The relativistic linear Boltzmann transport equation
International Nuclear Information System (INIS)
In this thesis the relativistic linear Boltzmann transport equation is applied to an experiment in pion production by 740 MeV protons incident on a variety of nuclei. This equation is solved by the Monte Carlo method of generating a single particle intranuclear cascade. The transport equation is derived starting with the N-body equation of motion for quantum mechanics in phase in order to determine under what conditions it is a valid approximation. It is shown that it should be a valid semi-classical approximation provided that: (1) The kinetic energy of the transport particle is much greater than its energy of interaction with the mean nuclear potential field. (2) The two-body collision interactions which make up the single particle intranuclear cascade take place over space and time intervals which are small relative to the internucleon space and time intervals for interactions within the nucleus and also compared to the space and time scales over which the probability distribution undergoes variation. In the pion production calculation condition (2) is only approximately met but reasonable agreement with the experimental data is obtained similar to that obtained in other theoretical calculations compared to this experiment
The Boltzmann equation theory of charged particle transport
International Nuclear Information System (INIS)
It is shown how a formally exact Kubo-like response theory equivalent to the Boltzmann equation theory of charged particle transport can be constructed. The response theory gives the general wavevector and time-dependent velocity distribution at any time in terms of an initial distribution function, to which is added the response induced by a generalized perturbation over the intervening time. The usual Kubo linear response result for the distribution function is recovered by choosing the initial velocity distribution to be Maxwellian. For completeness the response theory introduces an exponential convergence function into the response time integral. This is equivalent to using a modified Boltzmann equation but the general form of the transport theory is not changed. The modified transport theory can be used to advantage where possible convergence difficulties occur in numerical solutions of the Boltzmann equation. This paper gives a systematic development of the modified transport theory and shows how the response theory fits into the broader scheme of solving the Boltzmann equation. The discussion extends both the work of Kumar et al. (1980), where the distribution function is expanded out in terms of tensor functions, and the propagator description where the non-hydrodynamic time development of the distribution function is related to the wavevector dependent Green function of the Boltzmann equation
Heat conduction in multifunctional nanotrusses studied using Boltzmann transport equation
International Nuclear Information System (INIS)
Materials that possess low density, low thermal conductivity, and high stiffness are desirable for engineering applications, but most materials cannot realize these properties simultaneously due to the coupling between them. Nanotrusses, which consist of hollow nanoscale beams architected into a periodic truss structure, can potentially break these couplings due to their lattice architecture and nanoscale features. In this work, we study heat conduction in the exact nanotruss geometry by solving the frequency-dependent Boltzmann transport equation using a variance-reduced Monte Carlo algorithm. We show that their thermal conductivity can be described with only two parameters, solid fraction and wall thickness. Our simulations predict that nanotrusses can realize unique combinations of mechanical and thermal properties that are challenging to achieve in typical materials
International Nuclear Information System (INIS)
We consider the splitting of the straight-ahead Boltzmann transport equation in the Boltzmann-Fokker-Planck equation, decomposing the differential cross-section into a singular part, corresponding to small energy transfer events, and in a regular one, which corresponds to large energy transfer. The convergence of implantation profile, nuclear and electronic energy depositions, calculated from the Boltzmann-Fokker-Planck equation, to the respective exact distributions, calculated from Monte-Carlo method, was exanimate in a large-energy interval for various values of splitting parameter and for different ion-target mass relations. It is shown that for the universal potential there exists an optimal value of splitting parameter, for which range and deposited energy distributions, calculated from the Boltzmann-Fokker-Planck equation, accurately approximate the exact distributions and which minimizes the computational expenses
Boltzmann-Fourier transformed transport equation in half space
International Nuclear Information System (INIS)
Using eigenfunctions of the stationary transport equation and analytical expressions of semigroups generated by linear collision transport operator, an analytical solution of the transport equation in a compact form is being derived for semi-infinite medium. (author)
MULTI-FLUX FORMULATION OF THE BOLTZMANN EQUATION FOR CARRIER TRANSPORT IN SEMICONDUCTORS
Banoo, Kausar; Lundstrom, Mark
1998-01-01
This report describes how the Boltzmann Transport Equation for carrier transport in s~~miconductocrasn be formulated in a manner suit able for numerical simulation. It arose from an effort to generalise earlier work which used pre-computed scattering matrices to solve the Boltzmann Transport Equation. It also generalises the formulation used to treat neutron transport so that energy band-structure, scattering in semiconductors and electric fields can be treated. We present two different, but ...
Faghaninia, Alireza; Ager III, Joel W.; Lo, Cynthia S.
2015-01-01
Accurate models of carrier transport are essential for describing the electronic properties of semiconductor materials. To the best of our knowledge, the current models following the framework of the Boltzmann transport equation (BTE) either rely heavily on experimental data (i.e., semi-empirical), or utilize simplifying assumptions, such as the constant relaxation time approximation (BTE-cRTA). While these models offer valuable physical insights and accurate calculations of transport propert...
A generalized linear Boltzmann equation for non-classical particle transport
International Nuclear Information System (INIS)
This paper presents a derivation and initial study of a new generalized linear Boltzmann equation (GLBE), which describes particle transport for random statistically homogeneous systems in which the distribution function for chord lengths between scattering centers is non-exponential. Such problems have recently been proposed for the description of photon transport in atmospheric clouds; this paper is a first attempt to develop a Boltzmann-like equation for these and other related applications.
Punshon-Smith, Samuel; Smith, Scott
2016-01-01
This article studies the Cauchy problem for the Boltzmann equation with stochastic kinetic transport. Under a cut-off assumption on the collision kernel and a coloring hypothesis for the noise coefficients, we prove the global existence of renormalized (in the sense of DiPerna/Lions) martingale solutions to the Boltzmann equation for large initial data with finite mass, energy, and entropy. Our analysis includes a detailed study of weak martingale solutions to a class of linear stochastic kin...
Transition flow ion transport via integral Boltzmann equation
International Nuclear Information System (INIS)
A new approach is developed to solve the Integral Boltzmann Equation for the evolving velocity distribution of a source of ions, undergoing electrostatic acceleration through a neutral gas target. The theory is applicable to arbitrarily strong electric fields, any ion/neutral mass ratio greater than unity, and is not limited to spatially isotropic gas targets. A hard sphere collision model is used, with a provision for inelasticity. Both axial and radial velocity distributions are calculated for applications where precollision radial velocities are negligible, as is the case for ion beam extractions from high pressure sources. Theoretical predictions are tested through an experiment in which an atmospheric pressure ion source is coupled to a high vacuum energy analyser. Excellent agreement results for configurations in which the radial velocity remains small. Velocity distributions are applied to predicting the efficiency of coupling an atmospheric pressure ion source to a quadrupole mass spectrometer and results clearly indicate the most desirable extracting configuration. A method is devised to calculate ion-molecule hard sphere collision cross sections for easily fragmented organic ions
International Nuclear Information System (INIS)
Questions regarding accuracy and efficiency of deterministic transport methods are still on our mind today, even with modern supercomputers. The most versatile and widely used deterministic methods are the PN approximation, the SN method (discrete ordinates method) and their variants. In the discrete ordinates (SN) formulations of the transport equation, it is assumed that the linearized Boltzmann equation only holds for a set of distinct numerical values of the direction-of-motion variables. In this work, looking forward to confirm the capabilities of deterministic methods in obtaining accurate results, we present a general overview of deterministic methods to solve the Boltzmann transport equation for neutral and charged particles. First, we describe a review in the Laplace transform technique applied to SN two dimensional transport equation in a rectangular domain considering Compton scattering. Next, we solved the Fokker-Planck (FP) equation, an alternative approach for the Boltzmann transport equation, assuming a monoenergetic electron beam in a rectangular domain. The main idea relies on applying the PN approximation, a recent advance in the class of deterministic methods, in the angular variable, to the two dimensional Fokker-Planck equation and then applying the Laplace Transform in the spatial x-variable. Numerical results are given to illustrate the accuracy of deterministic methods presented. (author)
International Nuclear Information System (INIS)
A variational procedure is applied to a linearized Boltzmann equation to calculate electric conductivity, thermal conductivity and Seebeck coefficient. Interaction of electrons with vacancies and impurities as well as with magnetic ions and phonons are taken into consideration. As an example these three transport coefficients are evaluated for GdAl2 in the temperature range 0-300 0K. (G.Q.)
Radiative or neutron transport modeling using a lattice Boltzmann equation framework
Bindra, H.; Patil, D. V.
2012-07-01
In this paper, the lattice Boltzmann equation (LBE)-based framework is used to obtain the solution for the linear radiative or neutron transport equation. The LBE framework is devised for the integrodifferential forms of these equations which arise due to the inclusion of the scattering terms. The interparticle collisions are neglected, hence omitting the nonlinear collision term. Furthermore, typical representative examples for one-dimensional or two-dimensional geometries and inclusion or exclusion of the scattering term (isotropic and anisotropic) in the Boltzmann transport equation are illustrated to prove the validity of the method. It has been shown that the solution from the LBE methodology is equivalent to the well-known Pn and Sn methods. This suggests that the LBE can potentially provide a more convenient and easy approach to solve the physical problems of neutron and radiation transport.
Faghaninia, Alireza; Ager, Joel W.; Lo, Cynthia S.
2015-06-01
Accurate models of carrier transport are essential for describing the electronic properties of semiconductor materials. To the best of our knowledge, the current models following the framework of the Boltzmann transport equation (BTE) either rely heavily on experimental data (i.e., semiempirical), or utilize simplifying assumptions, such as the constant relaxation time approximation (BTE-cRTA). While these models offer valuable physical insights and accurate calculations of transport properties in some cases, they often lack sufficient accuracy—particularly in capturing the correct trends with temperature and carrier concentration. We present here a transport model for calculating low-field electrical drift mobility and Seebeck coefficient of n -type semiconductors, by explicitly considering relevant physical phenomena (i.e., elastic and inelastic scattering mechanisms). We first rewrite expressions for the rates of elastic scattering mechanisms, in terms of ab initio properties, such as the band structure, density of states, and polar optical phonon frequency. We then solve the linear BTE to obtain the perturbation to the electron distribution—resulting from the dominant scattering mechanisms—and use this to calculate the overall mobility and Seebeck coefficient. Therefore, we have developed an ab initio model for calculating mobility and Seebeck coefficient using the Boltzmann transport (aMoBT) equation. Using aMoBT, we accurately calculate electrical transport properties of the compound n -type semiconductors, GaAs and InN, over various ranges of temperature and carrier concentration. aMoBT is fully predictive and provides high accuracy when compared to experimental measurements on both GaAs and InN, and vastly outperforms both semiempirical models and the BTE-cRTA. Therefore, we assert that this approach represents a first step towards a fully ab initio carrier transport model that is valid in all compound semiconductors.
Shizgal, Bernie D.
