01
Full Text Available New full-statement 3D mathematical model of joint motion of thin liquid film and gas in a microchannel at local heating developed by taking into account the heat transfer by flows, evaporation and condensation, as well as the heat transfer at the gas-liquid interface is derived. The model is based on the full system of the Navier-Stokes equations, taking into account the convective terms of motion equations in the phases. Comparison of the numerical results obtained using the model based on the full Navier-Stokes equations and using the simplified model developed in the framework of the thin layer approximation has been performed. The comparison shows that at low Reynolds numbers, simplified model well describes all the main characteristics of the gas and liquid motion. With the gas Reynolds numbers significant increase difference between numerical results starts to grow.
Lattice Boltzmann modelling of intrinsic permeability
Li, Jun; Wu, Lei; Zhang, Yonghao
2016-01-01
Lattice Boltzmann method (LBM) has been applied to predict flow properties of porous media including intrinsic permeability, where it is implicitly assumed that the LBM is equivalent to the incompressible (or near incompressible) Navier-Stokes equation. However, in LBM simulations, high-order moments, which are completely neglected in the Navier-Stokes equation, are still available through particle distribution functions. To ensure that the LBM simulation is correctly working at the Navier-Stokes hydrodynamic level, the high-order moments have to be negligible. This requires that the Knudsen number (Kn) is small so that rarefaction effect can be ignored. In this technical note, we elaborate this issue in LBM modelling of porous media flows, which is particularly important for gas flows in ultra-tight media.
Yomba, Emmanuel; Zakeri, Gholam-Ali
2016-02-01
We investigate the existence of various solitary wave solutions in coupled Schrödinger equations with specific cubic and quintic nonlinearities. This system arises in wave propagation in fiber optics with focusing and defocusing with modulated nonlinearities. We obtain front-front, bright-bright, dark-dark, and dark-bright like solitons using a direct approach, and then, by reducing the system of equations to a single auxiliary equation of a Duffing-type ordinary differential equation, we provide a large class of Jacobi-elliptic solutions. These solutions are presented in the exact form and analyzed. We find a class of wide localized and snake-like (in both space and time) vector solitons. One of the novel aspects of this study is that we have shown that the qualitative behavior of the solutions is independent of the choice of similarity variables. Numerical results show that the solutions of the above system are stable with up to 10% white noises.
Enders, Craig K.
2008-01-01
Recent missing data studies have argued in favor of an "inclusive analytic strategy" that incorporates auxiliary variables into the estimation routine, and Graham (2003) outlined methods for incorporating auxiliary variables into structural equation analyses. In practice, the auxiliary variables often have missing values, so it is reasonable to…
Nauenberg, Michael
2005-03-01
In 1916 Einstein published a remarkable paper entitled ``On the Quantum Theory of Radiation''ootnotetextA. Einstein ``On the Quantum theory of Radiation,'' Phys. Zeitschrift 18 (1917) 121. First printed in Mitteilungender Physikalischen Gesellschaft Zurich. No 18, 1916. Translated into English in Van der Waerden ``Sources of Quantum Mechanics'' (North Holland 1967) pp. 63-77. in which he obtained Planck's formula for black-body radiation by introducing a new statistical hypothesis for the emmision and absorption of electromagneic radiation based on discrete bundles of energy and momentum which are now called photons. Einstein radiation theory replaced Maxwell's classical theory by a stochastic process which, when properly interpreted, also gives well known statistics of massless particles with even spin.^2 This quantum distribution, however, was not discovered by Einstein but was communicated to him by Bose in 1924. Like Boltzmann's classical counterpart, Einstein's statistical theory leads to an irreversible approach to thermal equilibrium, but because this violates time reversal, Einstein theory can not be regarded as a fundamental theory of physical process.ootnotetextM. Nauenberg ``The evolution of radiation towards thermal equilibrium: A soluble model which illustrates the foundations of statistical mechanics,'' American Journal of Physics 72 (2004) 313 Apparently Einstein and his contemporaries were unaware of this problem, and even today this problem is ignored in contemporary discussions of Einstein's treatment of the black-body spectrum.
Immersed Boundary-Lattice Boltzmann Method Using Two Relaxation Times
Directory of Open Access Journals (Sweden)
Kosuke Hayashi
2012-06-01
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.
Dafermos, Mihalis; Shlapentokh-Rothman, Yakov
2014-01-01
This paper concludes the series begun in [M. Dafermos and I. Rodnianski, Decay for solutions of the wave equation on Kerr exterior spacetimes I-II: the cases |a| << M or axisymmetry, arXiv:1010.5132], providing the complete proof of definitive boundedness and decay results for the scalar wave equation on Kerr backgrounds in the general subextremal |a| < M case without symmetry assumptions. The essential ideas of the proof (together with explicit constructions of the most difficult multiplier currents) have been announced in our survey [M. Dafermos and I. Rodnianski, The black hole stability problem for linear scalar perturbations, in Proceedings of the 12th Marcel Grossmann Meeting on General Relativity, T. Damour et al (ed.), World Scientific, Singapore, 2011, pp. 132-189, arXiv:1010.5137]. Our proof appeals also to the quantitative mode-stability proven in [Y. Shlapentokh-Rothman, Quantitative Mode Stability for the Wave Equation on the Kerr Spacetime, arXiv:1302.6902, to appear, Ann. Henri Poincar...
DEFF Research Database (Denmark)
Svec, Oldrich; Skoček, Jan
2013-01-01
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...
Lattice Boltzmann model for Coulomb-driven flows in dielectric liquids.
Luo, Kang; Wu, Jian; Yi, Hong-Liang; Tan, He-Ping
2016-02-01
In this paper, we developed a unified lattice Boltzmann model (LBM) to simulate electroconvection in a dielectric liquid induced by unipolar charge injection. Instead of solving the complex set of coupled Navier-Stokes equations, the charge conservation equation, and the Poisson equation of electric potential, three consistent lattice Boltzmann equations are formulated. Numerical results are presented for both strong and weak injection regimes, and different scenarios for the onset and evolution of instability, bifurcation, and chaos are tracked. All LBM results are found to be highly consistent with the analytical solutions and other numerical work.
Temperature based Restricted Boltzmann Machines.
Li, Guoqi; Deng, Lei; Xu, Yi; Wen, Changyun; Wang, Wei; Pei, Jing; Shi, Luping
2016-01-13
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.
Nomura, Yasunori
2015-01-01
Understanding the observed arrow of time is equivalent, under general assumptions, to explaining why Boltzmann brains do not overwhelm ordinary observers. It is usually thought that this provides a condition on the decay rate of every cosmologically accessible de Sitter vacuum, and that this condition is determined by the production rate of Boltzmann brains calculated using semiclassical theory built on each such vacuum. We argue, based on a recently developed picture of microscopic quantum gravitational degrees of freedom, that this thinking needs to be modified. In particular, depending on the structure of the fundamental theory, the decay rate of a de Sitter vacuum may not have to satisfy any condition except possibly the one imposed by the Poincare recurrence. The framework discussed here also addresses the question of whether a Minkowski vacuum may produce Boltzmann brains.
Immiscible multicomponent lattice Boltzmann model for fluids with high relaxation time ratio
Indian Academy of Sciences (India)
Tao Jiang; Qiwei Gong; Ruofan Qiu; Anlin Wang
2014-10-01
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, an interaction potential between particles is incorporated into the discrete lattice Boltzmann equation through the equilibrium velocity. Compared to the SC model, external forces in our model are discretized directly into the discrete lattice Boltzmann equation, as proposed by Guo et al. We develop it into a new multicomponent lattice Boltzmann (LB) model which has the ability to simulate immiscible multicomponent fluids with relaxation time ratio as large as 29.0 and to reduce `spurious velocity’. In this work, the improved model is validated and studied using the central bubble case and the rising bubble case. It finds good applications in both static and dynamic cases for multicomponent simulations with different relaxation time ratios.
Lattice Boltzmann model for incompressible flows through porous media.
Guo, Zhaoli; Zhao, T S
2002-09-01
In this paper a lattice Boltzmann model is proposed for isothermal incompressible flow in porous media. The key point is to include the porosity into the equilibrium distribution, and add a force term to the evolution equation to account for the linear and nonlinear drag forces of the medium (the Darcy's term and the Forcheimer's term). Through the Chapman-Enskog procedure, the generalized Navier-Stokes equations for incompressible flow in porous media are derived from the present lattice Boltzmann model. The generalized two-dimensional Poiseuille flow, Couette flow, and lid-driven cavity flow are simulated using the present model. It is found the numerical results agree well with the analytical and/or the finite-difference solutions.
Boltzmann transport calculation of collinear spin transport on short timescales
Nenno, Dennis M.; Kaltenborn, Steffen; Schneider, Hans Christian
2016-09-01
A spin-dependent Boltzmann transport equation is used to describe charge and spin dynamics resulting from the excitation of hot electrons in a ferromagnet/normal metal heterostructure. As the microscopic Boltzmann equation works with k -dependent distribution functions, it can describe far-from-equilibrium excitations, which are outside the scope of drift-diffusion theories. We study different scenarios for spin-dependent carrier injection into a nonmagnetic metal using an effectively two-dimensional phase space. While the charge signal is robust for various excitation schemes, the shape of the resulting spin current/density depends critically on the interplay between transport and scattering, and on the energetic distribution of the injected carriers. Our results imply that the energy dependence of the injected hot electrons has a decisive effect on the spin dynamics.
LATTICE-BOLTZMANN MODEL FOR COMPRESSIBLE PERFECT GASES
Institute of Scientific and Technical Information of China (English)
Sun Chenghai
2000-01-01
We present an adaptive lattice Boltzmann model to simulate super sonic flows. The particle velocities are determined by the mean velocity and internal energy. The adaptive nature of particle velocities permits the mean flow to have high Mach number. A particle potential energy is introduced so that the model is suitable for the perfect gas with arbitrary specific heat ratio. The Navier-Stokes equations are derived by the Chapman-Enskog method from the BGK Boltzmann equation.As preliminary tests, two kinds of simulations have been performed on hexagonal lattices. One is the one-dimensional simulation for sinusoidal velocity distributions.The velocity distributions are compared with the analytical solution and the mea sured viscosity is compared with the theoretical values. The agreements are basically good. However, the discretion error may cause some non-isotropic effects. The other simulation is the 29 degree shock reflection.
Lattice Boltzmann model with nearly constant density.
Fang, Hai-ping; Wan, Rong-zheng; Lin, Zhi-fang
2002-09-01
An improved lattice Boltzmann model is developed to simulate fluid flow with nearly constant fluid density. The ingredient is to incorporate an extra relaxation for fluid density, which is realized by introducing a feedback equation in the equilibrium distribution functions. The pressure is dominated by the moving particles at a node, while the fluid density is kept nearly constant and explicit mass conservation is retained as well. Numerical simulation based on the present model for the (steady) plane Poiseuille flow and the (unsteady) two-dimensional Womersley flow shows a great improvement in simulation results over the previous models. In particular, the density fluctuation has been reduced effectively while achieving a relatively large pressure gradient.
A lattice Boltzmann model for adsorption breakthrough
Energy Technology Data Exchange (ETDEWEB)
Agarwal, Saurabh; Verma, Nishith [Indian Institute of Technology Kanpur, Department of Chemical Engineering, Kanpur (India); Mewes, Dieter [Universitat Hannover, Institut fur Verfahrenstechnik, Hannover (Germany)
2005-07-01
A lattice Boltzmann model is developed to simulate the one-dimensional (1D) unsteady state concentration profiles, including breakthrough curves, in a fixed tubular bed of non-porous adsorbent particles. The lattice model solves the 1D time dependent convection-diffusion-reaction equation for an ideal binary gaseous mixture, with solute concentrations at parts per million levels. The model developed in this study is also able to explain the experimental adsorption/desorption data of organic vapours (toluene) on silica gel under varying conditions of temperature, concentrations and flowrates. Additionally, the programming code written for simulating the adsorption breakthrough is modified with minimum changes to successfully simulate a few flow problems, such as Poiseuille flow, Couette flow, and axial dispersion in a tube. The present study provides an alternative numerical approach to solving such types of mass transfer related problems. (orig.)