2011-05-01
The study of the solution of the linearized Boltzmann equation has a very long history arising from the classic work by Chapman and Cowling. For small departures from a Maxwellian, the nonlinear Boltzmann equation can be linearized and the transport coefficients calculated with the Chapman-Enskog approach. This procedure leads to a set of linear integral equations which are generally solved with the expansion of the departure from Maxwellian in Sonine polynomials. The method has been used successfully for many decades to compare experimental transport data in atomic gases with theory generally carried out for realistic atom-atom differential cross sections. There are alternate pseudospectral methods which involve the discretization of the distribution function on a discrete grid. This paper considers a pseudospectral method of solution of the linearized hard sphere Boltzmann equation for the viscosity in a simple gas. The relaxation of a small departure from a Maxwellian is also considered for the linear test particle problem with unit mass ratio which is compared with the relaxation for the linearized one component Boltzmann equation.
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.
International Nuclear Information System (INIS)
We propose to check and validate the approximations made in dissipative quantum transport (QT) simulations solved in the Non-equilibrium Green's Function formalism by comparing them with the exact solution of the linearized Boltzmann Transport Equation (LB) in the stationary regime. For that purpose, we calculate the phonon-limited electron and hole mobility in bulk Si and ultra-scaled Si nanowires for different crystal orientations 〈100〉, 〈110〉, and 〈111〉. In both QT and LB simulations, we use the same sp3d5s* tight-binding model to describe the electron/hole properties and the same valence-force-field approach to account for the phonon properties. It is found that the QT simplifications work well for electrons, but are less accurate for holes, where a renormalization of the phonon scattering strength is proved useful to improve the results
Rhyner, Reto; Luisier, Mathieu
2013-12-01
We propose to check and validate the approximations made in dissipative quantum transport (QT) simulations solved in the Non-equilibrium Green's Function formalism by comparing them with the exact solution of the linearized Boltzmann Transport Equation (LB) in the stationary regime. For that purpose, we calculate the phonon-limited electron and hole mobility in bulk Si and ultra-scaled Si nanowires for different crystal orientations ⟨100⟩, ⟨110⟩, and ⟨111⟩. In both QT and LB simulations, we use the same sp3d5s* tight-binding model to describe the electron/hole properties and the same valence-force-field approach to account for the phonon properties. It is found that the QT simplifications work well for electrons, but are less accurate for holes, where a renormalization of the phonon scattering strength is proved useful to improve the results.
International Nuclear Information System (INIS)
The Aron equation is a generalization of the Fokker-Planck equation allowing for diffusion motion with finite maximal velocity. The Aron equation can be regarded as a semi-phenomenological equation because it is based on phenomenological laws such as the Fick diffusion law. It is shown that the one-dimensional case of the Aron equation can be derived from the Boltzmann transport equation for particles in Zitterbewegung. The extension to the three-dimensional case, however, leads to an equation different from the Aron one
A simple Boltzmann transport equation for ballistic to diffusive transient heat transport
International Nuclear Information System (INIS)
Developing simplified, but accurate, theoretical approaches to treat heat transport on all length and time scales is needed to further enable scientific insight and technology innovation. Using a simplified form of the Boltzmann transport equation (BTE), originally developed for electron transport, we demonstrate how ballistic phonon effects and finite-velocity propagation are easily and naturally captured. We show how this approach compares well to the phonon BTE, and readily handles a full phonon dispersion and energy-dependent mean-free-path. This study of transient heat transport shows (i) how fundamental temperature jumps at the contacts depend simply on the ballistic thermal resistance, (ii) that phonon transport at early times approach the ballistic limit in samples of any length, and (iii) perceived reductions in heat conduction, when ballistic effects are present, originate from reductions in temperature gradient. Importantly, this framework can be recast exactly as the Cattaneo and hyperbolic heat equations, and we discuss how the key to capturing ballistic heat effects is to use the correct physical boundary conditions
International Nuclear Information System (INIS)
This paper describes a new second generation spherical wavelet method for discretising the angular dimension of the Boltzmann transport equation. The approximation scheme provides a spectrally accurate expansion of the angular domain using Chebyshev collocation polynomials mapped into a wavelet space. Our method extends the work in Buchan et al. [Buchan, A., Pain, C.C., Eaton, M.D., Smedley-Stevenson, R., Goddard, A., Oliveira, C.D., submitted for publication. Linear and quadratic hexahedral wavelets on the sphere for angular discretisations of the Boltzmann transport equation. Nucl. Sci. Eng.; Buchan, A., Pain, C.C., Eaton, M.D., Smedley-Stevenson, R., Goddard, A., Oliveira, C.D., 2005. Linear and quadratic octahedral wavelets on the sphere for angular discretisations of the Boltzmann transport equation. Ann. Nucl. Energy 32, 1224-1273] of using low order finite element based wavelets. Here we show the spectral wavelets can improve on these techniques by providing more accurate representation of the angular fluxes. This also implies the method can provide improved solutions to those of the established methods SN and PN by reducing ray-effects and possibly Gibbs oscillations. We demonstrate this using a set of demanding mono-energetic particle transport problems
International Nuclear Information System (INIS)
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. (paper)
Bouchard, Hugo; Bielajew, Alex
2015-07-01
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.
International Nuclear Information System (INIS)
Calculations and comparisons with experimental data indicate that the Boltzmann transport equation provides a comprehensive treatment of the general ion implantation problem. The primary ion distribution in a multilayer target can be calculated directly and is found to be in good agreement with experiments. The transport equation predicts the spatial distribution of recoils and thus provides the theoretical information needed to determine the fractional atomic displacement necessary for amorphization of silicon and the degree of stoichiometric imbalance that is produced when energetic ions are incident on a target having more than one type of host atom
Energy Technology Data Exchange (ETDEWEB)
Rhyner, Reto, E-mail: rhyner@iis.ee.ethz.ch; Luisier, Mathieu, E-mail: mluisier@iis.ee.ethz.ch [Integrated Systems Laboratory, ETH Zürich, Gloriastr. 35, 8092 Zürich (Switzerland)
2013-12-14
We propose to check and validate the approximations made in dissipative quantum transport (QT) simulations solved in the Non-equilibrium Green's Function formalism by comparing them with the exact solution of the linearized Boltzmann Transport Equation (LB) in the stationary regime. For that purpose, we calculate the phonon-limited electron and hole mobility in bulk Si and ultra-scaled Si nanowires for different crystal orientations 〈100〉, 〈110〉, and 〈111〉. In both QT and LB simulations, we use the same sp{sup 3}d{sup 5}s{sup *} tight-binding model to describe the electron/hole properties and the same valence-force-field approach to account for the phonon properties. It is found that the QT simplifications work well for electrons, but are less accurate for holes, where a renormalization of the phonon scattering strength is proved useful to improve the results.
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...
International Nuclear Information System (INIS)
This paper presents some elements of a new approach to solve analytically the linearized three-dimensional (3-D) transport equation of neutral particles. Since this task is of such special importance, we present some results of a paper that is still in progress. The most important is that using this transformation, an integro-differential equation with an analytical solution is obtained. For this purpose, a simplest 3-D equation is being considered which describes the transport process in an infinite medium. Until now, this equation has been analytically considered either using the Laplace transform with respect to time parameter t or applying the Fourier transform over the space coordinate. Both of them reduce the number of differential terms in the equation; however, evaluation of the inverse transformation is complicated. In this paper, we introduce for the first time a Fourier transform induced by the Boltzmann operator. For this, we use a complete set of 3-D eigenfunctions of the Boltzmann transport operator defined in a similar way as those that have been already used in 3-D transport theory as a basic set to transform the transport equation. This set consists of a continuous part and a discrete one with spectral measure. The density distribution equation shows the known form asymptotic behavior. Several applications are to be performed using this equation and compared to the benchmark one. Such an analysis certainly would be out of the available space
International Nuclear Information System (INIS)
In this paper, two new wavelet bases are developed for discretising the angular term of the first-order Boltzmann transport equation. The wavelets proposed are based on Sweldens second generation wavelets [Sweldens, W., 1993. The lifting scheme: a construction of second generation wavelets. SIAM J. Math. 1, 54], which are constructed through the lifting procedure [Sweldens, W., 1995. The lifting scheme: a new philosophy in biorthogonal wavelet construction. Wavelet Applications in Signal and Image Processing III]. In this paper, the wavelets are built on an octahedral domain, Fig. 2, and the angular flux approximation takes the form of finite element linear and quadratic representations. Full details of the meshing over the octahedron and derivation of the wavelet functions are given. The wavelets discussed are similar to the wavelets developed in Buchan [Buchan, A., 2003 c. Angular discretisation of the first order Boltzmann transport equation. Part 2: linear spherical wavelets. Technical Report, Imperial College, London, Dep. Earth Sci. Eng.] and [Buchan, A., 2003b. Angular discretisation of the first order Boltzmann transport equation. Part 3: quadratic spherical wavelets. Technical Report, Imperial College, London, Dep. Earth Sci. Eng.], in this paper the bases use a new fundamental amendment for mitigating the inaccuracies observed with the earlier bases. The performance of the new angular discretisation techniques are demonstrated using 2 one-dimensional and 4 two-dimensional test problems. These problems demonstrate the accuracy and susceptibility to ray effects of the proposed methods. Comparisons of all calculations are made with the conventional S N and P N approximations. Benchmark solutions are provided by the established code EVENT
Range profile calculations by direct numerical solution of linearized Boltzmann transport equations
International Nuclear Information System (INIS)
A new method to determine the depth distributions of implanted ions and recoil target atoms in amorphous targets is developed. Our procedure is based on the direct numerical solution of one-dimensional linearized Boltzmann transport equations for the scalar fluxes of the ions and the recoils. We consider characteristic examples of ion implantation into homogeneous and layered targets. The profiles calculated by the new method are compared with range distributions obtained from TRIM Monte Carlo simulations. Our program BOTE is up to two orders of magnitude faster than the TRIM calculations. (author)
Wave operators for the linearized Boltzmann equation in one-speed transport theory
International Nuclear Information System (INIS)
A dissipative integro-differential operator L arising in the linearization of Boltzmann's equation in one-speed particle transport theory is considered. Under assumptions ensuring that the point spectrum of L is finite a scalar multiple of the characteristic functions of L is found and a condition for the absence of spectral singularities is indicated. Using the techniques of non-stationary scattering theory and the Sz.-Nagy-Foias functional model direct and inverse wave operators with the completeness property are constructed. The structure of the operator L in the invariant subspace corresponding to its continuous spectrum is studied
Chiloyan, Vazrik; Zeng, Lingping; Huberman, Samuel; Maznev, Alexei A.; Nelson, Keith A.; Chen, Gang
2016-04-01
The phonon Boltzmann transport equation (BTE) is a powerful tool for studying nondiffusive thermal transport. Here, we develop a new universal variational approach to solving the BTE that enables extraction of phonon mean free path (MFP) distributions from experiments exploring nondiffusive transport. By utilizing the known Fourier heat conduction solution as a trial function, we present a direct approach to calculating the effective thermal conductivity from the BTE. We demonstrate this technique on the transient thermal grating experiment, which is a useful tool for studying nondiffusive thermal transport and probing the MFP distribution of materials. We obtain a closed form expression for a suppression function that is materials dependent, successfully addressing the nonuniversality of the suppression function used in the past, while providing a general approach to studying thermal properties in the nondiffusive regime.