Fernández-Pato, Javier; Caviedes-Voullième, Daniel; García-Navarro, Pilar
2016-05-01
One of the most difficult issues in the development of hydrologic models is to find a rigorous source of data and specific parameters to a given problem, on a given location that enable reliable calibration. In this paper, a distributed and physically based model (2D Shallow Water Equations) is used for surface flow and runoff calculations in combination with two infiltration laws (Horton and Green-Ampt) for estimating infiltration in a watershed. This technique offers the capability of assigning a local and time-dependent infiltration rate to each computational cell depending on the available surface water, soil type or vegetation. We investigate how the calibration of parameters is affected by transient distributed Shallow Water model and the complexity of the problem. In the first part of this work, we calibrate the infiltration parameters for both Horton and Green-Ampt models under flat ponded soil conditions. Then, by means of synthetic test cases, we perform a space-distributed sensitivity analysis in order to show that this calibration can be significantly affected by the introduction of topography or rainfall. In the second part, parameter calibration for a real catchment is addressed by comparing the numerical simulations with two different sets of experimental data, corresponding to very different events in terms of the rainfall volume. We show that the initial conditions of the catchment and the rainfall pattern have a special relevance in the quality of the adjustment. Hence, it is shown that the topography of the catchment and the storm characteristics affect the calibration of infiltration parameters.
Boltzmann-type control of opinion consensus through leaders.
Albi, G; Pareschi, L; Zanella, M
2014-11-13
The study of formations and dynamics of opinions leading to the so-called opinion consensus is one of the most important areas in mathematical modelling of social sciences. Following the Boltzmann-type control approach recently introduced by the first two authors, we consider a group of opinion leaders who modify their strategy accordingly to an objective functional with the aim of achieving opinion consensus. The main feature of the Boltzmann-type control is that, owing to an instantaneous binary control formulation, it permits the minimization of the cost functional to be embedded into the microscopic leaders' interactions of the corresponding Boltzmann equation. The related Fokker-Planck asymptotic limits are also derived, which allow one to give explicit expressions of stationary solutions. The results demonstrate the validity of the Boltzmann-type control approach and the capability of the leaders' control to strategically lead the followers' opinion. © 2014 The Author(s) Published by the Royal Society. All rights reserved.
Directory of Open Access Journals (Sweden)
K. Banoo
1998-01-01
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.
A note on the Lattice Boltzmann Method Beyond the Chapman Enskog Limits
Sbragaglia, M.; Succi, S.
2006-01-01
A non-perturbative analysis of the Bhatnagar-Gross-Krook (BGK) model kinetic equation for finite values of the Knudsen number is presented. This analysis indicates why discrete kinetic versions of the BGK equation, and notably the lattice Boltzmann method, can provide semi-quantitative results also
Extended lattice Boltzmann scheme for droplet combustion
Ashna, Mostafa; Rahimian, Mohammad Hassan; Fakhari, Abbas
2017-05-01
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.
Extended lattice Boltzmann scheme for droplet combustion.
Ashna, Mostafa; Rahimian, Mohammad Hassan; Fakhari, Abbas
2017-05-01
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.
Diffusive limit for a quantum linear Boltzmann dynamics
Clark, Jeremy
2010-01-01
We study the diffusive behavior for a quantum test particle interacting with a dilute background gas. The model we begin with is a reduced picture for the test particle dynamics given by a quantum linear Boltzmann equation in which the scattering with the gas particles is assumed to occur through a hard-sphere interaction. The state of the particle is represented by a density matrix evolving according to a translation-covariant Lindblad equation. Our main result is a proof that the particle diffuses for large times.
Topological Interactions in a Boltzmann-Type Framework
Blanchet, Adrien; Degond, Pierre
2016-04-01
We consider a finite number of particles characterised by their positions and velocities. At random times a randomly chosen particle, the follower, adopts the velocity of another particle, the leader. The follower chooses its leader according to the proximity rank of the latter with respect to the former. We study the limit of a system size going to infinity and, under the assumption of propagation of chaos, show that the limit equation is akin to the Boltzmann equation. However, it exhibits a spatial non-locality instead of the classical non-locality in velocity space. This result relies on the approximation properties of Bernstein polynomials. We illustrate the dynamics with numerical simulations.
Student understanding of the Boltzmann factor
Smith, Trevor I; Thompson, John R
2015-01-01
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 on student discussions about the Boltzmann f...
Are there Boltzmann brains in the vacuum
Davenport, Matthew
2010-01-01
"Boltzmann brains" are human brains that arise as thermal or quantum fluctuations and last at least long enough to think a few thoughts. In many scenarios involving universes of infinite size or duration, Boltzmann brains are infinitely more common than human beings who arise in the ordinary way. Thus we should expect to be Boltzmann brains, in contradiction to observation. We discuss here the question of whether Boltzmann brains can arise as quantum fluctuations in the vacuum. Such Boltzmann brains pose an even worse problem than those arising as fluctuations in the thermal state of an exponentially expanding universe. We give several arguments for and against inclusion of vacuum Boltzmann brains in the anthropic reference class, but find neither choice entirely satisfactory.
Quadrature-based Lattice Boltzmann Model for Relativistic Flows
Blaga, Robert
2016-01-01
A quadrature-based finite-difference lattice Boltzmann model is developed that is suitable for simulating relativistic flows of massless particles. We briefly review the relativistc Boltzmann equation and present our model. The quadrature is constructed such that the stress-energy tensor is obtained as a second order moment of the distribution function. The results obtained with our model are presented for a particular instance of the Riemann problem (the Sod shock tube). We show that the model is able to accurately capture the behavior across the whole domain of relaxation times, from the hydrodynamic to the ballistic regime. The property of the model of being extendable to arbitrarily high orders is shown to be paramount for the recovery of the analytical result in the ballistic regime.
Viscous QCD matter in a hybrid hydrodynamic+Boltzmann approach
Song, Huichao; Heinz, Ulrich W
2010-01-01
A hybrid transport approach for the bulk evolution of viscous QCD matter produced in ultra-relativistic heavy-ion collisions is presented. The expansion of the dense deconfined phase of the reaction is modeled with viscous hydrodynamics while the dilute late hadron gas stage is described microscopically by the Boltzmann equation. The advantages of such a hybrid approach lie in the improved capability of handling large dissipative corrections in the late dilute phase of the reaction, including a realistic treatment of the non-equilibrium hadronic chemistry and kinetic freeze-out. By varying the switching temperature at which the hydrodynamic output is converted to particles for further propagation with the Boltzmann cascade we test the ability of the macroscopic hydrodynamic approach to emulate the microscopic evolution during the hadronic stage and extract the temperature dependence of the effective shear viscosity of the hadron resonance gas produced in the collision. We find that the extracted values depend...
Lattice Boltzmann method with the cell-population equilibrium
Institute of Scientific and Technical Information of China (English)
Zhou Xiao-Yang; Cheng Bing; Shi Bao-Chang
2008-01-01
The central problem of the lattice Boltzmann method (LBM) is to construct a discrete equilibrium.In this paper,a multi-speed 1D cell-model of Boltzmann equation is proposed,in which the cell-population equilibrium,a direct nonnegative approximation to the continuous Maxwellian distribution,plays an important part.By applying the explicit one-order Chapman-Enskog distribution,the model reduces the transportation and collision,two basic evolution steps in LBM,to the transportation of the non-equilibrium distribution.Furthermore,1D dam-break problem is performed and the numerical results agree well with the analytic solutions.
Surface Tension of Acid Solutions: Fluctuations beyond the Nonlinear Poisson-Boltzmann Theory.
Markovich, Tomer; Andelman, David; Podgornik, Rudi
2017-01-10
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.
Boltzmann Transport in Hybrid PIC HET Modeling
2015-07-01
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...produced a variety of self-consistent electron swarm codes, such as the Magboltz code, focused on directly solving the steady Boltzmann trans-port...Std. 239.18 Boltzmann transport in hybrid PIC HET modeling IEPC-2015- /ISTS-2015-b- Presented at Joint Conference of 30th International
STUDY ON GAS KINETIC NUMERICAL ALGORITHM USING BOLTZMANN MODEL EQUATION%基于Boltzmann模型方程的气体运动论统一算法研究
Institute of Scientific and Technical Information of China (English)
李志辉; 张涵信
2005-01-01
从Boltzmann-Shakhov模型方程出发,研究确立含流态控制参数可描述不同流域气体流动特征的气体分子速度分布函数方程;研究发展气体运动论离散速度坐标法,借助非定常时间分裂数值计算方法和NND差分格式,结合DSMC方法关于分子运动与碰撞去耦技术,发展直接求解速度分布函数的气体运动论耦合迭代数值格式;研制可用于物理空间各点宏观流动取矩的离散速度数值积分方法,由此提出一套能有效模拟稀薄流到连续流不同流域气体流动问题统一算法.通过对不同Knudsen数下一维激波内流动、二维圆柱、三维球体绕流数值计算表明,计算结果与有关实验数据及其它途径研究结果(如DSMC模拟值、N-S数值解)吻合较好,证实气体运动论统一算法求解各流域气体流动问题的可行性.尝试将统一算法进行HPF并行化程序设计,基于对球体绕流及类"神舟"返回舱外形绕流问题进行HPF初步并行试算,显示出统一算法具有很好的并行可扩展性,可望建立起新型的能有效模拟各流域飞行器绕流HPF并行算法研究方向.通过将气体运动论统一算法推广应用于微槽道流动计算研究,已初步发展起可靠模拟二维短微槽道流动数值算法;通过对Couette流、Poiseuille流、压力驱动的二维短槽道流数值模拟,证实该算法对微槽道气体流动问题具有较强的模拟能力,可望发展起基于Boltzmann模型方程能可靠模拟MEMS微流动问题气体运动论数值计算方法研究途径.
Lattice-Boltzmann-based two-phase thermal model for simulating phase change
Kamali, M.R.; Gillissen, J.J.J.; Van den Akker, H.E.A.; Sundaresan, S.
2013-01-01
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 nonideal equation of state (EOS) is introduced by using a pseudopotential LB model. The evolution equation for the pseudotemperature variable is constructed in such a manner that in the continuum l...
Student Understanding of the Boltzmann Factor
Smith, Trevor I.; Mountcastle, Donald B.; Thompson, John R.
2015-01-01
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…
SIMPLE LATTICE BOLTZMANN MODEL FOR TRAFFIC FLOWS
Institute of Scientific and Technical Information of China (English)
Yan Guangwu; Hu Shouxin
2000-01-01
A lattice Boltzmann model with 5-bit lattice for traffic flows is proposed.Using the Chapman-Enskog expansion and multi-scale technique,we obtain the higher-order moments of equilibrium distribution function.A simple traffic light problem is simulated by using the present lattice Boltzmann model,and the result agrees well with analytical solution.
High-order hydrodynamics via lattice Boltzmann methods.
Colosqui, Carlos E
2010-02-01
In this work, closure of the Boltzmann-Bhatnagar-Gross-Krook (Boltzmann-BGK) moment hierarchy is accomplished via projection of the distribution function f onto a space H(N) spanned by N-order Hermite polynomials. While successive order approximations retain an increasing number of leading-order moments of f , the presented procedure produces a hierarchy of (single) N-order partial-differential equations providing exact analytical description of the hydrodynamics rendered by ( N-order) lattice Boltzmann-BGK (LBBGK) simulation. Numerical analysis is performed with LBBGK models and direct simulation Monte Carlo for the case of a sinusoidal shear wave (Kolmogorov flow) in a wide range of Weissenberg number Wi=taunuk(2) (i.e., Knudsen number Kn=lambdak=square root Wi); k is the wave number, [corrected] tau is the relaxation time of the system, and lambda approximately tauc(s) is the mean-free path, where c(s) is the speed of sound. The present results elucidate the applicability of LBBGK simulation under general nonequilibrium conditions.
Filter-matrix lattice Boltzmann model for microchannel gas flows.
Zhuo, Congshan; Zhong, Chengwen
2013-11-01
The lattice Boltzmann method has been shown to be successful for microscale gas flows, and it has attracted significant research interest. In this paper, the recently proposed filter-matrix lattice Boltzmann (FMLB) model is first applied to study the microchannel gas flows, in which a Bosanquet-type effective viscosity is used to capture the flow behaviors in the transition regime. A kinetic boundary condition, the combined bounce-back and specular-reflection scheme with the second-order slip scheme, is also designed for the FMLB model. By analyzing a unidirectional flow, the slip velocity and the discrete effects related to the boundary condition are derived within the FMLB model, and a revised scheme is presented to overcome such effects, which have also been validated through numerical simulations. To gain an accurate simulation in a wide range of Knudsen numbers, covering the slip and the entire transition flow regimes, a set of slip coefficients with an introduced fitting function is adopted in the revised second-order slip boundary condition. The periodic and pressure-driven microchannel flows have been investigated by the present model in this study. The numerical results, including the velocity profile and the mass flow rate, as well as the nonlinear pressure distribution along the channel, agree fairly well with the solutions of the linearized Boltzmann equation, the direct simulation Monte Carlo results, the experimental data, and the previous results of the multiple effective relaxation lattice Boltzmann model. Also, the present results of the velocity profile and the mass flow rate show that the present model with the fitting function can yield improved predictions for the microchannel gas flow with higher Knudsen numbers in the transition flow regime.
Bohinc, Klemen; Shrestha, Ahis; Brumen, Milan; May, Sylvio
2012-03-01
In the classical mean-field description of the electric double layer, known as the Poisson-Boltzmann model, ions interact exclusively through their Coulomb potential. Ion specificity can arise through solvent-mediated, nonelectrostatic interactions between ions. We employ the Yukawa pair potential to model the presence of nonelectrostatic interactions. The combination of Yukawa and Coulomb potential on the mean-field level leads to the Poisson-Helmholtz-Boltzmann model, which employs two auxiliary potentials: one electrostatic and the other nonelectrostatic. In the present work we apply the Poisson-Helmholtz-Boltzmann model to ionic mixtures, consisting of monovalent cations and anions that exhibit different Yukawa interaction strengths. As a specific example we consider a single charged surface in contact with a symmetric monovalent electrolyte. From the minimization of the mean-field free energy we derive the Poisson-Boltzmann and Helmholtz-Boltzmann equations. These nonlinear equations can be solved analytically in the weak perturbation limit. This together with numerical solutions in the nonlinear regime suggests an intricate interplay between electrostatic and nonelectrostatic interactions. The structure and free energy of the electric double layer depends sensitively on the Yukawa interaction strengths between the different ion types and on the nonelectrostatic interactions of the mobile ions with the surface.
A Multicoefficient Slip-Corrected Reynolds Equation for Micro-Thin Film Gas Lubrication
Directory of Open Access Journals (Sweden)
Ng Eddie Yin-Kwee
2005-01-01
Full Text Available This work investigates and analyzes the performance of conventional slip models among various regimes of Knudsen number and developes a new multicoefficient slip-velocity model, by using Taguchi quality control techniques and numerical analysis. A modified Reynolds equation is also derived based on the new slip-flow model. The multicoefficient slip model and its slip-corrected Reynolds equation are suitable to a wide Knudsen range from slip to transition regime. In comparison with other conventional slip models, it is found that the current results have a better agreement with the solution obtained from the linearized Boltzmann equation and direct simulation of Monte Carlo method (DSMC.
Least-squares finite-element lattice Boltzmann method.
Li, Yusong; LeBoeuf, Eugene J; Basu, P K
2004-06-01
A new numerical model of the lattice Boltzmann method utilizing least-squares finite element in space and Crank-Nicolson method in time is presented. The new method is able to solve problem domains that contain complex or irregular geometric boundaries by using finite-element method's geometric flexibility and numerical stability, while employing efficient and accurate least-squares optimization. For the pure advection equation on a uniform mesh, the proposed method provides for fourth-order accuracy in space and second-order accuracy in time, with unconditional stability in the time domain. Accurate numerical results are presented through two-dimensional incompressible Poiseuille flow and Couette flow.
Hydrodynamic behaviour of Lattice Boltzmann and Lattice BGK models
Behrend, O; Warren, P
1993-01-01
Abstract: We present a numerical analysis of the validity of classical and generalized hydrodynamics for Lattice Boltzmann Equation (LBE) and Lattice BGK methods in two and three dimensions, as a function of the collision parameters of these models. Our analysis is based on the wave-number dependence of the evolution operator. Good ranges of validity are found for BGK models as long as the relaxation time is chosen smaller than or equal to unity. The additional freedom in the choice of collision parameters for LBE models does not seem to give significant improvement.
A lattice Boltzmann method for dilute polymer solutions.
Singh, Shiwani; Subramanian, Ganesh; Ansumali, Santosh
2011-06-13
We present a lattice Boltzmann approach for the simulation of non-Newtonian fluids. The method is illustrated for the specific case of dilute polymer solutions. With the appropriate local equilibrium distribution, phase-space dynamics on a lattice, driven by a Bhatnagar-Gross-Krook (BGK) relaxation term, leads to a solution of the Fokker-Planck equation governing the probability density of polymer configurations. Results for the bulk rheological characteristics for steady and start-up shear flow are presented, and compare favourably with those obtained using Brownian dynamics simulations. The new method is less expensive than stochastic simulation techniques, particularly in the range of small to moderate Weissenberg numbers (Wi).
Static contact angle in lattice Boltzmann models of immiscible fluids.
Latva-Kokko, M; Rothman, Daniel H
2005-10-01
We study numerically the capillary rise between two horizontal plates and in a rectangular tube, using a lattice Boltzmann (LB) method. We derive an equation for the static fluid-solid contact angle as a function of the wetting tendency of the walls and test its validity. We show that the generalized Laplace law with two independent radii of curvature is followed in capillary rise in rectangular tubes. Our method removes the history dependence of the fluid-solid contact angle that had been present in earlier LB schemes.
On space enrichment estimator for nonlinear Poisson-Boltzmann
Randrianarivony, Maharavo
2013-10-01
We consider the mathematical aspect of the nonlinear Poisson-Boltzmann equation which physically governs the ionic interaction between solute and solvent media. The presented a-posteriori estimates can be computed locally in a very efficient manner. The a-posteriori error is based upon hierarchical space enrichment which ensures its efficiency and reliability. A brief survey of the solving of the nonlinear system resulting from the FEM discretization is reported. To corroborate the analysis, we report on a few numerical results for illustrations. We numerically examine some values of the constants encountered in the theoretical study.
Moving charged particles in lattice Boltzmann-based electrokinetics.
Kuron, Michael; Rempfer, Georg; Schornbaum, Florian; Bauer, Martin; Godenschwager, Christian; Holm, Christian; de Graaf, Joost
2016-12-07
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.
Directory of Open Access Journals (Sweden)
Pål Johan From
2012-04-01
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.
Moving Charged Particles in Lattice Boltzmann-Based Electrokinetics
Kuron, Michael; Schornbaum, Florian; Bauer, Martin; Godenschwager, Christian; Holm, Christian; de Graaf, Joost
2016-01-01
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 (LB) algorithm for the simulation of hydrodynamic flow and exploits computational locality. However, thus far, a description of how to incorporate moving boundary conditions, which are needed to simulate moving colloids, into the Capuani scheme has been lacking. In this paper, we detail how to introduce such moving boundaries, based on an analogue to the moving boundary method for the pure LB solver. The key ingredients in our method are mass and charge conservation for the solute spec...
A Nonlinera Krylov Accelerator for the Boltzmann k-Eigenvalue Problem
Calef, Matthew T; Warsa, James S; Berndt, Markus; Carlson, Neil N
2011-01-01
We compare variants of Anderson Mixing with the Jacobian-Free Newton-Krylov and Broyden methods applied to the k-eigenvalue formulation of the linear Boltzmann transport equation. We present evidence that one variant of Anderson Mixing finds solutions in the fewest number of iterations. We examine and strengthen theoretical results of Anderson Mixing applied to linear problems.
Lattice-Boltzmann-based two-phase thermal model for simulating phase change
Kamali, M.R.; Gillissen, J.J.J.; Van den Akker, H.E.A.; Sundaresan, S.
2013-01-01
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
Reprint of : The Boltzmann--Langevin approach: A simple quantum-mechanical derivation
Nagaev, K. E.
2016-08-01
We present a simple quantum-mechanical derivation of correlation function of Langevin sources in the semiclassical Boltzmann-Langevin equation. The specific case of electron-phonon scattering is considered. It is shown that the assumption of weak scattering leads to the Poisson nature of the scattering fluxes.
Lattice-Boltzmann-based two-phase thermal model for simulating phase change
Kamali, M.R.; Gillissen, J.J.J.; Van den Akker, H.E.A.; Sundaresan, S.
2013-01-01
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
Thermal cascaded lattice Boltzmann method
Fei, Linlin
2016-01-01
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...
An Infinite Restricted Boltzmann Machine.
Côté, Marc-Alexandre; Larochelle, Hugo
2016-07-01
We present a mathematical construction for the restricted Boltzmann machine (RBM) that does not require specifying the number of hidden units. In fact, the hidden layer size is adaptive and can grow during training. This is obtained by first extending the RBM to be sensitive to the ordering of its hidden units. Then, with a carefully chosen definition of the energy function, we show that the limit of infinitely many hidden units is well defined. As with RBM, approximate maximum likelihood training can be performed, resulting in an algorithm that naturally and adaptively adds trained hidden units during learning. We empirically study the behavior of this infinite RBM, showing that its performance is competitive to that of the RBM, while not requiring the tuning of a hidden layer size.
Directory of Open Access Journals (Sweden)
M. A. Aziz
2014-01-01
Full Text Available The design of a transonic-fan rotor is optimized using numerical computations of the full three-dimensional Navier-Stokes equations. The CFDRC-ACE multiphysics module, which is a pressure-based solver, is used for the numerical simulation. The code is coupled with simplex optimization algorithm. The optimization process is started from a suitable design point obtained using low fidelity analytical methods that is based on experimental correlations for the pressure losses and blade deviation angle. The fan blade shape is defined by its stacking line and airfoil shape which are considered the optimization parameters. The stacking line is defined by lean, sweep, and skews, while blade airfoil shape is modified considering the thickness and camber distributions. The optimization has been performed to maximize the rotor total pressure ratio while keeping the rotor efficiency and surge margin above certain required values. The results obtained are verified with the experimental data of Rotor 67. In addition, the results of the optimized fan indicate that the optimum design is found to be leaned in the direction of rotation and has a forward sweep from the hub to mean section and backward sweep to the tip. The pressure ratio increases from 1.427 to 1.627 at the design speed and mass flow rate.