Quadratic inner element subgrid scale discretisation of the Boltzmann transport equation
International Nuclear Information System (INIS)
This paper explores the application of the inner element subgrid scale method to the Boltzmann transport equation using quadratic basis functions. Previously, only linear basis functions for both the coarse scale and the fine scale were considered. This paper, therefore, analyses the advantages of using different coarse and subgrid basis functions for increasing the accuracy of the subgrid scale method. The transport of neutral particle radiation may be described by the Boltzmann transport equation (BTE) which, due to its 7 dimensional phase space, is computationally expensive to resolve. Multi-scale methods offer an approach to efficiently resolve the spatial dimensions of the BTE by separating the solution into its coarse and fine scales and formulating a solution whereby only the computationally efficient coarse scales need to be solved. In previous work an inner element subgrid scale method was developed that applied a linear continuous and discontinuous finite element method to represent the solution’s coarse and fine scale components. This approach was shown to generate efficient and stable solutions, and so this article continues its development by formulating higher order quadratic finite element expansions over the continuous and discontinuous scales. Here it is shown that a solution’s convergence can be improved significantly using higher order basis functions. Furthermore, by using linear finite elements to represent coarse scales in combination with quadratic fine scales, convergence can also be improved with only a modest increase in computational expense.
International Nuclear Information System (INIS)
An improved solution of the Boltzmann transport equation was developed for calculating the results of ion implantation into a multilayer target. A multiple pass scheme is used to integrate the coupled, linearized transport equations describing the momentum distributions of the implanted ion and the recoil particles. The multiple-pass approach correctly treats the case of ions scattered by more than 900, whereas in previous calculations these ions were assumed to be stopped at the scattering point. The accurate treatment of these ions is essential for calculations involving light ions and/or low ion energies, and also an essential prerequisite for two-dimensional calculations. The nuclear cross section used is improved over previous TE calculations by removing the small-angle approximation in the LSS formation of nuclear scattering. Implanted and recoil ion range and damage distributions can be calculated directly for multilayer targets, including stoichiometric disturbances in compounds and recoil yields between target layers
Boltzmann equation and hydrodynamic fluctuations.
Colangeli, Matteo; Kröger, Martin; Ottinger, Hans Christian
2009-11-01
We apply the method of invariant manifolds to derive equations of generalized hydrodynamics from the linearized Boltzmann equation and determine exact transport coefficients, obeying Green-Kubo formulas. Numerical calculations are performed in the special case of Maxwell molecules. We investigate, through the comparison with experimental data and former approaches, the spectrum of density fluctuations and address the regime of finite Knudsen numbers and finite frequencies hydrodynamics. PMID:20364972
Boltzmann equation and hydrodynamic fluctuations
Colangeli, M.; Kroger, M.; Ottinger, H. C.
2009-01-01
We apply the method of invariant manifolds to derive equations of generalized hydrodynamics from the linearized Boltzmann equation and determine exact transport coefficients, obeying Green-Kubo formulas. Numerical calculations are performed in the special case of Maxwell molecules. We investigate, through the comparison with experimental data and former approaches, the spectrum of density fluctuations and address the regime of finite Knudsen numbers and finite frequencies hydrodynamics.
Romano, Giuseppe; Esfarjani, Keivan; Strubbe, David A.; Broido, David; Kolpak, Alexie M.
2016-01-01
Nanostructured materials exhibit low thermal conductivity because of the additional scattering due to phonon-boundary interactions. As these interactions are highly sensitive to the mean free path (MFP) of phonons, MFP distributions in nanostructures can be dramatically distorted relative to bulk. Here we calculate the MFP distribution in periodic nanoporous Si for different temperatures, using the recently developed MFP-dependent Boltzmann transport equation. After analyzing the relative contribution of each phonon branch to thermal transport in nanoporous Si, we find that at room temperature optical phonons contribute 17 % to heat transport, compared to 5 % in bulk Si. Interestingly, we observe a constant thermal conductivity over the range 200 K engineering, in which the bulk material and geometry are optimized concurrently.
Solution of the Boltzmann-Fokker-Planck transport equation using exponential nodal schemes
International Nuclear Information System (INIS)
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 S4 with expansions of the dispersion cross sections until P3 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)
Radtke, Gregg A; Hadjiconstantinou, Nicolas G
2009-05-01
We present an efficient variance-reduced particle simulation technique for solving the linearized Boltzmann transport equation in the relaxation-time approximation used for phonon, electron, and radiative transport, as well as for kinetic gas flows. The variance reduction is achieved by simulating only the deviation from equilibrium. We show that in the limit of small deviation from equilibrium of interest here, the proposed formulation achieves low relative statistical uncertainty that is also independent of the magnitude of the deviation from equilibrium, in stark contrast to standard particle simulation methods. Our results demonstrate that a space-dependent equilibrium distribution improves the variance reduction achieved, especially in the collision-dominated regime where local equilibrium conditions prevail. We also show that by exploiting the physics of relaxation to equilibrium inherent in the relaxation-time approximation, a very simple collision algorithm with a clear physical interpretation can be formulated. PMID:19518597
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
International Nuclear Information System (INIS)
Starting from the phase space formulation of quantum mechanics, a relativistic transport equation describing the propagation of a particle through a medium is developed. In the second part of the paper, a transport equation of this type is solved using the Monte Carlo method (C.P.)
Numerical and analytic solutions of the Boltzmann equation for cosmic ray transport
International Nuclear Information System (INIS)
A method for accurately determining the longitudinal transport of cosmic rays in a disordered, diverging magnetic field is described. Eigenfunctions of the operator in the Boltzmann equation which describes the effects of adiabatic focusing and pitch angle scattering are defined and numerically evaluated. When the particle distribution function is expressed as a series of these focusing eigenfunctions, the Boltzmann equation is transformed into a matrix equation. A computer program was written which uses this matrix representation to calculate the distribution function as a function of distance, time, and pitch angle. In addition, an analytic expression for pseudodiffusion, which replaces classical diffusion when the guiding magnetic field is not rectilinear, is derived. It is shown that the theory of focused transport predicts many of the observed features of solar cosmic ray events. Application of the theory to extragalactic radio sources is also considered. To enable other investigators to apply this method, documented listings of the requisite computer programs are included. In a separate analysis, the effect that energy loss due to synchrotron radiation has on electron propagation in a rectilinear magnetic field is considered. It is shown that a synchrotron radiation region, defined as the region in which the density of electrons of a specific energy is nonzero, has a double-lobed structure even when the electrons are injected into the field isotropically. It is also shown that, because the electrons decay to a given energy almost simultaneously over a large area, a synchrotron radiation region can expand with a superluminal velocity. Application of the theory to radio galaxies and quasars is considered
Ordonez-Miranda, J.; Yang, Ronggui; Alvarado-Gil, J. J.
2011-04-01
A constitutive equation for heat conduction is derived from the exact solution of the Boltzmann transport equation under the relaxation time approximation. This is achieved by a series expansion on multiple space derivatives of the temperature and introducing the concept of thermal multipoles, where the thermal conductivity defined under the framework of the Fourier law of heat conduction is just the first thermal pole. It is shown that this equation generalizes the Fourier law and Cattaneo equation of heat conduction, and it depends strongly on the relative values of the length and time scales compared with the mean-free path and mean-free time of the energy carriers, respectively. In the limiting case of steady-state heat conduction, it is shown that the heat flux vector depends on a spatial scale ratio whose effects are remarkable in the micro-scale spatial domains. By applying a first-order approximation of the obtained thermal multipole expansion to the problem of transient heat conduction across a thin film and comparing the results with the predictions for the same problem using the Fourier, Cattaneo and Boltzmann transport equations, it is shown that our results could be useful in the study of the heat transport in short as well as in long scales of space and time. The common and different features of the multipole expansion compared with the Ballistic-diffusive model of heat conduction are also discussed. Special emphasis is put to the cases where the physical scales of space and time are comparable to the mean-free path and mean-free time of the energy carriers.
International Nuclear Information System (INIS)
The interaction between drifting carriers and traveling electromagnetic waves is considered within the context of the classical Boltzmann transport equation to compute the Casimir-Lifshitz force between media with small density of charge carriers, including dielectrics and intrinsic semiconductors. We expand upon our previous work (Phys. Rev. Lett. 2008, in press; arXiv:0805.1676) and derive in some detail the frequency-dependent reflection amplitudes in this theory and compute the corresponding Casimir free energy for a parallel plate configuration. We critically discuss the the issue of verification of the Nernst theorem of thermodynamics in Casimir physics, and explicitly show that our theory satisfies that theorem. Finally, we show how the theory of drifting carriers connects to previous computations of Casimir forces using spatial dispersion for the material boundaries.
Cumulant solution of the elastic Boltzmann transport equation in an infinite uniform medium
International Nuclear Information System (INIS)
We consider an analytical solution of the time-dependent elastic Boltzmann transport equation in an infinite uniform isotropic medium with an arbitrary phase function. We obtain (1) the exact distribution in angle, (2) the exact first and second spatial cumulants at any angle, and (3) an approximate combined distribution in position and angle and a spatial distribution whose central position and half-width of spread are always exact. The resulting Gaussian distribution has a center that advances in time, and an ellipsoidal contour that grows and changes shape providing a clear picture of the time evolution of the particle migration from near ballistic, through snakelike and into the final diffusive regime. (c) 2000 The American Physical Society
Low-variance Monte Carlo Solutions of the Boltzmann Transport Equation
Hadjiconstantinou, Nicolas G; Baker, Lowell L
2009-01-01
We present and discuss a variance-reduced stochastic particle method for simulating the relaxation-time model of the Boltzmann transport equation. The present paper focuses on the dilute gas case, although the method is expected to directly extend to all fields (carriers) for which the relaxation-time approximation is reasonable. The variance reduction, achieved by simulating only the deviation from equilibrium, results in a significant computational efficiency advantage compared to traditional stochastic particle methods in the limit of small deviation from equilibrium. More specifically, the proposed method can efficiently simulate arbitrarily small deviations from equilibrium at a computational cost that is independent of the deviation from equilibrium, which is in sharp contrast to traditional particle methods.
International Nuclear Information System (INIS)
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 PN (spherical harmonic), SN (discrete ordinate) and LWN (linear octahedral wavelet) angular expansions. These methods are applied to a series of demanding two-dimensional radiation transport problems
Energy Technology Data Exchange (ETDEWEB)
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)
Inelastic Quantum Transport in Superlattices: Success and Failure of the Boltzmann Equation
DEFF Research Database (Denmark)
Wacker, Andreas; Jauho, Antti-Pekka; Rott, Stephan;
1999-01-01
the whole held range from linear response to negative differential conductivity. The quantum results are compared with the respective results obtained from a Monte Carlo solution of the Boltzmann equation. Our analysis thus sets the limits of validity for the semiclassical theory in a nonlinear...