Institute of Scientific and Technical Information of China (English)
蔡新景; 王新新; 邹晓兵; 鲁志伟
2016-01-01
The propagation properties of streamer could be obtained accurately by using the fluid model to simulate, provided that precise electron transport and reaction coefficients were input the model. Presently, there were two methods to calculate the transport coefficients: interpolation method based on the data of swarm experiment or two-term approximation of the Boltzmann equation. It was showed that interpolation method could be merely used in low reduced electric condition, and for the second method the reasonableness of the isotropic assumption and the accuracy of the results were a question. In view of the above problems, multi-term approximation of the Boltzmann equation to obtain electron transport coefficients was proposed by Nesset al. Here two improvements are provided: firstly, the expansion sequence is adjusted to derive the unified hierarchy in the hydrodynamic and non-hydrodynamic limit; secondly, the collision integral is evaluated based on the Gauss-Kronrod rule instead of Gauss-Laguerre rule as used in Ness''s works. In the end hard sphere model and Reid''s ramp inelastic model are considered. It is shown that it is more accurate to evaluate the collision integral based on Gauss-Kronrod integration method than Gauss-Laguerre method. Furthermore, it is demonstrated that the electron velocity distribution is anisotropic even only undergoing conservative collision, so there is a great error if the two -term approximation is used to obtain the electron transport coefficients.%获得准确的电子输运和反应系数是采用流体模型准确仿真流注传播特性的前提.目前,电子输运系数主要有两种计算方法:一是用电子群实验数据进行插值;二是用两项近似方法解玻尔兹曼方程.方法一只能用在约化场强很小的场合,方法二的理论基础即各相同性假设是否成立和数据准确度尚无定论.针对以上问题,Ness 等人采用了多项近似法解玻尔兹曼方程计算电子输运系数.该
Study of the Dynamics of a Condensing Bubble Using Lattice Boltzmann Method
Directory of Open Access Journals (Sweden)
Shahnawaz Ahmed
2015-06-01
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.
Pruning Boltzmann networks and hidden Markov models
DEFF Research Database (Denmark)
Pedersen, Morten With; Stork, D.
1996-01-01
We present sensitivity-based pruning algorithms for general Boltzmann networks. Central to our methods is the efficient calculation of a second-order approximation to the true weight saliencies in a cross-entropy error. Building upon previous work which shows a formal correspondence between linear...... Boltzmann chains and hidden Markov models (HMMs), we argue that our method can be applied to HMMs as well. We illustrate pruning on Boltzmann zippers, which are equivalent to two HMMs with cross-connection links. We verify that our second-order approximation preserves the rank ordering of weight saliencies...
Multiphase lattice Boltzmann methods theory and application
Huang, Haibo; Lu, Xiyun
2015-01-01
Theory and Application of Multiphase Lattice Boltzmann Methods presents a comprehensive review of all popular multiphase Lattice Boltzmann Methods developed thus far and is aimed at researchers and practitioners within relevant Earth Science disciplines as well as Petroleum, Chemical, Mechanical and Geological Engineering. Clearly structured throughout, this book will be an invaluable reference on the current state of all popular multiphase Lattice Boltzmann Methods (LBMs). The advantages and disadvantages of each model are presented in an accessible manner to enable the reader to choose the
Held, M
2015-01-01
A lattice Boltzmann method (LBM) approach to the Charney-Hasegawa-Mima (CHM) model for adiabatic drift wave turbulence in magnetised plasmas, is implemented. The CHM-LBM model contains a barotropic equation of state for the potential, a force term including a cross-product analogous to the Coriolis force in quasigeostrophic models, and a density gradient source term. Expansion of the resulting lattice Boltzmann model equations leads to cold-ion fluid continuity and momentum equations, which resemble CHM dynamics under drift ordering. The resulting numerical solutions of standard test cases (monopole propagation, stable drift modes and decaying turbulence) are compared to results obtained by a conventional finite difference scheme that directly discretizes the CHM equation. The LB scheme resembles characteristic CHM dynamics apart from an additional shear in the density gradient direction. The occuring shear reduces with the drift ratio and is ascribed to the compressible limit of the underlying LBM.
Lattice Boltzmann model for incompressible axisymmetric thermal flows through porous media.
Grissa, Kods; Chaabane, Raoudha; Lataoui, Zied; Benselama, Adel; Bertin, Yves; Jemni, Abdelmajid
2016-10-01
The present work proposes a simple lattice Boltzmann model for incompressible axisymmetric thermal flows through porous media. By incorporating forces and source terms into the lattice Boltzmann equation, the incompressible Navier-Stokes equations are recovered through the Chapman-Enskog expansion. It is found that the added terms are just the extra terms in the governing equations for the axisymmetric thermal flows through porous media compared with the Navier-Stokes equations. Four numerical simulations are performed to validate this model. Good agreement is obtained between the present work and the analytic solutions and/or the results of previous studies. This proves its efficacy and simplicity regarding other methods. Also, this approach provides guidance for problems with more physical phenomena and complicated force forms.
Ion-Conserving Modified Poisson-Boltzmann Theory Considering a Steric Effect in an Electrolyte
Sugioka, Hideyuki
2016-12-01
The modified Poisson-Nernst-Planck (MPNP) and modified Poisson-Boltzmann (MPB) equations are well known as fundamental equations that consider a steric effect, which prevents unphysical ion concentrations. However, it is unclear whether they are equivalent or not. To clarify this problem, we propose an improved free energy formulation that considers a steric limit with an ion-conserving condition and successfully derive the ion-conserving modified Poisson-Boltzmann (IC-MPB) equations that are equivalent to the MPNP equations. Furthermore, we numerically examine the equivalence by comparing between the IC-MPB solutions obtained by the Newton method and the steady MPNP solutions obtained by the finite-element finite-volume method. A surprising aspect of our finding is that the MPB solutions are much different from the MPNP (IC-MPB) solutions in a confined space. We consider that our findings will significantly contribute to understanding the surface science between solids and liquids.
Simpson, Jamesina J.
2009-04-01
Advances in computing technologies in recent decades have provided a means of generating and performing highly sophisticated computational simulations of electromagnetic phenomena. In particular, just after the turn of the twenty-first century, improvements to computing infrastructures provided for the first time the opportunity to conduct advanced, high-resolution three-dimensional full-vector Maxwell’s equations investigations of electromagnetic propagation throughout the global Earth-ionosphere spherical volume. These models, based on the finite-difference time-domain (FDTD) method, are capable of including such details as the Earth’s topography and bathymetry, as well as arbitrary horizontal/vertical geometrical and electrical inhomogeneities and anisotropies of the ionosphere, lithosphere, and oceans. Studies at this level of detail simply are not achievable using analytical methods. The goal of this paper is to provide an historical overview and future prospectus of global FDTD computational research for both natural and man-made electromagnetic phenomena around the world. Current and future applications of global FDTD models relating to lightning sources and radiation, Schumann resonances, hypothesized earthquake precursors, remote sensing, and space weather are discussed.
Jang, Jinwoo; Smyth, Andrew W.
2017-01-01
The objective of structural model updating is to reduce inherent modeling errors in Finite Element (FE) models due to simplifications, idealized connections, and uncertainties of material properties. Updated FE models, which have less discrepancies with real structures, give more precise predictions of dynamic behaviors for future analyses. However, model updating becomes more difficult when applied to civil structures with a large number of structural components and complicated connections. In this paper, a full-scale FE model of a major long-span bridge has been updated for improved consistency with real measured data. Two methods are applied to improve the model updating process. The first method focuses on improving the agreement of the updated mode shapes with the measured data. A nonlinear inequality constraint equation is used to an optimization procedure, providing the capability to regulate updated mode shapes to remain within reasonable agreements with those observed. An interior point algorithm deals with nonlinearity in the objective function and constraints. The second method finds very efficient updating parameters in a more systematic way. The selection of updating parameters in FE models is essential to have a successful updating result because the parameters are directly related to the modal properties of dynamic systems. An in-depth sensitivity analysis is carried out in an effort to precisely understand the effects of physical parameters in the FE model on natural frequencies. Based on the sensitivity analysis, cluster analysis is conducted to find a very efficient set of updating parameters.
Student understanding of the Boltzmann factor
Smith, Trevor I.; Mountcastle, Donald B.; Thompson, John R.
2015-12-01
[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.
Boltzmann and the art of flying
Dahmen, Silvio R
2007-01-01
One of the less known facets of Ludwig Boltzmann was that of an advocate of Aviation, one of the most challenging technological problems of his times. Boltzmann followed closely the studies of pioneers like Otto Lilienthal in Berlin, and during a lecture on a prestigious conference he vehemently defended further investments in the area. In this article I discuss his involvement with Aviation, his role in its development and his correspondence with two flight pioneers, Otto Lilienthal e Wilhelm Kress.
Matrix-valued Quantum Lattice Boltzmann Method
Mendl, Christian B
2013-01-01
We develop a numerical framework for the quantum analogue of the "classical" lattice Boltzmann method (LBM), with the Maxwell-Boltzmann distribution replaced by the Fermi-Dirac function. To accommodate the spin density matrix, the distribution functions become 2x2-matrix valued. We show that the efficient, commonly used BGK approximation of the collision operator is valid in the present setting. The framework could leverage the principles of LBM for simulating complex spin systems, with applications to spintronics.
Comment on ‘A low-uncertainty measurement of the Boltzmann constant’
Macnaughton, Donald B.
2016-02-01
The International Committee for Weights and Measures has projected a major revision of the International System of Units in which all the base units will be defined by fixing the values of certain fundamental constants of nature. To assist, de Podesta et al recently experimentally obtained a precise new estimate of the Boltzmann constant. This estimate is proposed as a basis for the redefinition of the unit of temperature, the kelvin. The present paper reports a reanalysis of de Podesta et al’s data that reveals systematic non-random patterns in the residuals of the key fitted model equation. These patterns violate the assumptions underlying the analysis and thus they raise questions about the validity of de Podesta et al’s estimate of the Boltzmann constant. An approach is discussed to address these issues, which should lead to an accurate estimate of the Boltzmann constant with a lower uncertainty.
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
2016-11-02
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.
Three-dimensional lattice Boltzmann model for electrodynamics.
Mendoza, M; Muñoz, J D
2010-11-01
In this paper we introduce a three-dimensional Lattice-Boltzmann model that recovers in the continuous limit the Maxwell equations in materials. In order to build conservation equations with antisymmetric tensors, like the Faraday law, the model assigns four auxiliary vectors to each velocity vector. These auxiliary vectors, when combined with the distribution functions, give the electromagnetic fields. The evolution is driven by the usual Bhatnager-Gross-Krook (BGK) collision rule, but with a different form for the equilibrium distribution functions. This lattice Bhatnager-Gross-Krook (LBGK) model allows us to consider for both dielectrics and conductors with realistic parameters, and therefore it is adequate to simulate the most diverse electromagnetic problems, like the propagation of electromagnetic waves (both in dielectric media and in waveguides), the skin effect, the radiation pattern of a small dipole antenna and the natural frequencies of a resonant cavity, all with 2% accuracy. Actually, it shows to be one order of magnitude faster than the original Finite-difference time-domain (FDTD) formulation by Yee to reach the same accuracy. It is, therefore, a valuable alternative to simulate electromagnetic fields and opens lattice Boltzmann for a broad spectrum of new applications in electrodynamics.
Lattice Boltzmann methods for global linear instability analysis
Pérez, José Miguel; Aguilar, Alfonso; Theofilis, Vassilis
2016-11-01
Modal global linear instability analysis is performed using, for the first time ever, the lattice Boltzmann method (LBM) to analyze incompressible flows with two and three inhomogeneous spatial directions. Four linearization models have been implemented in order to recover the linearized Navier-Stokes equations in the incompressible limit. Two of those models employ the single relaxation time and have been proposed previously in the literature as linearization of the collision operator of the lattice Boltzmann equation. Two additional models are derived herein for the first time by linearizing the local equilibrium probability distribution function. Instability analysis results are obtained in three benchmark problems, two in closed geometries and one in open flow, namely the square and cubic lid-driven cavity flow and flow in the wake of the circular cylinder. Comparisons with results delivered by classic spectral element methods verify the accuracy of the proposed new methodologies and point potential limitations particular to the LBM approach. The known issue of appearance of numerical instabilities when the SRT model is used in direct numerical simulations employing the LBM is shown to be reflected in a spurious global eigenmode when the SRT model is used in the instability analysis. Although this mode is absent in the multiple relaxation times model, other spurious instabilities can also arise and are documented herein. Areas of potential improvements in order to make the proposed methodology competitive with established approaches for global instability analysis are discussed.
Lattice Boltzmann Method for Spacecraft Propellant Slosh Simulation
Orr, Jeb S.; Powers, Joseph F.; Yang, Hong Q
2015-01-01
A scalable computational approach to the simulation of propellant tank sloshing dynamics in microgravity is presented. In this work, we use the lattice Boltzmann equation (LBE) to approximate the behavior of two-phase, single-component isothermal flows at very low Bond numbers. Through the use of a non-ideal gas equation of state and a modified multiple relaxation time (MRT) collision operator, the proposed method can simulate thermodynamically consistent phase transitions at temperatures and density ratios consistent with typical spacecraft cryogenic propellants, for example, liquid oxygen. Determination of the tank forces and moments is based upon a novel approach that relies on the global momentum conservation of the closed fluid domain, and a parametric wall wetting model allows tuning of the free surface contact angle. Development of the interface is implicit and no interface tracking approach is required. A numerical example illustrates the method's application to prediction of bulk fluid behavior during a spacecraft ullage settling maneuver.
Simulation of a Microfluidic Gradient Generator using Lattice Boltzmann Methods
Simon, Tanaka
2013-01-01
Microfluidics provides a powerful and versatile technology to accurately control spatial and temporal conditions for cell culturing and can therefore be used to study cellular responses to gradients. Here we use Lattice Boltzmann methods (LBM) to solve both the Navier-Stokes equation (NSE) for the fluid and the coupled convection-diffusion equation (CDE) for the compounds that form the diffusion-based gradient. The design of a microfluidic chamber for diffusion-based gradients must avoid flow through the cell chamber. This can be achieved by alternately opening the source and the sink channels. The fast toggling of microfluidic valves requires switching between different boundary conditions. We demonstrate that the LBM is a powerful method for handling complex geometries, high Peclet number conditions, discontinuities in the boundary conditions, and multiphysics coupling.
An integrated Boltzmann+hydrodynamics approach to heavy ion collisions
Energy Technology Data Exchange (ETDEWEB)
Petersen, Hannah
2009-04-22
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
Numerical Comparison of Solutions of Kinetic Model Equations
Directory of Open Access Journals (Sweden)
A. A. Frolova
2015-01-01
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
KOMPUTASI DISTRIBUSI NEUTRON DALAM STATISTIK MAXWELL BOLTZMANN
Directory of Open Access Journals (Sweden)
Tuti Purwoningsih
2013-03-01
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
Influence of asperities on fluid and thermal flow in a fracture: a coupled Lattice Boltzmann study
Neuville, Amélie; Toussaint, Renaud
2013-01-01
The characteristics of the hydro-thermal flow which occurs when a cold fluid is injected into a hot fractured bedrock depend on the morphology of the fracture. We consider a sharp triangular asperity, invariant in one direction, perturbing an otherwise flat fracture. We investigate its influence on the macroscopic hydraulic transmissivity and heat transfer efficiency, at fixed low Reynolds number. In this study, numerical simulations are done with a coupled lattice Boltzmann method that solves both the complete Navier-Stokes and advection-diffusion equations in three dimensions. The results are compared with those obtained under lubrication approximations which rely on many hypotheses and neglect the three-dimensional (3D) effects. The lubrication results are obtained by analytically solving the Stokes equation and a two-dimensional (integrated over the thickness) advection-diffusion equation. We use a lattice Boltzmann method with a double distribution (for mass and energy transport) on hypercubic and cubic ...
Furusawa, S.; Togashi, H.; Nagakura, H.; Sumiyoshi, K.; Yamada, S.; Suzuki, H.; Takano, M.
2017-09-01
We have constructed a nuclear equation of state (EOS) that includes a full nuclear ensemble for use in core-collapse supernova simulations. It is based on the EOS for uniform nuclear matter that two of the authors derived recently, applying a variational method to realistic two- and three-body nuclear forces. We have extended the liquid drop model of heavy nuclei, utilizing the mass formula that accounts for the dependences of bulk, surface, Coulomb and shell energies on density and/or temperature. As for light nuclei, we employ a quantum-theoretical mass evaluation, which incorporates the Pauli- and self-energy shifts. In addition to realistic nuclear forces, the inclusion of in-medium effects on the full ensemble of nuclei makes the new EOS one of the most realistic EOSs, which covers a wide range of density, temperature and proton fraction that supernova simulations normally encounter. We make comparisons with the FYSS EOS, which is based on the same formulation for the nuclear ensemble but adopts the relativistic mean field theory with the TM1 parameter set for uniform nuclear matter. The new EOS is softer than the FYSS EOS around and above nuclear saturation densities. We find that neutron-rich nuclei with small mass numbers are more abundant in the new EOS than in the FYSS EOS because of the larger saturation densities and smaller symmetry energy of nuclei in the former. We apply the two EOSs to 1D supernova simulations and find that the new EOS gives lower electron fractions and higher temperatures in the collapse phase owing to the smaller symmetry energy. As a result, the inner core has smaller masses for the new EOS. It is more compact, on the other hand, due to the softness of the new EOS and bounces at higher densities. It turns out that the shock wave generated by core bounce is a bit stronger initially in the simulation with the new EOS. The ensuing outward propagations of the shock wave in the outer core are very similar in the two simulations, which
Progress in developing Poisson-Boltzmann equation solvers
Li, Chuan; Li, Lin; Petukh, Marharyta; Alexov, Emil
2013-01-01
This review outlines the recent progress made in developing more accurate and efficient solutions to model electrostatics in systems comprised of bio-macromolecules and nano-objects, the last one referring to objects that do not have biological function themselves but nowadays are frequently used in biophysical and medical approaches in conjunction with bio-macromolecules. The problem of modeling macromolecular electrostatics is reviewed from two different angles: as a mathematical task provided the specific definition of the system to be modeled and as a physical problem aiming to better capture the phenomena occurring in the real experiments. In addition, specific attention is paid to methods to extend the capabilities of the existing solvers to model large systems toward applications of calculations of the electrostatic potential and energies in molecular motors, mitochondria complex, photosynthetic machinery and systems involving large nano-objects. PMID:24199185
Goal Adaptive Discretization of a One-Dimensional Boltzmann Equation
Hoitinga, W.
2011-01-01
Fluid-flow problems in the transitional molecular/continuum regime play an important role in many engineering applications. Such problems are gaining further prominence with the perpetual trend towards miniaturization in science and engineering. The numerical simulation of flows in the transitional
Progress in developing Poisson-Boltzmann equation solvers.
Li, Chuan; Li, Lin; Petukh, Marharyta; Alexov, Emil
2013-03-01
This review outlines the recent progress made in developing more accurate and efficient solutions to model electrostatics in systems comprised of bio-macromolecules and nano-objects, the last one referring to objects that do not have biological function themselves but nowadays are frequently used in biophysical and medical approaches in conjunction with bio-macromolecules. The problem of modeling macromolecular electrostatics is reviewed from two different angles: as a mathematical task provided the specific definition of the system to be modeled and as a physical problem aiming to better capture the phenomena occurring in the real experiments. In addition, specific attention is paid to methods to extend the capabilities of the existing solvers to model large systems toward applications of calculations of the electrostatic potential and energies in molecular motors, mitochondria complex, photosynthetic machinery and systems involving large nano-objects.
Computational Aeroacoustics Using the Generalized Lattice Boltzmann Equation Project
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...
Niu, Xiao-Dong; Hyodo, Shi-Aki; Munekata, Toshihisa; Suga, Kazuhiko
2007-09-01
It is well known that the Navier-Stokes equations cannot adequately describe gas flows in the transition and free-molecular regimes. In these regimes, the Boltzmann equation (BE) of kinetic theory is invoked to govern the flows. However, this equation cannot be solved easily, either by analytical techniques or by numerical methods. Hence, in order to efficiently maneuver around this equation for modeling microscale gas flows, a kinetic lattice Boltzmann method (LBM) has been introduced in recent years. This method is regarded as a numerical approach for solving the BE in discrete velocity space with Gauss-Hermite quadrature. In this paper, a systematic description of the kinetic LBM, including the lattice Boltzmann equation, the diffuse-scattering boundary condition for gas-surface interactions, and definition of the relaxation time, is provided. To capture the nonlinear effects due to the high-order moments and wall boundaries, an effective relaxation time and a modified regularization procedure of the nonequilibrium part of the distribution function are further presented based on previous work [Guo et al., J. Appl. Phys. 99, 074903 (2006); Shan et al., J. Fluid Mech. 550, 413 (2006)]. The capability of the kinetic LBM of simulating microscale gas flows is illustrated based on the numerical investigations of micro Couette and force-driven Poiseuille flows.
Watari, Minoru
2009-06-01
Two problems exist in the current studies on the application of the lattice Boltzmann method (LBM) to rarefied gas dynamics. First, most studies so far are applications of two-dimensional models. The numbers of velocity particles are small. Consequently, the boundary-condition methods of these studies are not directly applicable to a multispeed finite-difference lattice Boltzmann method (FDLBM) that has many velocity particles. Second, the LBM and FDLBM share their origins with the Boltzmann equation. Therefore, the results of LBM and FDLBM studies should be verified by the results of the continuous Boltzmann equation. In my review to date on the LBM studies, it appears that such verifications were seldom done. In this study, velocity slip and temperature jump simulations in the slip-flow regime were conducted using a three-dimensional FDLBM model. The results were compared with preceding theoretical studies based on the continuous Boltzmann equation. The results agreed with the theory with errors of a few percent. To further improve the accuracy of the FDLBM, it seems necessary to increase the number of velocity particles.
Directory of Open Access Journals (Sweden)
Mandana Samari Kermani
2016-01-01
Full Text Available The interaction of spherical solid particles with turbulent eddies in a 3-D turbulent channel flow with friction Reynolds number was studied. A generalized lattice Boltzmann equation (GLBE was used for computation of instantaneous turbulent flow field for which large eddy simulation (LES was employed. The sub-grid-scale (SGS turbulence effects were simulated through a shear-improved Smagorinsky model (SISM, which can predict turbulent near wall region without any wall function. Statistical properties of particles behavior such as root mean square (RMS velocities were studied as a function of dimensionless particle relaxation time ( by using a Lagrangian approach. Combination of SISM in GLBE with particle tracking analysis in turbulent channel flow is novelty of the present work. Both GLBE and SISM solve the flow field equations locally. This is an advantage of this method and makes it easy implementing. Comparison of the present results with previous available data indicated that SISM in GLBE is a reliable method for simulation of turbulent flows which is a key point to predict particles behavior correctly.
Lu, Benzhuo; Cheng, Xiaolin; Huang, Jingfang; McCammon, J. Andrew
2013-11-01
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 the 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 http://lsec.cc.ac.cn/~lubz/afmpb.html and a mirror site at http://mccammon.ucsd.edu/. This paper is a brief summary of the program: the algorithms, the implementation and the usage. Restrictions: Only three or six significant digits options are provided in this version. Unusual features: Most of the codes are in Fortran77 style. Memory allocation functions from Fortran90 and above are used in a few subroutines. Additional comments: The current version of the codes is designed and written for single core/processor desktop machines. Check http://lsec.cc.ac.cn/lubz/afmpb.html for updates and changes. Running time: The running time varies with the number of discretized elements (N) in the system and their distributions. In most cases, it scales linearly as a function of N.
Non-Boltzmann Modeling for Air Shock-Layer Radiation at Lunar-Return Conditions
Johnston, Christopher O.; Hollis, Brian R.; Sutton, Kenneth
2008-01-01
This paper investigates the non-Boltzmann modeling of the radiating atomic and molecular electronic states present in lunar-return shock-layers. The Master Equation is derived for a general atom or molecule while accounting for a variety of excitation and de-excitation mechanisms. A new set of electronic-impact excitation rates is compiled for N, O, and N2+, which are the main radiating species for most lunar-return shock-layers. Based on these new rates, a novel approach of curve-fitting the non-Boltzmann populations of the radiating atomic and molecular states is developed. This new approach provides a simple and accurate method for calculating the atomic and molecular non-Boltzmann populations while avoiding the matrix inversion procedure required for the detailed solution of the Master Equation. The radiative flux values predicted by the present detailed non-Boltzmann model and the approximate curve-fitting approach are shown to agree within 5% for the Fire 1634 s case.