International Nuclear Information System (INIS)
Purpose: To investigate the use of the linear Boltzmann transport equation as a dose calculation tool which can account for interface effects, while still having faster computation times than Monte Carlo methods. In particular, we introduce a forward scattering approximation, in hopes of improving calculation time without a significant hindrance to accuracy. Methods: Two coupled Boltzmann transport equations were constructed, one representing the fluence of photons within the medium, and the other, the fluence of electrons. We neglect the scattering term within the electron transport equation, resulting in an extreme forward scattering approximation to reduce computational complexity. These equations were then solved using a numerical technique for solving partial differential equations, known as a finite difference scheme, where the fluence at each discrete point in space is calculated based on the fluence at the previous point in the particle's path. Using this scheme, it is possible to develop a solution to the Boltzmann transport equations by beginning with boundary conditions and iterating across the entire medium. The fluence of electrons can then be used to find the dose at any point within the medium. Results: Comparisons with Monte Carlo simulations indicate that even simplistic techniques for solving the linear Boltzmann transport equation yield expected interface effects, which many popular dose calculation algorithms are not capable of predicting. Implementation of a forward scattering approximation does not appear to drastically reduce the accuracy of this algorithm. Conclusion: Optimized implementations of this algorithm have been shown to be very accurate when compared with Monte Carlo simulations, even in build up regions where many models fail. Use of a forward scattering approximation could potentially give a reasonably accurate dose distribution in a shorter amount of time for situations where a completely accurate dose distribution is not
High Order Finite Volume Nonlinear Schemes for the Boltzmann Transport Equation
Energy Technology Data Exchange (ETDEWEB)
Bihari, B L; Brown, P N
2005-03-29
The authors apply the nonlinear WENO (Weighted Essentially Nonoscillatory) scheme to the spatial discretization of the Boltzmann Transport Equation modeling linear particle transport. The method is a finite volume scheme which ensures not only conservation, but also provides for a more natural handling of boundary conditions, material properties and source terms, as well as an easier parallel implementation and post processing. It is nonlinear in the sense that the stencil depends on the solution at each time step or iteration level. By biasing the gradient calculation towards the stencil with smaller derivatives, the scheme eliminates the Gibb's phenomenon with oscillations of size O(1) and reduces them to O(h{sup r}), where h is the mesh size and r is the order of accuracy. The current implementation is three-dimensional, generalized for unequally spaced meshes, fully parallelized, and up to fifth order accurate (WENO5) in space. For unsteady problems, the resulting nonlinear spatial discretization yields a set of ODE's in time, which in turn is solved via high order implicit time-stepping with error control. For the steady-state case, they need to solve the non-linear system, typically by Newton-Krylov iterations. There are several numerical examples presented to demonstrate the accuracy, non-oscillatory nature and efficiency of these high order methods, in comparison with other fixed-stencil schemes.
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
Classical non-Markovian Boltzmann equation
Energy Technology Data Exchange (ETDEWEB)
Alexanian, Moorad, E-mail: alexanian@uncw.edu [Department of Physics and Physical Oceanography, University of North Carolina Wilmington, Wilmington, North Carolina 28403-5606 (United States)
2014-08-01
The modeling of particle transport involves anomalous diffusion, (x²(t) ) ∝ t{sup α} with α ≠ 1, with subdiffusive transport corresponding to 0 < α < 1 and superdiffusive transport to α > 1. These anomalies give rise to fractional advection-dispersion equations with memory in space and time. The usual Boltzmann equation, with only isolated binary collisions, is Markovian and, in particular, the contributions of the three-particle distribution function are neglected. We show that the inclusion of higher-order distribution functions give rise to an exact, non-Markovian Boltzmann equation with resulting transport equations for mass, momentum, and kinetic energy with memory in both time and space. The two- and the three-particle distribution functions are considered under the assumption that the two- and the three-particle correlation functions are translationally invariant that allows us to obtain advection-dispersion equations for modeling transport in terms of spatial and temporal fractional derivatives.
A high-order Petrov-Galerkin method for the Boltzmann transport equation
International Nuclear Information System (INIS)
We describe a new Petrov-Galerkin method using high-order terms to introduce dissipation in a residual-free formulation. The method is developed following both a Taylor series analysis and a variational principle, and the result has much in common with traditional Petrov-Galerkin, Self Adjoint Angular Flux (SAAF) and Even Parity forms of the Boltzmann transport equation. In addition, we consider the subtleties in constructing appropriate boundary conditions. In sub-grid scale (SGS) modelling of fluids the advantages of high-order dissipation are well known. Fourth-order terms, for example, are commonly used as a turbulence model with uniform dissipation. They have been shown to have superior properties to SGS models based upon second-order dissipation or viscosity. Even higher-order forms of dissipation (e.g. 16.-order) can offer further advantages, but are only easily realised by spectral methods because of the solution continuity requirements that these higher-order operators demand. Higher-order operators are more effective, bringing a higher degree of representation to the solution locally. Second-order operators, for example, tend to relax the solution to a linear variation locally, whereas a high-order operator will tend to relax the solution to a second-order polynomial locally. The form of the dissipation is also important. For example, the dissipation may only be applied (as it is in this work) in the streamline direction. While for many problems, for example Large Eddy Simulation (LES), simply adding a second or fourth-order dissipation term is a perfectly satisfactory SGS model, it is well known that a consistent residual-free formulation is required for radiation transport problems. This motivated the consideration of a new Petrov-Galerkin method that is residual-free, but also benefits from the advantageous features that SGS modelling introduces. We close with a demonstration of the advantages of this new discretization method over standard Petrov
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.
Energy Technology Data Exchange (ETDEWEB)
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
International Nuclear Information System (INIS)
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/cm3) 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 support
Rukolaine, Sergey A.
2016-05-01
In classical kinetic models a particle free path distribution is exponential, but this is more likely to be an exception than a rule. In this paper we derive a generalized linear Boltzmann equation (GLBE) for a general free path distribution in the framework of Alt's model. In the case that the free path distribution has at least first and second finite moments we construct an asymptotic solution to the initial value problem for the GLBE for small mean free paths. In the special case of the one-speed transport problem the asymptotic solution results in a diffusion approximation to the GLBE.
International Nuclear Information System (INIS)
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+ variational principle for slab geometry. The program has a core K+ 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 102 has been achieved using the new approach in some cases
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.
The Milne problem for the Boltzmann equation
International Nuclear Information System (INIS)
Existence, uniqueness and asymptotic properties are proved for the solution of the Milne problem for the Boltzmann equation, in which the incoming velocity distribution and the total mass flux are specified arbitrarily. The collision law corresponds to a hard sphere gas. The solution uses energy estimates and is similar to that of Bardos, Santos and Sentis for neutron transport. From the Milne problem one can then easily deduce the solution of the Kramers problem
Cekmen, Z. C.; Dincer, M. S.
2009-07-01
The effective ionization coefficients and transport parameters such as electron mean energy drift velocity and transverse diffusion coefficient in binary and ultradilute SF6-Ar gas mixtures have been calculated for density reduced electric field strength E/N values from 10 to 400 Td. These calculations have been performed by using the two-term spherical harmonic expansion to obtain the numerical solution of the Boltzmann transport equation based on the finite element method under steady-state Townsend condition. In order to confirm the model and code developed in this study, the Reid ramp model has been used as a benchmark test and then effective ionization coefficients and transport parameters have been evaluated for SF6 contents of 1%, 10%, 25%, 50%, 70% and 100% in the binary mixture. Finally SF6 contents in the ultradilute mixtures of 0.1%, 0.3%, 0.5% and 0.7% are taken into account with the evaluated effective ionizations and transport parameters of electron mean energy, drift velocity and transverse diffusion coefficients.
International Nuclear Information System (INIS)
This article presents a method for goal-based anisotropic adaptive methods for the finite element method applied to the Boltzmann transport equation. The neutron multiplication factor, keff, is used as the goal of the adaptive procedure. The anisotropic adaptive algorithm requires error measures for keff with directional dependence. General error estimators are derived for any given functional of the flux and applied to keff to acquire the driving force for the adaptive procedure. The error estimators require the solution of an appropriately formed dual equation. Forward and dual error indicators are calculated by weighting the Hessian of each solution with the dual and forward residual respectively. The Hessian is used as an approximation of the interpolation error in the solution which gives rise to the directional dependence. The two indicators are combined to form a single error metric that is used to adapt the finite element mesh. The residual is approximated using a novel technique arising from the sub-grid scale finite element discretisation. Two adaptive routes are demonstrated: (i) a single mesh is used to solve all energy groups, and (ii) a different mesh is used to solve each energy group. The second method aims to capture the benefit from representing the flux from each energy group on a specifically optimised mesh. The keff goal-based adaptive method was applied to three examples which illustrate the superior accuracy in criticality problems that can be obtained
International Nuclear Information System (INIS)
The kinetic theory of charged particles in gases has come a long way in the last 60 years or so, but many of the advances have yet to find their way into contemporary studies of low-temperature plasmas. This review explores the way in which this gap might be bridged, and focuses in particular on the analytic framework and numerical techniques for the solution of Boltzmann's equation for both electrons and ions, as well as on the development of fluid models and semi-empirical formulae. Both hydrodynamic and non-hydrodynamic regimes are considered and transport properties are calculated in various configurations of dc and ac electric and magnetic fields. We discuss in particular the duality in transport coefficients arising from non-conservative collisions (attachment, ionization). (review article)
International Nuclear Information System (INIS)
In this paper a new method for the discretization of the radiation transport equation is presented, based on a discontinuous Galerkin method in space and angle that allows for local refinement in angle where any spatial element can support its own angular discretization. To cope with the discontinuous spatial nature of the solution, a generalized Riemann procedure is required to distinguish between incoming and outgoing contributions of the numerical fluxes. A new consistent framework is introduced that is based on the solution of a generalized eigenvalue problem. The resulting numerical fluxes for the various possible cases where neighboring elements have an equal, higher or lower level of refinement in angle are derived based on tensor algebra and the resulting expressions have a very clear physical interpretation. The choice of discontinuous trial functions not only has the advantage of easing local refinement, it also facilitates the use of efficient sweep-based solvers due to decoupling of unknowns on a large scale thereby approaching the efficiency of discrete ordinates methods with local angular resolution. The approach is illustrated by a series of numerical experiments. Results show high orders of convergence for the scalar flux on angular refinement. The generalized Riemann upwinding procedure leads to stable and consistent solutions. Further the sweep-based solver performs well when used as a preconditioner for a Krylov method
Kópházi, József; Lathouwers, Danny
2015-09-01
In this paper a new method for the discretization of the radiation transport equation is presented, based on a discontinuous Galerkin method in space and angle that allows for local refinement in angle where any spatial element can support its own angular discretization. To cope with the discontinuous spatial nature of the solution, a generalized Riemann procedure is required to distinguish between incoming and outgoing contributions of the numerical fluxes. A new consistent framework is introduced that is based on the solution of a generalized eigenvalue problem. The resulting numerical fluxes for the various possible cases where neighboring elements have an equal, higher or lower level of refinement in angle are derived based on tensor algebra and the resulting expressions have a very clear physical interpretation. The choice of discontinuous trial functions not only has the advantage of easing local refinement, it also facilitates the use of efficient sweep-based solvers due to decoupling of unknowns on a large scale thereby approaching the efficiency of discrete ordinates methods with local angular resolution. The approach is illustrated by a series of numerical experiments. Results show high orders of convergence for the scalar flux on angular refinement. The generalized Riemann upwinding procedure leads to stable and consistent solutions. Further the sweep-based solver performs well when used as a preconditioner for a Krylov method.
Energy Technology Data Exchange (ETDEWEB)
Kópházi, József, E-mail: j.kophazi@imperial.ac.uk; Lathouwers, Danny, E-mail: d.lathouwers@tudelft.nl
2015-09-15
In this paper a new method for the discretization of the radiation transport equation is presented, based on a discontinuous Galerkin method in space and angle that allows for local refinement in angle where any spatial element can support its own angular discretization. To cope with the discontinuous spatial nature of the solution, a generalized Riemann procedure is required to distinguish between incoming and outgoing contributions of the numerical fluxes. A new consistent framework is introduced that is based on the solution of a generalized eigenvalue problem. The resulting numerical fluxes for the various possible cases where neighboring elements have an equal, higher or lower level of refinement in angle are derived based on tensor algebra and the resulting expressions have a very clear physical interpretation. The choice of discontinuous trial functions not only has the advantage of easing local refinement, it also facilitates the use of efficient sweep-based solvers due to decoupling of unknowns on a large scale thereby approaching the efficiency of discrete ordinates methods with local angular resolution. The approach is illustrated by a series of numerical experiments. Results show high orders of convergence for the scalar flux on angular refinement. The generalized Riemann upwinding procedure leads to stable and consistent solutions. Further the sweep-based solver performs well when used as a preconditioner for a Krylov method.
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.
International Nuclear Information System (INIS)
Purpose: Accurate radiotherapy dose calculation algorithms are essential to any successful radiotherapy program, considering the high level of dose conformity and modulation in many of today’s treatment plans. As technology continues to progress, such as is the case with novel MRI-guided radiotherapy systems, the necessity for dose calculation algorithms to accurately predict delivered dose in increasingly challenging scenarios is vital. To this end, a novel deterministic solution has been developed to the first order linear Boltzmann transport equation which accurately calculates x-ray based radiotherapy doses in the presence of magnetic fields. Methods: The deterministic formalism discussed here with the inclusion of magnetic fields is outlined mathematically using a discrete ordinates angular discretization in an attempt to leverage existing deterministic codes. It is compared against the EGSnrc Monte Carlo code, utilizing the emf-macros addition which calculates the effects of electromagnetic fields. This comparison is performed in an inhomogeneous phantom that was designed to present a challenging calculation for deterministic calculations in 0, 0.6, and 3 T magnetic fields oriented parallel and perpendicular to the radiation beam. The accuracy of the formalism discussed here against Monte Carlo was evaluated with a gamma comparison using a standard 2%/2 mm and a more stringent 1%/1 mm criterion for a standard reference 10 × 10 cm2 field as well as a smaller 2 × 2 cm2 field. Results: Greater than 99.8% (94.8%) of all points analyzed passed a 2%/2 mm (1%/1 mm) gamma criterion for all magnetic field strengths and orientations investigated. All dosimetric changes resulting from the inclusion of magnetic fields were accurately calculated using the deterministic formalism. However, despite the algorithm’s high degree of accuracy, it is noticed that this formalism was not unconditionally stable using a discrete ordinate angular discretization. Conclusions: The
Priimak, Dmitri
2014-01-01
We present a finite difference numerical algorithm for solving two dimensional spatially homogeneous Boltzmann transport equation which describes electron transport in a semiconductor superlattice subject to crossed time dependent electric and constant magnetic fields. The algorithm is implemented both in C language targeted to CPU and in CUDA C language targeted to commodity NVidia GPU. We compare performances and merits of one implementation versus another and discuss various software optim...
International Nuclear Information System (INIS)
We present a finite difference numerical algorithm for solving two dimensional spatially homogeneous Boltzmann transport equation which describes electron transport in a semiconductor superlattice subject to crossed time dependent electric and constant magnetic fields. The algorithm is implemented both in C language targeted to CPU and in CUDA C language targeted to commodity NVidia GPU. We compare performances and merits of one implementation versus another and discuss various software optimisation techniques
Paussa, A.; Esseni, D.
2013-03-01
This paper revisits the problem of the linearized Boltzmann transport equation (BTE), or, equivalently, of the momentum relaxation time, momentum relaxation time (MRT), for the calculation of low field mobility, which in previous works has been almost universally solved in approximated forms. We propose an energy driven discretization method that allows an exact determination of the relaxation time by solving a linear, algebraic problem, where multiple scattering mechanisms are naturally accounted for by adding the corresponding scattering rates before the calculation of the MRT, and without resorting to the semi-empirical Matthiessen's rule for the relaxation times. The application of our rigorous solution of the linearized BTE to a graphene bilayer reveals that, for a non monotonic energy relation, the relaxation time can legitimately take negative values with no unphysical implications. We finally compare the mobility calculations provided by an exact solution of the MRT problem with the results obtained with some of the approximations most frequently employed in the literature and so discuss their accuracy.
International Nuclear Information System (INIS)
The fundamental process for determining the electric discharge phenomena of gas which take various forms depending on the kinds of gas, gas pressure, relative position of electrodes and applied voltage, is the mutual collision of electrons, atoms, molecular ions and neutral atoms and molecules. The initial stage before the establishment of electric discharge seems to be in Townsend discharge region where the collision of electrons with neutral molecules and atoms mainly occurs, being the weakly ionized condition at low gas temperature. Recently, the breakdown phenomena of N2-O2 gas mixture is being examined for the purpose of clarifying the impulse break mechanism in low pressure air, and the energy distribution of electrons and the transport coefficients in N2, O2 and N2-O2 mixed gases are required to investigate closely the results. Here, the energy distribution and the transport coefficients of electrons in steady Townsend discharge region in N2 and O2 gases respectively were analyzed by using Boltzmann equation, as a preparatory stage. The analyzed results and the discussions on them are reported in each paragraph of the energy distribution and the mean energy of electrons, ionization coefficients and adhesion coefficients, electron drift speed and diffusion coefficients, and excitation frequencies for various electron levels. It was confirmed that each collision process for electrons and the cross-section used for the analysis were properly selected. The excitation frequencies of electrons for N2 and O2 gases concerning the band spectra emitted from discharge channels and the electron energy distribution at 200 V/cm-Torr or below of E/P0 were newly calculated, where E is electric field, and P0 is gas pressure at 0 deg C. (Wakatsuki, Y.)
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.
International Nuclear Information System (INIS)
Purpose: To evaluate the dose distributions of an 192Ir source (model VS2000) in homogeneous water geometry calculated using a deterministic grid-based Boltzmann transport equation solver (GBBS) in the commercial treatment planning system (TPS) (BRACHYVISION-ACUROS v8.8). Methods: Using percent dose differences (%ΔD), the GBBS (BV-ACUROS) was compared to the (1) published TG-43 data, (2) MCNPX Monte Carlo (MC) simulations of the 192Ir source centered in a 15 cm radius water sphere, and (3) TG-43 output from the TPS using vendor supplied (BV-TG43-vendor) and user extended (BV-TG43-extended) 2D anisotropy functions F(r,θ). BV-ACUROS assumes 1 mm of NiTi cable, while the TPS TG-43 algorithm uses data based on a 15 cm cable. MC models of various cable lengths were simulated. Results: The MC simulations resulted in >20% dose deviations along the cable for 1, 2, and 3 mm cable lengths relative to 15 cm. BV-ACUROS comparisons with BV-TG43-vendor and BV-TG43-extended yielded magnitude of differences, consistent with those seen in MC simulations. However, differences >20% extended further (θ≤10 deg.) when using the vendor supplied anisotropy function Fven(r,θ). These differences were also seen in comparisons of F(r,θ) derived from the TPS output. Conclusions: The results suggest that %ΔD near the cable region is larger than previously estimated. The spatial distribution of the dose deviation is highly dependent on the reference TG-43 data used to compare to GBBS. The differences observed, while important to realize, should not have an impact on clinical dosimetry in homogeneous water.
de Urquijo, Jaime; Basurto, E.; Juarez, A. M.; Ness, Kevin; Robson, Robert; Brunger, Michael; White, Ron
2014-10-01
The drift velocity of electrons in mixtures of gaseous water with helium and argon are measured over the range of reduced electric fields from 0--300 Td using a pulsed-Townsend technique. Small admixtures of water to both helium and argon are found to produce negative differential conductivity (NDC), despite NDC being absent from the pure gases. Comparison of the measured drift velocities with those calculated from a multi-term solution of Boltzmann's equation provides a further discriminative assessment on the accuracy and completeness of electron water vapour cross-sections. Funding acknowledgements: ARC, Mexican govt (PAPIIT IN 111014).
Existence of the scattering operator for the linear Boltzmann equation
International Nuclear Information System (INIS)
Existence theorems are proven in a study of the scattering problem for the linear Boltzmann equation (transport equation), describing the motion of a cloud of nonself-interacting particles (neutrons) in phase space. Also Simon's weak coupling result is discussed, and a meaningful wave operator in the presence of trapped particles is defined and its existence proven. 7 references
The Quantum Boltzmann Equation in Semiconductor Physics
Snoke, D. W.
2010-01-01
The quantum Boltzmann equation, or Fokker-Planck equation, has been used to successfully explain a number of experiments in semiconductor optics in the past two decades. This paper reviews some of the developments of this work, including models of excitons in bulk materials, electron-hole plasmas, and polariton gases.
Péraud, Jean-Philippe M.; Hadjiconstantinou, Nicolas G.
2015-01-01
We derive the continuum equations and boundary conditions governing phonon-mediated heat transfer in the limit of small but finite mean free path from asymptotic solution of the linearized Boltzmann equation in the relaxation time approximation. Our approach uses the ratio of the mean free path to the characteristic system lengthscale, also known as the Knudsen number, as the expansion parameter to study the effects of boundaries on the breakdown of the Fourier descrition. We show that, in th...
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 Boltzmann equation in the difference formulation
International Nuclear Information System (INIS)
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.
Pair Production in the Quantum Boltzmann Equation
Rau, Jochen
1994-01-01
A source term in the quantum Boltzmann equation, which accounts for the spontaneous creation of $e^+e^-$-pairs in external electric fields, is derived from first principles and evaluated numerically. Careful analysis of time scales reveals that this source term is generally non-Markovian. This implies in particular that there may be temporary violations of the $H$-theorem.