Polyelectrolyte Microcapsules: Ion Distributions from a Poisson-Boltzmann Model
Tang, Qiyun; Denton, Alan R.; Rozairo, Damith; Croll, Andrew B.
2014-03-01
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.
High order spectral difference lattice Boltzmann method for incompressible hydrodynamics
Li, Weidong
2017-09-01
This work presents a lattice Boltzmann equation (LBE) based high order spectral difference method for incompressible flows. In the present method, the spectral difference (SD) method is adopted to discretize the convection and collision term of the LBE to obtain high order (≥3) accuracy. Because the SD scheme represents the solution as cell local polynomials and the solution polynomials have good tensor-product property, the present spectral difference lattice Boltzmann method (SD-LBM) can be implemented on arbitrary unstructured quadrilateral meshes for effective and efficient treatment of complex geometries. Thanks to only first oder PDEs involved in the LBE, no special techniques, such as hybridizable discontinuous Galerkin method (HDG), local discontinuous Galerkin method (LDG) and so on, are needed to discrete diffusion term, and thus, it simplifies the algorithm and implementation of the high order spectral difference method for simulating viscous flows. The proposed SD-LBM is validated with four incompressible flow benchmarks in two-dimensions: (a) the Poiseuille flow driven by a constant body force; (b) the lid-driven cavity flow without singularity at the two top corners-Burggraf flow; and (c) the unsteady Taylor-Green vortex flow; (d) the Blasius boundary-layer flow past a flat plate. Computational results are compared with analytical solutions of these cases and convergence studies of these cases are also given. The designed accuracy of the proposed SD-LBM is clearly verified.
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)
2007-10-15
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.
Lattice Boltzmann Simulation for Complex Flow in a Solar Wall
Institute of Scientific and Technical Information of China (English)
CHEN Rou; Shao Jiu-Gu; ZHENG You-Qu; YU Hui-Dan; XU You-Sheng
2013-01-01
In this letter,we present a lattice Boltzmann simulation for complex flow in a solar wall system which includes porous media flow and heat transfer,specifically for solar energy utilization through an unglazed transpired solar air collector (UTC).Besides the lattice Boltzmann equation (LBE) for time evolution of particle distribution function for fluid field,we introduce an analogy,LBE for time evolution of distribution function for temperature.Both temperature fields of fluid (air) and solid (porous media) are modeled.We study the effects of fan velocity,solar radiation intensity,porosity,etc.on the thermal performance of the UTC.In general,our simulation results are in good agreement with what in literature.With the current system setting,both fan velocity and solar radiation intensity have significant effect on the thermal performance of the UTC.However,it is shown that the porosity has negligible effect on the heat collector indicating the current system setting might not be realistic.Further examinations of thermal performance in different UTC systems are ongoing.The results are expected to present in near future.
Phantom cosmology and Boltzmann brains problem
Astashenok, Artyom V; Yurov, Valerian V
2013-01-01
We consider the well-known Boltzmann brains problem in frames of simple phantom energy models with little rip, big rip and big freeze singularity. It is showed that these models (i) satisfy to observational data and (ii) may be free from Boltzmann brains problem. The human observers in phantom models can exist only in during for a certain period $t
Huang, Haiping
2017-05-01
Revealing hidden features in unlabeled data is called unsupervised feature learning, which plays an important role in pretraining a deep neural network. Here we provide a statistical mechanics analysis of the unsupervised learning in a restricted Boltzmann machine with binary synapses. A message passing equation to infer the hidden feature is derived, and furthermore, variants of this equation are analyzed. A statistical analysis by replica theory describes the thermodynamic properties of the model. Our analysis confirms an entropy crisis preceding the non-convergence of the message passing equation, suggesting a discontinuous phase transition as a key characteristic of the restricted Boltzmann machine. Continuous phase transition is also confirmed depending on the embedded feature strength in the data. The mean-field result under the replica symmetric assumption agrees with that obtained by running message passing algorithms on single instances of finite sizes. Interestingly, in an approximate Hopfield model, the entropy crisis is absent, and a continuous phase transition is observed instead. We also develop an iterative equation to infer the hyper-parameter (temperature) hidden in the data, which in physics corresponds to iteratively imposing Nishimori condition. Our study provides insights towards understanding the thermodynamic properties of the restricted Boltzmann machine learning, and moreover important theoretical basis to build simplified deep networks.
An exact energy conservation property of the quantum lattice Boltzmann algorithm
Energy Technology Data Exchange (ETDEWEB)
Dellar, Paul J., E-mail: dellar@maths.ox.ac.uk [OCIAM, Mathematical Institute, 24-29 St Giles' , Oxford OX1 3LB (United Kingdom)
2011-11-28
The quantum lattice Boltzmann algorithm offers a unitary and readily parallelisable discretisation of the Dirac equation that is free of the fermion-doubling problem. The expectation of the discrete time-advance operator is an exact invariant of the algorithm. Its imaginary part determines the expectation of the Hamiltonian operator, the energy of the solution, with an accuracy that is consistent with the overall accuracy of the algorithm. In the one-dimensional case, this accuracy may be increased from first to second order using a variable transformation. The three-dimensional quantum lattice Boltzmann algorithm uses operator splitting to approximate evolution under the three-dimensional Dirac equation by a sequence of solutions of one-dimensional Dirac equations. The three-dimensional algorithm thus inherits the energy conservation property of the one-dimensional algorithm, although the implementation shown remains only first-order accurate due to the splitting error. -- Highlights: ► The quantum lattice Boltzmann algorithm approximates the Dirac equation. ► It has an exact invariant: the expectation of the discrete time-advance operator. ► The invariant consistently approximates the energy of the continuous system. ► We achieve second-order accuracy through a variable transformation.
Grid refinement for entropic lattice Boltzmann models.
Dorschner, B; Frapolli, N; Chikatamarla, S S; Karlin, I V
2016-11-01
We propose a multidomain grid refinement technique with extensions to entropic incompressible, thermal, and compressible lattice Boltzmann models. Its validity and accuracy are assessed by comparison to available direct numerical simulation and experiment for the simulation of isothermal, thermal, and viscous supersonic flow. In particular, we investigate the advantages of grid refinement for the setups of turbulent channel flow, flow past a sphere, Rayleigh-Bénard convection, as well as the supersonic flow around an airfoil. Special attention is paid to analyzing the adaptive features of entropic lattice Boltzmann models for multigrid simulations.
Grid refinement for entropic lattice Boltzmann models
Dorschner, B; Chikatamarla, S S; Karlin, I V
2016-01-01
We propose a novel multi-domain grid refinement technique with extensions to entropic incompressible, thermal and compressible lattice Boltzmann models. Its validity and accuracy are accessed by comparison to available direct numerical simulation and experiment for the simulation of isothermal, thermal and viscous supersonic flow. In particular, we investigate the advantages of grid refinement for the set-ups of turbulent channel flow, flow past a sphere, Rayleigh-Benard convection as well as the supersonic flow around an airfoil. Special attention is payed to analyzing the adaptive features of entropic lattice Boltzmann models for multi-grid simulations.
Fast lattice Boltzmann solver for relativistic hydrodynamics.
Mendoza, M; Boghosian, B M; Herrmann, H J; Succi, S
2010-07-01
A lattice Boltzmann formulation for relativistic fluids is presented and numerically validated through quantitative comparison with recent hydrodynamic simulations of relativistic fluids. In order to illustrate its capability to handle complex geometries, the scheme is also applied to the case of a three-dimensional relativistic shock wave, generated by a supernova explosion, impacting on a massive interstellar cloud. This formulation opens up the possibility of exporting the proven advantages of lattice Boltzmann methods, namely, computational efficiency and easy handling of complex geometries, to the context of (mildly) relativistic fluid dynamics at large, from quark-gluon plasmas up to supernovae with relativistic outflows.
Lattice-Boltzmann Method for Geophysical Plastic Flows
Leonardi, Alessandro; Mendoza, Miller; Herrmann, Hans J
2015-01-01
We explore possible applications of the Lattice-Boltzmann Method for the simulation of geophysical flows. This fluid solver, while successful in other fields, is still rarely used for geotechnical applications. We show how the standard method can be modified to represent free-surface realization of mudflows, debris flows, and in general any plastic flow, through the implementation of a Bingham constitutive model. The chapter is completed by an example of a full-scale simulation of a plastic fluid flowing down an inclined channel and depositing on a flat surface. An application is given, where the fluid interacts with a vertical obstacle in the channel.
Thrombosis modeling in intracranial aneurysms: a lattice Boltzmann numerical algorithm
Ouared, R.; Chopard, B.; Stahl, B.; Rüfenacht, D. A.; Yilmaz, H.; Courbebaisse, G.
2008-07-01
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.
Multispeed models in off-lattice Boltzmann simulations
Bardow, A.; Karlin, I.V.; Gusev, A.A.
2008-01-01
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.
Ou, Yaobin
2017-01-01
The vacuum free boundary problem of one-dimensional non-isentropic compressible Navier-Stokes equations with large initial data is investigated in this paper. The fluid is initially assumed to occupy a finite interval and connect to the vacuum continuously at the free boundary, which is often considered in the gas-vacuum interface problem. Using the method of Lagrangian particle path, we derive some point-wise estimates and weighted spatial and time energy estimates for the classical solutions. Then the global existence and uniqueness of classical solutions are shown, and the expanding speed for the free boundary is proved to be finite. The main difficulty of this problem is the degeneracy of the system near the free boundary. Previous results are only for the solutions with low regularity (cf. [G. Q. Chen and M. Kratka, Commun. Partial Differ. Equations. 27 907-943 (2002)]).
Chubykalo, A E; Chubykalo, Andrew E; Smirnov-Rueda, Roman
1995-01-01
The inadequacy of Li\\'{e}nard-Wiechert potentials is demonstrated as one of the examples related to the nonconsistence of the conventional classical electrodynamics. The insufficiency of the Faraday-Maxwell concept to describe the whole electromagnetic phenomena and the incompleteness of a set of solutions of Maxwell equations are discussed and mathematically proved. Reasons of the introduction of the so-called ``electrodynamics dualism concept" (simultaneous coexistence of instantaneous Newton long-range and Faraday-Maxwell short-range interactions) have been displayed. It is strictly shown that the new concept presents itself as the direct consequence of the complete set of Maxwell equations and makes it possible to consider classical electrodynamics as a self-consistent and complete theory, devoid of inward contradictions. In the framework of the new approach, all main concepts of classical electrodynamics are reconsidered. In particular, a limited class of motion is revealed when accelerated charges do no...
Beyond Poisson-Boltzmann: fluctuations and fluid structure in a self-consistent theory.
Buyukdagli, S; Blossey, R
2016-09-01
Poisson-Boltzmann (PB) theory is the classic approach to soft matter electrostatics and has been applied to numerous physical chemistry and biophysics problems. Its essential limitations are in its neglect of correlation effects and fluid structure. Recently, several theoretical insights have allowed the formulation of approaches that go beyond PB theory in a systematic way. In this topical review, we provide an update on the developments achieved in the self-consistent formulations of correlation-corrected Poisson-Boltzmann theory. We introduce a corresponding system of coupled non-linear equations for both continuum electrostatics with a uniform dielectric constant, and a structured solvent-a dipolar Coulomb fluid-including non-local effects. While the approach is only approximate and also limited to corrections in the so-called weak fluctuation regime, it allows us to include physically relevant effects, as we show for a range of applications of these equations.
Reis, T.
2010-09-06
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.
Beyond Poisson-Boltzmann: fluctuations and fluid structure in a self-consistent theory
Buyukdagli, S.; Blossey, R.