Li, Wu
2015-08-01
We demonstrate the ab initio electrical transport calculation limited by electron-phonon coupling by using the full solution of the Boltzmann transport equation (BTE), which applies equally to metals and semiconductors. Numerical issues are emphasized in this work. We show that the simple linear interpolation of the electron-phonon coupling matrix elements from a relatively coarse grid to an extremely fine grid can ease the calculational burden, which makes the calculation feasible in practice. For the Brillouin zone (BZ) integration of the transition probabilities involving one δ function, the Gaussian smearing method with a physical choice of locally adaptive broadening parameters is employed. We validate the calculation in the cases of n -type Si and Al. The calculated conductivity and mobility are in good agreement with experiments. In the metal case we also demonstrate that the Gaussian smearing method with locally adaptive broadening parameters works excellently for the BZ integration with double δ functions involved in the Eliashberg spectral function and its transport variant. The simpler implementation is the advantage of the Gaussian smearing method over the tetrahedron method. The accuracy of the relaxation time approximation and the approximation made by Allen [Phys. Rev. B 17, 3725 (1978), 10.1103/PhysRevB.17.3725] has been examined by comparing with the exact solution of BTE. We also apply our method to n -type monolayer MoS2, for which a mobility of 150 cm2 v-1 s-1 is obtained at room temperature. Moreover, the mean free paths are less than 9 nm, indicating that in the presence of grain boundaries the mobilities should not be effectively affected if the grain boundary size is tens of nanometers or larger. The ab initio approach demonstrated in this paper can be directly applied to other materials without the need for any a priori knowledge about the electron-phonon scattering processes, and can be straightforwardly extended to study cases with
Lattice Boltzmann Model and Geophysical Hydrodynamic Equation
Institute of Scientific and Technical Information of China (English)
冯士德; 杨京龙; 郜宪林; 季仲贞
2002-01-01
A lattice Boltzmann equation model in a rotating system is developed by introducing the Coriolis force effect.The geophysical hydrodynamic equation can be derived from this model. Numerical computations are performed to simulate the cylindrical annulus experiment and Benard convection. The numerical results have shown the flow behaviour of large-scale geostrophic current and Benard convection cells, which verifies the applicability of this model to both theory and experiment.
Rigorous Navier-Stokes Limit of the Lattice Boltzmann Equation
Junk, Michael; Yong, Wen-An
2001-01-01
Here we riqorously investigate the diffusive limit of a velocity-discrete Boltzmann equation which is used in the lattice Boltzmann method to construct approximate solutions of the incompressible Navier-Stokes equation.
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...
Test of Information Theory on the Boltzmann Equation
Hyeon-Deuk, Kim; Hayakawa, Hisao
2002-01-01
We examine information theory using the steady-state Boltzmann equation. In a nonequilibrium steady-state system under steady heat conduction, the thermodynamic quantities from information theory are calculated and compared with those from the steady-state Boltzmann equation. We have found that information theory is inconsistent with the steady-state Boltzmann equation.
Test of Information Theory on the Boltzmann Equation
Kim, Hyeon-Deuk; Hayakawa, Hisao
2003-01-01
We examine information theory using the steady-state Boltzmann equation. In a nonequilibrium steady-state system under steady heat conduction, the thermodynamic quantities from information theory are calculated and compared with those from the steady-state Boltzmann equation. We have found that information theory is inconsistent with the steady-state Boltzmann equation.
Energy Technology Data Exchange (ETDEWEB)
Wilcox, T.P. Jr.; Lent, E.M.
1988-12-02
COG is a Monte Carlo computer code designed to solve the Boltzmann equation for transporting neutrons and photons and in future versions, charged particles. Sixty-four different problems were run using the current versions of the COG code on Cray-1 and Cray/X-MP computers. In all cases, the calculated COG results either agree with the values known analytically for some problems or are within the statistical and uncertainties determined experimentally for the others. Problems such as these are referred to benchmark problems and form an important part of the validation of any new computer code. Benchmark problems are of value in that they are used to: check that the code works correctly; check that the physical data used in the code are correct; check that the user has learned to run the code properly; and understand the inherent errors associated with the calculated results. 22 refs., 21 figs., 10 tabs.
Péraud, Jean-Philippe M.; Hadjiconstantinou, Nicolas G.
2016-01-01
We derive the continuum equations and boundary conditions governing phonon-mediated heat transfer in the limit of a small but finite mean-free path from the asymptotic solution of the linearized Boltzmann equation in the relaxation time approximation. Our approach uses the ratio of the mean-free path to the characteristic system length scale, also known as the Knudsen number, as the expansion parameter to study the effects of boundaries on the breakdown of the Fourier description. We show that, in the bulk, the traditional heat conduction equation using Fourier's law as a constitutive relation is valid at least up to second order in the Knudsen number for steady problems and first order for time-dependent problems. However, this description does not hold within distances on the order of a few mean-free paths from the boundary; this breakdown is a result of kinetic effects that are always present in the boundary vicinity and require solution of a Boltzmann boundary layer problem to be determined. Matching the inner, boundary layer solution to the outer, bulk solution yields boundary conditions for the Fourier description as well as additive corrections in the form of universal kinetic boundary layers; both are found to be proportional to the bulk-solution gradients at the boundary and parametrized by the material model and the phonon-boundary interaction model (Boltzmann boundary condition). Our derivation shows that the traditional no-jump boundary condition for prescribed temperature boundaries and the no-flux boundary condition for diffusely reflecting boundaries are appropriate only to zeroth order in the Knudsen number; at higher order, boundary conditions are of the jump type. We illustrate the utility of the asymptotic solution procedure by demonstrating that it can be used to predict the Kapitza resistance (and temperature jump) associated with an interface between two materials. All results are validated via comparisons with low-variance deviational Monte
Maginot, Peter G.; Morel, Jim E.; Ragusa, Jean C.
2012-08-01
We present a new nonlinear spatial finite-element method for the linearized Boltzmann transport equation with Sn angular discretization in 1-D and 2-D Cartesian geometries. This method has two central characteristics. First, it is equivalent to the linear-discontinuous (LD) Galerkin method whenever that method yields a strictly non-negative solution. Second, it always satisfies both the zeroth and first spatial moment equations. Because it yields the LD solution when that solution is non-negative, one might interpret our method as a classical fix-up to the LD scheme. However, fix-up schemes for the LD equations derived in the past have given up solution of the first moment equations when the LD solution is negative in order to satisfy positivity in a simple manner. We present computational results comparing our method in 1-D to the strictly non-negative linear exponential-discontinuous method and to the LD method. We present computational results in 2-D comparing our method to a recently developed LD fix-up scheme and to the LD scheme. It is demonstrated that our method is a valuable alternative to existing methods.
Energy Technology Data Exchange (ETDEWEB)
Prinja, A.K.
1995-08-01
We have developed and successfully implemented a two-dimensional bilinear discontinuous in space and time, used in conjunction with the S{sub N} angular approximation, to numerically solve the time dependent, one-dimensional, one-speed, slab geometry, (ion) transport equation. Numerical results and comparison with analytical solutions have shown that the bilinear-discontinuous (BLD) scheme is third-order accurate in the space ad time dimensions independently. Comparison of the BLD results with diamond-difference methods indicate that the BLD method is both quantitavely and qualitatively superior to the DD scheme. We note that the form of the transport operator is such that these conclusions carry over to energy dependent problems that include the constant-slowing-down-approximation term, and to multiple space dimensions or combinations thereof. An optimized marching or inversion scheme or a parallel algorithm should be investigated to determine if the increased accuracy can compensate for the extra overhead required for a BLD solution, and then could be compared to other discretization methods such as nodal or characteristic schemes.
International Nuclear Information System (INIS)
We have developed and successfully implemented a two-dimensional bilinear discontinuous in space and time, used in conjunction with the SN angular approximation, to numerically solve the time dependent, one-dimensional, one-speed, slab geometry, (ion) transport equation. Numerical results and comparison with analytical solutions have shown that the bilinear-discontinuous (BLD) scheme is third-order accurate in the space ad time dimensions independently. Comparison of the BLD results with diamond-difference methods indicate that the BLD method is both quantitavely and qualitatively superior to the DD scheme. We note that the form of the transport operator is such that these conclusions carry over to energy dependent problems that include the constant-slowing-down-approximation term, and to multiple space dimensions or combinations thereof. An optimized marching or inversion scheme or a parallel algorithm should be investigated to determine if the increased accuracy can compensate for the extra overhead required for a BLD solution, and then could be compared to other discretization methods such as nodal or characteristic schemes
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.
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. PMID:18643046
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.
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.
International Nuclear Information System (INIS)
This paper presents a new multigrid method applied to the most common Sn discretizations (Petrov-Galerkin, diamond-differenced, corner-balanced, and discontinuous Galerkin) of the mono-energetic Boltzmann transport equation in the optically thick and thin regimes, and with strong anisotropic scattering. Unlike methods that use scalar DSA diffusion preconditioners for the source iteration, this multigrid method is applied directly to an integral equation for the scalar flux. Thus, unlike the former methods that apply a multigrid strategy to the scalar DSA diffusion operator, this method applies a multigrid strategy to the integral source iteration operator, which is an operator for 5 independent variables in spatial 3-d (3 in space and 2 in angle) and 4 independent variables in spatial 2-d (2 in space and 2 in angle). The core smoother of this multigrid method involves applications of the integral operator. Since the kernel of this integral operator involves the transport sweeps, applying this integral operator requires a transport sweep (an inversion of an upper triangular matrix) for each of the angles used. As the equation is in 5-space or 4-space, the multigrid approach in this paper coarsens in both angle and space, effecting efficient applications of the coarse integral operators. Although each V-cycle of this method is more expensive than a V-cycle for the DSA preconditioner, since the DSA equation does not have angular dependence, the overall computational efficiency is about the same for problems where DSA preconditioning is effective. This new method also appears to be more robust over all parameter regimes than DSA approaches. Moreover, this new method is applicable to a variety of Sn spatial discretizations, to problems involving a combination of optically thick and thin regimes, and more importantly, to problems with anisotropic scattering cross-sections, all of which DSA approaches perform poorly or not applicable at all. This multigrid approach is
Jet propagation within a Linearized Boltzmann Transport Model
Luo, Tan; Wang, Xin-Nian; Zhu, Yan
2015-01-01
A Linear Boltzmann Transport (LBT) model has been developed for the study of jet propagation inside a quark-gluon plasma. Both leading and thermal recoiled partons are transported according to the Boltzmann equations to account for jet-induced medium excitations. In this talk, we present our study within the LBT model in which we implement the complete set of elastic parton scattering processes. We investigate elastic parton energy loss and their energy and length dependence. We further investigate elastic energy loss and transverse shape of reconstructed jets. Contributions from the recoiled thermal partons are found to have significant influences on the jet energy loss and transverse profile.
Scattering theory for the linearized Boltzmann equation
International Nuclear Information System (INIS)
Scattering theory for a cloud of mutually non-interacting particles that in its passage through R3 undergoes absorption and production in a region D is contained in R3 through interaction with the medium in D is investigated. The motion for such a model is given by the linearized Boltzmann equation. Let n(x,v,t) denote the particle density in phase space at time t. The dynamics is described by a 1-parameter semigroup, W(t), which is in general not isometric. The existence of the wave operators Ω/sub +/ = s - lim W0(-t)W(t) (t →+infinity) and Ω/sub -/ = s - lim W(-t)W0(t) (t →-infinity), where W0(t) is the free dynamics, is examined at length
Asymptotic-preserving Boltzmann model equations for binary gas mixture
Liu, Sha; Liang, Yihua
2016-02-01
An improved system of Boltzmann model equations is developed for binary gas mixture. This system of model equations has a complete asymptotic preserving property that can strictly recover the Navier-Stokes equations in the continuum limit with the correct constitutive relations and the correct viscosity, thermal conduction, diffusion, and thermal diffusion coefficients. In this equation system, the self- and cross-collision terms in Boltzmann equations are replaced by single relaxation terms. In monocomponent case, this system of equations can be reduced to the commonly used Shakhov equation. The conservation property and the H theorem which are important for model equations are also satisfied by this system of model equations.
A hybrid method for the solution of linear Boltzmann equation
International Nuclear Information System (INIS)
Highlights: • The paper presents a novel method for the solution of linear Boltzmann equation. • The hybrid method, based on multiple collisions, combines transport with diffusion. • The physical basis of the method is discussed together with the mathematical model. • Results show its performance in terms of accuracy and computational time. • The extension of the method to more general configurations is discussed. - Abstract: This paper presents a novel approach devised to solve the transport of neutral particles in scattering and absorbing media. The solution to the linear Boltzmann equation is sought starting from a multi-collision approach of the integro-differential equation which is combined with an approximate model for the description of the residue after truncation of the Neumann series. In the paper, the theoretical basis of such hybrid method is discussed together with the physical intuition at the basis of the methodology. Results for both steady-state and transient problems are presented and an extension to general multi-dimensional, anisotropic problem is reported
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...
Supersymmetric electroweak baryogenesis, nonequilibrium field theory and quantum Boltzmann equations
Riotto, Antonio
1998-01-01
The closed time-path (CPT) formalism is a powerful Green's function formulation to describe nonequilibrium phenomena in field theory and it leads to a complete nonequilibrium quantum kinetic theory. In this paper we make use of the CPT formalism to write down a set of quantum Boltzmann equations describing the local number density asymmetries of the particles involved in supersymmetric electroweak baryogenesis. These diffusion equations automatically and self-consistently incorporate the CP-violating sources which fuel baryogenesis when transport properties allow the CP-violating charges to diffuse in front of the bubble wall separating the broken from the unbroken phase at the electroweak phase transition. This is a significant improvement with respect to recent approaches where the CP-violating sources are inserted by hand into the diffusion equations. Furthermore, the CP-violating sources and the particle number changing interactions manifest ``memory'' effects which are typical of the quantum transp ort t...
Exact results for the Boltzmann equation and Smoluchowski's coagulation equation
International Nuclear Information System (INIS)
Almost no analytical solutions have been found for realistic intermolecular forces, largely due to the complicated structure of the collision term which calls for the construction of simplified models, in which as many physical properties are maintained as possible. In the first three chapters of this thesis such model Boltzmann equations are studied. Only spatially homogeneous gases with isotropic distribution functions are considered. Chapter I considers transition kernels, chapter II persistent scattering models and chapter III very hard particles. The second part of this dissertation deals with Smoluchowski's coagulation equation for the size distribution function in a coagulating system, with chapters devoted to the following topics: kinetics of gelation and universality, coagulation equations with gelation and exactly soluble models of nucleation. (Auth./C.F.)
Linearized Boltzmann Equation and Hydrodynamics for Granular Gases
Brey, J. Javier; Dufty, James W.; Ruiz-Montero, M. J.
2003-01-01
The linearized Boltzmann equation is considered to describe small spatial perturbations of the homogeneous cooling state. The corresponding macroscopic balance equations for the density, temperature, and flow velocity are derived from it as the basis for a hydrodynamic description. Hydrodynamics is defined in terms of the spectrum of the generator for the dynamics of the linearized Boltzmann equation. The hydrodynamic eigenfunctions and eigenvalues are calculated in the long wavelength limit....
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.
Thermal equation of state for lattice Boltzmann gases
Ran, Zheng
2009-06-01
The Galilean invariance and the induced thermo-hydrodynamics of the lattice Boltzmann Bhatnagar-Gross-Krook model are 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.
A probabilistic view on the general relativistic Boltzmann equation
Bailleul, Ismael
2011-01-01
A new probalistic approach to general relativistic kinetic theory is proposed. The general relativistic Boltzmann equation is linked to a new Markov process in a completely intrinsic way. This treatment is then used to prove the causal character of the relativistic Boltzmann model.
Langevin theory of fluctuations in the discrete Boltzmann equation
Gross, M; Varnik, F; Adhikari, R
2010-01-01
The discrete Boltzmann equation for both the ideal and a non-ideal fluid is extended by adding Langevin noise terms in order to incorporate the effects of thermal fluctuations. After casting the fluctuating discrete Boltzmann equation in a form appropriate to the Onsager-Machlup theory of linear fluctuations, the statistical properties of the noise are determined by invoking a fluctuation-dissipation theorem at the kinetic level. By integrating the fluctuating discrete Boltzmann equation, the fluctuating lattice Boltzmann equation is obtained, which provides an efficient way to solve the equations of fluctuating hydrodynamics for ideal and non-ideal fluids. Application of the framework to a generic force-based non-ideal fluid model leads to ideal gas-type thermal noise. Simulation results indicate proper thermalization of all degrees of freedom.
On the linearized relativistic Boltzmann equation. II. Existence of hydrodynamics
International Nuclear Information System (INIS)
Solutions are analyzed of the linearized relativistic Boltzmann equation for initial data from L2(r, p) in long-time and/or small-mean-free-path limits. In both limits solutions of this equation converge to approximate ones constructed with solutions of the set of differential equations called the equations of relativistic hydrodynamics
Second-order Boltzmann equation: gauge dependence and gauge invariance
International Nuclear Information System (INIS)
In the context of cosmological perturbation theory, we derive the second-order Boltzmann equation describing the evolution of the distribution function of radiation without a specific gauge choice. The essential steps in deriving the Boltzmann equation are revisited and extended given this more general framework: (i) the polarization of light is incorporated in this formalism by using a tensor-valued distribution function; (ii) the importance of a choice of the tetrad field to define the local inertial frame in the description of the distribution function is emphasized; (iii) we perform a separation between temperature and spectral distortion, both for the intensity and polarization for the first time; (iv) the gauge dependence of all perturbed quantities that enter the Boltzmann equation is derived, and this enables us to check the correctness of the perturbed Boltzmann equation by explicitly showing its gauge-invariance for both intensity and polarization. We finally discuss several implications of the gauge dependence for the observed temperature. (paper)
Second order Boltzmann equation : gauge dependence and gauge invariance
Naruko, Atsushi; Koyama, Kazuya; Sasaki, Misao
2013-01-01
In the context of cosmological perturbation theory, we derive the second order Boltzmann equation describing the evolution of the distribution function of radiation without a specific gauge choice. The essential steps in deriving the Boltzmann equation are revisited and extended given this more general framework: i) the polarisation of light is incorporated in this formalism by using a tensor-valued distribution function; ii) the importance of a choice of the tetrad field to define the local inertial frame in the description of the distribution function is emphasized; iii) we perform a separation between temperature and spectral distortion, both for the intensity and for polarisation for the first time; iv) the gauge dependence of all perturbed quantities that enter the Boltzmann equation is derived, and this enables us to check the correctness of the perturbed Boltzmann equation by explicitly showing its gauge-invariance for both intensity and polarization. We finally discuss several implications of the gaug...
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 $...
Philippi, P C; Surmas, R; Philippi, Paulo Cesar; Santos, Luis Orlando Emerich dos; Surmas, Rodrigo
2005-01-01
The particles model, the collision model, the polynomial development used for the equilibrium distribution, the time discretization and the velocity discretization are factors that let the lattice Boltzmann framework (LBM) far away from its conceptual support: the continuous Boltzmann equation (BE). Most collision models are based on the BGK, single parameter, relaxation-term leading to constant Prandtl numbers. The polynomial expansion used for the equilibrium distribution introduces an upper-bound in the local macroscopic speed. Most widely used time discretization procedures give an explicit numerical scheme with second-order time step errors. In thermal problems, quadrature did not succeed in giving discrete velocity sets able to generate multi-speed regular lattices. All these problems, greatly, difficult the numerical simulation of LBM based algorithms. In present work, the systematic derivation of lattice-Boltzmann models from the continuous Boltzmann equation is discussed. The collision term in the li...
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.
The Boltzmann-Hamel Equations for Optimal Control
Maruskin, Jared M.; Bloch, Anthony M.
2007-01-01
We extend the Boltzmann-Hamel equations to the optimal control setting, producing a set of equations for both kinematic and dynamic nonholonomic optimal control problems. In particular, we will show the dynamic optimal control problem can be written as a minimal set of 4n-2m first order differential equations of motion.
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...
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.
Kapitza conductance, temperature gradients, and solutions to the Boltzmann equation
International Nuclear Information System (INIS)
In the belief that the study of heat transport requires the study of the transport equation, we present an approach to the problem of the Kapitza conductance h/subK/ between two materials which involves the solutions of the Boltzmann equation. One of our purposes is to investigate the origin of the apparent temperature discontinuity ΔT that is associated with this phenomenon. The hydrodynamic solutions of the Boltzmann equation, which (by definition) are describable in terms of local thermohydrodynamic variables, can transfer heat but are not at all responsible for ΔT; whereas the nonhydrodynamic solutions are completely responsible for ΔT but do not transfer heat. An effective temperature T tilde is defined which approaches the thermodynamic temperature T far from the interface, and which is assumed to be continuous across the interface. With this assumption, formal expressions for ΔT and h/subK/ are derived. In the limit as the properties of the two materials become identical, R/subK/ (=h/subK//sup -1/) approaches zero, as should be the case. Further, this approach has a natural generalization to finite frequencies and includes lifetime effects. It is pointed out that thermometers do not measure T tilde but rather T/subR/ which reflects, in a complicated fashion, the presence of the nonhydrodynamic modes, whose amplitudes fall off exponentially as one moves from the interface. In He II, determination of the exponential damping lengths (as a function of temperature and pressure) would provide information about phonon dispersion and phonon interactions which is at least as detailed as could be obtained by other means
The non-linear Boltzmann equation and its application to time and space dependent problems
International Nuclear Information System (INIS)
This thesis is divided into two parts which both involve finding solutions of the Boltzmann Equation. The motivation behind Part 1 is laser fusion where energy transport is by electrons but the temperature gradients are so large in relation to their mean free paths that classical conduction theory breaks down. In this treatment the electron distribution function is found from an appropriate space-dependent Boltzmann Equation and thus physical quantities (in particular heat flux) are calculated for typical parameters from laser fusion. In part 2, an analytic solution of a certain non-linear one-dimensional Boltzmann Equation is obtained which describes the temporal relaxation to equilibrium of a system of particles. Solutions to the corresponding linearised equation and two E.G.K models (with energy-dependent and ''averaged'' collision times) are also derived and compared with that of the non-linear equation. (author)
Lattice Boltzmann Equation On a 2D Rectangular Grid
Bouzidi, MHamed; DHumieres, Dominique; Lallemand, Pierre; Luo, Li-Shi; Bushnell, Dennis M. (Technical Monitor)
2002-01-01
We construct a multi-relaxation lattice Boltzmann model on a two-dimensional rectangular grid. The model is partly inspired by a previous work of Koelman to construct a lattice BGK model on a two-dimensional rectangular grid. The linearized dispersion equation is analyzed to obtain the constraints on the isotropy of the transport coefficients and Galilean invariance for various wave propagations in the model. The linear stability of the model is also studied. The model is numerically tested for three cases: (a) a vortex moving with a constant velocity on a mesh periodic boundary conditions; (b) Poiseuille flow with an arbitrasy inclined angle with respect to the lattice orientation: and (c) a cylinder &symmetrically placed in a channel. The numerical results of these tests are compared with either analytic solutions or the results obtained by other methods. Satisfactory results are obtained for the numerical simulations.