2016-09-01
Poisson-Boltzmann (PB) theory is the classic approach to soft matter electrostatics and has been applied to numerous physical chemistry and biophysics problems. Its essential limitations are in its neglect of correlation effects and fluid structure. Recently, several theoretical insights have allowed the formulation of approaches that go beyond PB theory in a systematic way. In this topical review, we provide an update on the developments achieved in the self-consistent formulations of correlation-corrected Poisson-Boltzmann theory. We introduce a corresponding system of coupled non-linear equations for both continuum electrostatics with a uniform dielectric constant, and a structured solvent—a dipolar Coulomb fluid—including non-local effects. While the approach is only approximate and also limited to corrections in the so-called weak fluctuation regime, it allows us to include physically relevant effects, as we show for a range of applications of these equations.
Kinetic theory of correlated fluids: from dynamic density functional to Lattice Boltzmann methods.
Marconi, Umberto Marini Bettolo; Melchionna, Simone
2009-07-07
Using methods of kinetic theory and liquid state theory we propose a description of the nonequilibrium behavior of molecular fluids, which takes into account their microscopic structure and thermodynamic properties. The present work represents an alternative to the recent dynamic density functional theory, which can only deal with colloidal fluids and is not apt to describe the hydrodynamic behavior of a molecular fluid. The method is based on a suitable modification of the Boltzmann transport equation for the phase space distribution and provides a detailed description of the local structure of the fluid and its transport coefficients. Finally, we propose a practical scheme to solve numerically and efficiently the resulting kinetic equation by employing a discretization procedure analogous to the one used in the Lattice Boltzmann method.
Investigation of Resistivity of Saturated Porous Media with Lattice Boltzmann Method
Institute of Scientific and Technical Information of China (English)
YUE Wen-Zheng; TAO Guo; ZHU Ke-Qin
2004-01-01
The lattice Boltzmann method is employed to study the electrical transport properties of saturated porous media.Electrical current flow through the porous media is simulated and the relationship between resistivity index and water saturation is derived. It is found that this kind of relation is not a straight line as described by the Archie equation with the parameter n being a constant in a log-log scale. A new equation is thus developed to formulate this relation with n being a function of porosity and water saturation. The comparisons between the results by lattice Boltzmann and by the laboratory experiments on rock samples demonstrate that this numerical method can provide an alternative way for the expensive laboratory experiments to investigate the electrical transport properties of saturated porous media and can be used to explore micro mechanisms more conveniently.
Slits, plates, and Poisson-Boltzmann theory in a local formulation of nonlocal electrostatics.
Paillusson, Fabien; Blossey, Ralf
2010-11-01
Polar liquids like water carry a characteristic nanometric length scale, the correlation length of orientation polarizations. Continuum theories that can capture this feature commonly run under the name of "nonlocal" electrostatics since their dielectric response is characterized by a scale-dependent dielectric function ε(q), where q is the wave vector; the Poisson(-Boltzmann) equation then turns into an integro-differential equation. Recently, "local" formulations have been put forward for these theories and applied to water, solvated ions, and proteins. We review the local formalism and show how it can be applied to a structured liquid in slit and plate geometries, and solve the Poisson-Boltzmann theory for a charged plate in a structured solvent with counterions. Our results establish a coherent picture of the local version of nonlocal electrostatics and show its ease of use when compared to the original formulation.
Poisson-Boltzmann Calculations: van der Waals or Molecular Surface?
Pang, Xiaodong; Zhou, Huan-Xiang
2013-01-01
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.
Lattice Boltzmann based discrete simulation for gas-solid fluidization
Wang, Limin; Wang, Xiaowei; Ge, Wei
2013-01-01
Discrete particle simulation, a combined approach of computational fluid dynamics and discrete methods such as DEM (Discrete Element Method), SPH (Smoothed Particle Hydrodynamics), PIC (Particle-In-Cell), etc., is becoming a practical tool for exploring lab-scale gas-solid systems owing to the fast development of its parallel computation. However, the gas-solid coupling and the corresponding fluid flow solver remain immature. In this work, we presented a modified lattice Boltzmann approach to consider the effect of both the local solid volume fraction and the local relative velocity between the particles and the fluid, which was different from the traditional volume-averaged Navier-Stokes equations. This approach is combined with a time-driven hard sphere algorithm to simulate the motion of individual particles in which particles interact with each other via hard-sphere collisions but the collision detection and motion of the particle are performed at constant time intervals, and the EMMS (energy minimization...
Chemical-potential-based Lattice Boltzmann Method for Nonideal Fluids
Wen, Binghai; He, Bing; Zhang, Chaoying; Fang, Haiping
2016-01-01
Chemical potential is an effective way to drive phase transition or express wettability. In this letter, we present a chemical-potential-based lattice Boltzmann model to simulate multiphase flows. The nonideal force is directly evaluated by a chemical potential. The model theoretically satisfies thermodynamics and Galilean invariance. The computational efficiency is improved owing to avoiding the calculation of pressure tensor. We have derived several chemical potentials of the popular equations of state from the free-energy density function. An effective chemical-potential boundary condition is implemented to investigate the wettability of a solid surface. Remarkably, the numerical results show that the contact angle can be linearly tuned by the surface chemical potential.
Double MRT thermal lattice Boltzmann method for simulating convective flows
Energy Technology Data Exchange (ETDEWEB)
Mezrhab, Ahmed, E-mail: mezrhab@fso.ump.m [Laboratoire de Mecanique and Energetique, Departement de Physique, Faculte des Sciences, Universite Mohammed 1er, 60000 Oujda (Morocco); Amine Moussaoui, Mohammed; Jami, Mohammed [Laboratoire de Mecanique and Energetique, Departement de Physique, Faculte des Sciences, Universite Mohammed 1er, 60000 Oujda (Morocco); Naji, Hassan [Universite Lille Nord de France, F-59000 Lille, and LML UMR CNRS 8107, F-59655 Villeneuve d' Ascq cedex (France); Bouzidi, M' hamed [Universite Clermont 2, LaMI EA 3867, IUT de Montlucon, Av. A. Briand, BP 2235, F-03101 Montlucon cedex (France)
2010-07-26
A two-dimensional double Multiple Relaxation Time-Thermal Lattice Boltzmann Equation (2-MRT-TLBE) method is developed for predicting convective flows in a square differentially heated cavity filled with air (Pr=0.71). In this Letter, we propose a numerical scheme to solve the flow and the temperature fields using the MRT-D2Q9 model and the MRT-D2Q5 model, respectively. Thus, the main objective of this study is to show the effectiveness of such model to predict thermodynamics for heat transfer. This model is validated by the numerical simulations of the 2-D convective square cavity flow. Excellent agreements are obtained between numerical predictions. These results demonstrate the accuracy and the effectiveness of the proposed methodology.
Stefan-Boltzmann law for massive photons
Moreira, E S
2015-01-01
Thirty years ago a paper appeared in the literature generalizing the Stefan-Boltzmann law to include massive photons. The paper suffers from a flaw though: it assumes that a massive photon travels at the speed of (massless) light. The present work fixes the mistake and presents the correct formula for the radiance.
Stefan-Boltzmann Law for Massive Photons
Moreira, E. S.; Ribeiro, T. G.
2016-08-01
This paper generalizes the Stefan-Boltzmann law to include massive photons. A crucial ingredient to obtain the correct formula for the radiance is to realize that a massive photon does not travel at the speed of (massless) light. It follows that, contrary to what could be expected, the radiance is not proportional to the energy density times the speed of light.
Boltzmann und das Ende des mechanistischen Weltbildes
Renn, Jürgen
2007-01-01
Der Wissenschaftshistoriker und Physiker Jürgen Renn untersucht die Rolle des österreichischen Physikers und Philosophen Ludwig Boltzmann (18441906) bei der Entwicklung der modernen Physik. Boltzmann war einer der letzen Vertreter des mechanistischen Weltbildes und stand somit am Ende eines Zeitalters. Renn porträtiert den Wissenschaftler aber als einen Pionier der modernen Physik, dessen Beschäftigung mit den inneren Spannungen der klassischen Physik ihn visionär zukünftige Fragestellungen aufgreifen ließ. So befasste sich Boltzmann etwa mit den Grenzproblemen zwischen Mechanik und Thermodynamik, die ihn zur Entwicklung immer raffinierterer Instrumente der statistischen Physik antrieb, die schließlich zu Schlüsselinstrumenten der modernen Physik wurden. Boltzmanns Werk steht somit am Übergang vom mechanistischen Weltbild zur Relativitäts- und Quantentheorie. Der Aussage des viel bekannteren Physikers Albert Einstein, dass Fantasie wichtiger sei als Wissen, hält Jürgen Renn im Hinblick auf Leben ...
Geometric variations of the Boltzmann entropy
Kalogeropoulos, Nikos
2008-01-01
We perform a calculation of the first and second order infinitesimal variations, with respect to energy, of the Boltzmann entropy of constant energy hypersurfaces of a system with a finite number of degrees of freedom. We comment on the stability interpretation of the second variation in this framework.
Boltzmann Samplers for Colored Combinatorial Objects
Bodini, Olivier
2009-01-01
In this paper, we give a general framework for the Boltzmann generation of colored objects belonging to combinatorial constructible classes. We propose an intuitive notion called profiled objects which allows the sampling of size-colored objects (and also of k-colored objects) although the corresponding class cannot be described by an analytic ordinary generating function.
Dynamically adaptive Lattice Boltzmann simulation of shallow water flows with the Peano framework
Neumann, Philipp
2015-09-01
© 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.
Tjong, Harianto; Zhou, Huang-Xiang
2006-11-28
The Poisson-Boltzmann equation gives the electrostatic free energy of a solute molecule (with dielectric constant epsilon(l)) solvated in a continuum solvent (with dielectric constant epsilon(s)). Here a simple formula is presented that accurately predicts the electrostatic free energy for all combinations of epsilon(l) and epsilon(s) from the calculation on a single set of epsilon(l) and epsilon(s) values.
Boltzmann-Gibbs Distribution of Fortune and Broken Time-Reversible Symmetry in Econodynamics
Ao, P
2005-01-01
Within the description of stochastic differential equations it is argued that the existence of Boltzmann-Gibbs type distribution in economy is independent of the time reversal symmetry in econodynamics. Both power law and exponential distributions can be accommodated by it. The demonstration is based on a mathematical structure discovered during a study in gene regulatory network dynamics. Further possible analogy between equilibrium economy and thermodynamics is explored.
Lattice Boltzmann formulation for conjugate heat transfer in heterogeneous media.
Karani, Hamid; Huber, Christian
2015-02-01
In this paper, we propose an approach for studying conjugate heat transfer using the lattice Boltzmann method (LBM). The approach is based on reformulating the lattice Boltzmann equation for solving the conservative form of the energy equation. This leads to the appearance of a source term, which introduces the jump conditions at the interface between two phases or components with different thermal properties. The proposed source term formulation conserves conductive and advective heat flux simultaneously, which makes it suitable for modeling conjugate heat transfer in general multiphase or multicomponent systems. The simple implementation of the source term approach avoids any correction of distribution functions neighboring the interface and provides an algorithm that is independent from the topology of the interface. Moreover, our approach is independent of the choice of lattice discretization and can be easily applied to different advection-diffusion LBM solvers. The model is tested against several benchmark problems including steady-state convection-diffusion within two fluid layers with parallel and normal interfaces with respect to the flow direction, unsteady conduction in a three-layer stratified domain, and steady conduction in a two-layer annulus. The LBM results are in excellent agreement with analytical solution. Error analysis shows that our model is first-order accurate in space, but an extension to a second-order scheme is straightforward. We apply our LBM model to heat transfer in a two-component heterogeneous medium with a random microstructure. This example highlights that the method we propose is independent of the topology of interfaces between the different phases and, as such, is ideally suited for complex natural heterogeneous media. We further validate the present LBM formulation with a study of natural convection in a porous enclosure. The results confirm the reliability of the model in simulating complex coupled fluid and thermal dynamics
Polarizable Atomic Multipole Solutes in a Poisson-Boltzmann Continuum
Schnieders, Michael J.; Baker, Nathan A.; Ren, Pengyu; Ponder, Jay W.
2008-01-01
Modeling the change in the electrostatics of organic molecules upon moving from vacuum into solvent, due to polarization, has long been an interesting problem. In vacuum, experimental values for the dipole moments and polarizabilities of small, rigid molecules are known to high accuracy; however, it has generally been difficult to determine these quantities for a polar molecule in water. A theoretical approach introduced by Onsager used vacuum properties of small molecules, including polarizability, dipole moment and size, to predict experimentally known permittivities of neat liquids via the Poisson equation. Since this important advance in understanding the condensed phase, a large number of computational methods have been developed to study solutes embedded in a continuum via numerical solutions to the Poisson-Boltzmann equation (PBE). Only recently have the classical force fields used for studying biomolecules begun to include explicit polarization in their functional forms. Here we describe the theory underlying a newly developed Polarizable Multipole Poisson-Boltzmann (PMPB) continuum electrostatics model, which builds on the Atomic Multipole Optimized Energetics for Biomolecular Applications (AMOEBA) force field. As an application of the PMPB methodology, results are presented for several small folded proteins studied by molecular dynamics in explicit water as well as embedded in the PMPB continuum. The dipole moment of each protein increased on average by a factor of 1.27 in explicit water and 1.26 in continuum solvent. The essentially identical electrostatic response in both models suggests that PMPB electrostatics offers an efficient alternative to sampling explicit solvent molecules for a variety of interesting applications, including binding energies, conformational analysis, and pKa prediction. Introduction of 150 mM salt lowered the electrostatic solvation energy between 2–13 kcal/mole, depending on the formal charge of the protein, but had only a
Simulation of Magnetorheological Fluids Based on Lattice Boltzmann Method with Double Meshes
Directory of Open Access Journals (Sweden)
Xinhua Liu
2012-01-01
Full Text Available In order to study the rheological characteristics of magnetorheological fluids, a novel approach based on the two-component Lattice Boltzmann method with double meshes was proposed, and the micro-scale structures of magnetorheological fluids in different strength magnetic fields were simulated. The framework composed of three steps for the simulation of magnetorheological fluids was addressed, and the double meshes method was elaborated. Moreover, the various internal and external forces acting on the magnetic particles were analyzed and calculated. The two-component Lattice Boltzmann model was set up, and the flowchart for the simulation of magnetorheological fluids based on the two-component Lattice Boltzmann method with double meshes was designed. Finally, a physics experiment was carried out, and the simulation examples were provided. The comparison results indicated that the proposed approach was feasible, efficient, and outperforming others.
A generalized Poisson and Poisson-Boltzmann solver for electrostatic environments
Energy Technology Data Exchange (ETDEWEB)
Fisicaro, G., E-mail: giuseppe.fisicaro@unibas.ch; 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)
2016-01-07
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.
Directory of Open Access Journals (Sweden)
Savita Agrawal
2015-11-01
Full Text Available In the last decades, image segmentation has proved its applicability in various areas like satellite image processing, medical image processing and many more. In the present scenario the researchers tries to develop hybrid image segmentation techniques to generates efficient segmentation. Due to the development of the parallel programming, the lattice Boltzmann method (LBM has attracted much attention as a fast alternative approach for solving partial differential equations. In this project work, first designed an energy functional based on the fuzzy c-means objective function which incorporates the bias field that accounts for the intensity in homogeneity of the real-world image. Using the gradient descent method, corresponding level set equations are obtained from which we deduce a fuzzy external force for the LBM solver based on the model by Zhao. The method is speedy, robust for denoise, and does not dependent on the position of the initial contour, effective in the presence of intensity in homogeneity, highly parallelizable and can detect objects with or without edges. For the implementation of segmentation techniques defined for gray images, most of the time researchers determines single channel segments of the images and superimposes the single channel segment information on color images. This idea leads to provide color image segmentation using single channel segments of multi channel images. Though this method is widely adopted but doesn’t provide complete true segmentation of multichannel ie color images because a color image contains three different channels for Red, green and blue components. Hence segmenting a color image, by having only single channel segments information, will definitely loose important segment regions of color images. To overcome this problem this paper work starts with the development of Enhanced Level Set Segmentation for single channel Images Using Fuzzy Clustering and Lattice Boltzmann Method. For the
Directory of Open Access Journals (Sweden)
Savita Agrawal
2014-05-01
Full Text Available In the last decades, image segmentation has proved its applicability in various areas like satellite image processing, medical image processing and many more. In the present scenario the researchers tries to develop hybrid image segmentation techniques to generates efficient segmentation. Due to the development of the parallel programming, the lattice Boltzmann met hod (LBM has attracted much attention as a fast alternative approach for solving partial differential equations. In this project work, first designed an energy functional based on the fuzzy c-means objective function which incorporates the bias field that accounts for the intensity in homogeneity of the real-world image. Using the gradient descent method, corresponding level set equations are obtained from which we deduce a fuzzy external force for the LBM solver based on the model by Zhao. The method is speedy, robust for denoise, and does not dependent on the position of the initial contour, effective in the presence of intensity in homogeneity, highly parallelizable and can detect objects with or without edges. For the implementation of segmentation techniques defined for gr ay images, most of the time researchers determines single channel segments of the images and superimposes the single channel segment information on color images. This idea leads to provide color image segmentation using single channel segments of multi chann el images. Though this method is widely adopted but doesn’t provide complete true segmentation of multichannel ie color images because a color image contains three different channels for Red, green and blue components. Hence segmenting a color image, b y having only single channel segments information, will definitely loose important segment regions of color images. To overcome this problem this paper work starts with the development of Enhanced Level Set Segmentation for single channel Images Using Fuzzy Clustering and Lattice Boltzmann Method. For the
Macroscopic model and truncation error of discrete Boltzmann method
Hwang, Yao-Hsin
2016-10-01
A derivation procedure to secure the macroscopically equivalent equation and its truncation error for discrete Boltzmann method is proffered in this paper. Essential presumptions of two time scales and a small parameter in the Chapman-Enskog expansion are disposed of in the present formulation. Equilibrium particle distribution function instead of its original non-equilibrium form is chosen as key variable in the derivation route. Taylor series expansion encompassing fundamental algebraic manipulations is adequate to realize the macroscopically differential counterpart. A self-contained and comprehensive practice for the linear one-dimensional convection-diffusion equation is illustrated in details. Numerical validations on the incurred truncation error in one- and two-dimensional cases with various distribution functions are conducted to verify present formulation. As shown in the computational results, excellent agreement between numerical result and theoretical prediction are found in the test problems. Straightforward extensions to more complicated systems including convection-diffusion-reaction, multi-relaxation times in collision operator as well as multi-dimensional Navier-Stokes equations are also exposed in the Appendix to point out its expediency in solving complicated flow problems.
Energy-Dependent Octagonal Lattice Boltzmann Modeling for Compressible Flows
Pavlo, Pavol; Vahala, Linda; Vahala, George
2000-10-01
There has been much interest in thermal lattice Boltzmann modeling (TLBM) for compressible flows because of their inherent parallelizeability. Instead of applying CFD techniques to the nonlinear conservation equations, one instead solves a linear BGK kinetic equation. To reduce storage requirements, the velocity space is discretized and lattice geometries are so chosen to minimize the number of degrees of freedom that must be retained in the Chapman-Enskog recovery of the original macroscopic equations. The simplest (and most efficient) TLBM runs at a CFL=1, so that no numerical diffusion or dissipation is introduced. The algorithm involves Lagrangian streaming (shift operator) and purely local operations. Because of the underlying discrete lattice symmetry, the relaxation distributions cannot be Maxwellian and hence the inherent numerical instability problem in TLBM. We are investigating the use of energy-dependent lattices so as to allow simulation of problems of interest in divertor physics, The appeal of TLBM is that it can provide a unified representation for both strongly collisional (‘fluid’) and weakly collisional (‘Monte Carlo’) regimes. Moreover, our TLBM code is more efficiently solved on mulit-PE platforms than the corresponding CFD codes and is readily extended to 3D. MHD can also be handled by TLBM.
Lattice Boltzmann Simulation of Permeability and Tortuosity for Flow through Dense Porous Media
Directory of Open Access Journals (Sweden)
Ping Wang
2014-01-01
Full Text Available Discrete element method (DEM is used to produce dense and fixed porous media with rigid mono spheres. Lattice Boltzmann method (LBM is adopted to simulate the fluid flow in interval of dense spheres. To simulating the same physical problem, the permeability is obtained with different lattice number. We verify that the permeability is irrelevant to the body force and the media length along flow direction. The relationships between permeability, tortuosity and porosity, and sphere radius are researched, and the results are compared with those reported by other authors. The obtained results indicate that LBM is suited to fluid flow simulation of porous media due to its inherent theoretical advantages. The radius of sphere should have ten lattices at least and the media length along flow direction should be more than twenty radii. The force has no effect on the coefficient of permeability with the limitation of slow fluid flow. For mono spheres porous media sample, the relationship of permeability and porosity agrees well with the K-C equation, and the tortuosity decreases linearly with increasing porosity.
Directory of Open Access Journals (Sweden)
Lloyd K. Williams
1987-01-01
Full Text Available In this paper we find closed form solutions of some Riccati equations. Attention is restricted to the scalar as opposed to the matrix case. However, the ones considered have important applications to mathematics and the sciences, mostly in the form of the linear second-order ordinary differential equations which are solved herewith.
Avoiding Boltzmann Brain domination in holographic dark energy models
Directory of Open Access Journals (Sweden)
R. Horvat
2015-11-01
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.
Nonequilibrium thermodynamics of restricted Boltzmann machines
Salazar, Domingos S. P.
2017-08-01
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.
Lattice-Boltzmann simulations of droplet evaporation
Ledesma-Aguilar, Rodrigo
2014-09-04
© 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
Prandtl number effects in MRT lattice Boltzmann models for shocked and unshocked compressible fluids
Institute of Scientific and Technical Information of China (English)
无
2011-01-01
This paper constructs a new multiple relaxation time lattice Boltzmann model which is not only for the shocked compressible fluids,but also for the unshocked compressible fluids.To make the model work for unshocked compressible fluids,a key step is to modify the collision operators of energy flux so that the viscous coefficient in momentum equation is consistent with that in energy equation even in the unshocked system.The unnecessity of the modification for systems under strong shock is analyzed.The model ...
Hybrid lattice Boltzmann method on overlapping grids.
Di Ilio, G; Chiappini, D; Ubertini, S; Bella, G; Succi, S
2017-01-01
In this work, a hybrid lattice Boltzmann method (HLBM) is proposed, where the standard lattice Boltzmann implementation based on the Bhatnagar-Gross-Krook (LBGK) approximation is combined together with an unstructured finite-volume lattice Boltzmann model. The method is constructed on an overlapping grid system, which allows the coexistence of a uniform lattice nodes spacing and a coordinate-free lattice structure. The natural adaptivity of the hybrid grid system makes the method particularly suitable to handle problems involving complex geometries. Moreover, the provided scheme ensures a high-accuracy solution near walls, given the capability of the unstructured submodel of achieving the desired level of refinement in a very flexible way. For these reasons, the HLBM represents a prospective tool for solving multiscale problems. The proposed method is here applied to the benchmark problem of a two-dimensional flow past a circular cylinder for a wide range of Reynolds numbers and its numerical performances are measured and compared with the standard LBGK ones.
Dimensional Equations of Entropy
Sparavigna, Amelia Carolina
2015-01-01
Entropy is a quantity which is of great importance in physics and chemistry. The concept comes out of thermodynamics, proposed by Rudolf Clausius in his analysis of Carnot cycle and linked by Ludwig Boltzmann to the number of specific ways in which a physical system may be arranged. Any physics classroom, in its task of learning physics, has therefore to face this crucial concept. As we will show in this paper, the lectures can be enriched by discussing dimensional equations linked to the entropy of some physical systems.