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.
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.
Tsumura, Kyosuke; Kikuchi, Yuta; Kunihiro, Teiji
2015-01-01
We derive the second-order hydrodynamic equation and the microscopic formulae of the relaxation times as well as the transport coefficients systematically from the relativistic Boltzmann equation. Our derivation is based on a novel development of the renormalization-group method, a powerful reduction theory of dynamical systems, which has been applied successfully to derive the non-relativistic second-order hydrodynamic equation Our theory nicely gives a compact expression of the deviation of...
From the Boltzmann Equation to the Euler Equations in the Presence of Boundaries
Golse, François
2011-01-01
The fluid dynamic limit of the Boltzmann equation leading to the Euler equations for an incompressible fluid with constant density in the presence of material boundaries shares some important features with the better known inviscid limit of the Navier-Stokes equations. The present paper slightly extends recent results from [C. Bardos, F. Golse, L. Paillard, Comm. Math. Sci., 10 (2012), 159--190] to the case of boundary conditions for the Boltzmann equation more general than Maxwell's accomodation condition.
From the Boltzmann Equation to the Euler Equations in the Presence of Boundaries
Golse, François
2011-01-01
The fluid dynamic limit of the Boltzmann equation leading to the Euler equations for an incompressible fluid with constant density in the presence of material boundaries shares some important features with the better known inviscid limit of the Navier-Stokes equations. The present paper slightly extends recent results from [C. Bardos, F. Golse, L. Paillard, Comm. Math. Sci., 10 (2012), 159--190] to the case of boundary conditions for the Boltzmann equation more general than Maxwell's accomoda...
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.
Weighted particle method for solving the Boltzmann equation
International Nuclear Information System (INIS)
We propose a new, deterministic, method of solution of the nuclear Boltzmann equation. In this Weighted Particle Method two-body collisions are treated by a Master equation for an occupation probability of each numerical particle. We apply the method to the quadrupole motion of 12C. A comparison with usual stochastic methods is made. Advantages and disadvantages of the Weighted Particle Method are discussed
Derivation of anisotropic dissipative fluid dynamics from the Boltzmann equation
Molnar, E.; Niemi, H.; Rischke, D. H.
2016-01-01
Fluid-dynamical equations of motion can be derived from the Boltzmann equation in terms of an expansion around a single-particle distribution function which is in local thermodynamical equilibrium, i.e., isotropic in momentum space in the rest frame of a fluid element. However, in situations where the single-particle distribution function is highly anisotropic in momentum space, such as the initial stage of heavy-ion collisions at relativistic energies, such an expansion is bound to break dow...
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.
On generalized Boltzmann equations for reacting systems
Veguillas, Juan; Rivas, Martin
A quantum-statistical treatment of chemical kinetics is presented which does not differ between non-reactive scattering and rearrangement processes. This treatment is done in such a way that the standard methods of nonequilibrium statistical mechanics become applicable. Kinetics equations of the Waldmann-Snider, and Wang Chang and Uhlenbeck type are derived for the reduced density operator of different species related to an homo-geneous, dilute gaseous system of the type AB+C⇌2AC+B⇌2BC+A. Global rate coefficients for the different rearrangement processes are defined and derived when starting with Waldmann-Snider type equations.
International Nuclear Information System (INIS)
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. (paper)
Shock-wave structure using nonlinear model Boltzmann equations.
Segal, B. M.; Ferziger, J. H.
1972-01-01
The structure of strong plane shock waves in a perfect monatomic gas was studied using four nonlinear models of the Boltzmann equation. The models involved the use of a simplified collision operator with velocity-independent collision frequency, in place of the complicated Boltzmann collision operator. The models employed were the BGK and ellipsoidal models developed by earlier authors, and the polynomial and trimodal gain function models developed during the work. An exact set of moment equations was derived for the density, velocity, temperature, viscous stress, and heat flux within the shock. This set was reduced to a pair of coupled nonlinear integral equations and solved using specially adapted numerical techniques. A new and simple Gauss-Seidel iteration was developed during the work and found to be as efficient as the best earlier iteration methods.
Transport Equations for Oscillating Neutrinos
Zhang, Yunfan
2013-01-01
We derive a suite of generalized Boltzmann equations, based on the density-matrix formalism, that incorporates the physics of neutrino oscillations for two- and three-flavor oscillations, matter refraction, and self-refraction. The resulting equations are straightforward extensions of the classical transport equations that nevertheless contain the full physics of quantum oscillation phenomena. In this way, our broadened formalism provides a bridge between the familiar neutrino transport algorithms employed by supernova modelers and the more quantum-heavy approaches frequently employed to illuminate the various neutrino oscillation effects. We also provide the corresponding angular-moment versions of this generalized equation set. Our goal is to make it easier for astrophysicists to address oscillation phenomena in a language with which they are familiar. The equations we derive are simple and practical, and are intended to facilitate progress concerning oscillation phenomena in the context of core-collapse su...
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.
Solving the Homogeneous Boltzmann Equation with Arbitrary Scattering Kernel
Hohenegger, A
2008-01-01
With applications in astroparticle physics in mind, we generalize a method for the solution of the nonlinear, space homogeneous Boltzmann equation with isotropic distribution function to arbitrary matrix elements. The method is based on the expansion of the matrix element in terms of two cosines of the "scattering angles". The scattering functions used by previous authors in particle physics for matrix elements in Fermi-approximation are retrieved as lowest order results in this expansion. The method is designed for the unified treatment of reactive mixtures of particles obeying different scattering laws, including the quantum statistical terms for blocking or stimulated emission, in possibly large networks of Boltzmann equations. Although our notation is the relativistic one, as it is used in astroparticle physics, the results can also be applied in the classical case.
Solving the homogeneous Boltzmann equation with arbitrary scattering kernel
International Nuclear Information System (INIS)
With applications in astroparticle physics in mind, we generalize a method for the solution of the nonlinear, space-homogeneous Boltzmann equation with an isotropic distribution function to arbitrary matrix elements. The method is based on the expansion of the scattering kernel in terms of two cosines of the 'scattering angles'. The scattering functions used by previous authors in particle physics for matrix elements in the Fermi approximation are retrieved as lowest order results in this expansion. The method is designed for the unified treatment of reactive mixtures of particles obeying different scattering laws, including the quantum statistical terms for blocking or stimulated emission, in possibly large networks of Boltzmann equations. Although our notation is the relativistic one, as it is used in astroparticle physics, the results can also be applied in the classical case.
Solving the Homogeneous Boltzmann Equation with Arbitrary Scattering Kernel
Hohenegger, A.
2008-01-01
With applications in astroparticle physics in mind, we generalize a method for the solution of the nonlinear, space homogeneous Boltzmann equation with isotropic distribution function to arbitrary matrix elements. The method is based on the expansion of the matrix element in terms of two cosines of the "scattering angles". The scattering functions used by previous authors in particle physics for matrix elements in Fermi-approximation are retrieved as lowest order results in this expansion. Th...
Non-linear effects in the Boltzmann equation
International Nuclear Information System (INIS)
The Boltzmann equation is studied by defining an integral transformation of the energy distribution function for an isotropic and homogeneous gas. This transformation may be interpreted as a linear superposition of equilibrium states with variable temperatures. It is shown that the temporal evolution features of the distribution function are determined by the singularities of said transformation. This method is applied to Maxwell and Very Hard Particle interaction models. For the latter, the solution of the Boltzmann equation with the solution of its linearized version is compared, finding out many basic discrepancies and non-linear effects. This gives a hint to propose a new rational approximation method with a clear physical meaning. Applying this technique, the relaxation features of the BKW (Bobylev, Krook anf Wu) mode is analyzed, finding a conclusive counter-example for the Krook and Wu conjecture. The anisotropic Boltzmann equation for Maxwell models is solved as an expansion in terms of the eigenfunctions of the corresponding linearized collision operator, finding interesting transient overpopulation and underpopulation effects at thermal energies as well as a new preferential spreading effect. By analyzing the initial collision, a criterion is established to deduce the general features of the final approach to equilibrium. Finally, it is shown how to improve the convergence of the eigenfunction expansion for high energy underpopulated distribution functions. As an application of this theory, the linear cascade model for sputtering is analyzed, thus finding out that many differences experimentally observed are due to non-linear effects. (M.E.L.)
On half-space problems for the discrete Boltzmann equation
International Nuclear Information System (INIS)
We study typical half-space problems of rarefied gas dynamics, including the problems of Milne and Kramer, for the discrete Boltzmann equation (a general discrete velocity model, DVM, with an arbitrary finite number of velocities). Then the discrete Boltzmann equation reduces to a system of Odes. The data for the outgoing particles at the boundary are assigned, possibly linearly depending on the data for the incoming particles. A classification of well-posed half-space problems for the homogeneous, as well as the inhomogeneous, linearized discrete Boltzmann equation is made. In the non-linear case the solutions are assumed to tend to an assigned Maxwellian at infinity. The conditions on the data at the boundary needed for the existence of a unique (in a neighborhood of the assigned Maxwellian) solution of the problem are investigated. In the non-degenerate case (corresponding, in the continuous case, to the case when the Mach number at the Maxwellian at infinity is different of (1, 0 and 1) implicit conditions are found. Furthermore, under certain assumptions explicit conditions are found, both in the non-degenerate and degenerate cases. An application to axially symmetric models is also studied.
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.
Marcozzi, M.; Nota, A.
2016-03-01
We consider a test particle moving in a random distribution of obstacles in the plane, under the action of a uniform magnetic field, orthogonal to the plane. We show that, in a weak coupling limit, the particle distribution behaves according to the linear Landau equation with a magnetic transport term. Moreover, we show that, in a low density regime, when each obstacle generates an inverse power law potential, the particle distribution behaves according to the linear Boltzmann equation with a magnetic transport term. We provide an explicit control of the error in the kinetic limit by estimating the contributions of the configurations which prevent the Markovianity. We compare these results with those ones obtained for a system of hard disks in Bobylev et al. (Phys Rev Lett 75:2, 1995), which show instead that the memory effects are not negligible in the Boltzmann-Grad limit.
Derivation of anisotropic dissipative fluid dynamics from the Boltzmann equation
Molnár, Etele; Niemi, Harri; Rischke, Dirk H.
2016-06-01
Fluid-dynamical equations of motion can be derived from the Boltzmann equation in terms of an expansion around a single-particle distribution function which is in local thermodynamical equilibrium, i.e., isotropic in momentum space in the rest frame of a fluid element. However, in situations where the single-particle distribution function is highly anisotropic in momentum space, such as the initial stage of heavy-ion collisions at relativistic energies, such an expansion is bound to break down. Nevertheless, one can still derive a fluid-dynamical theory, called anisotropic dissipative fluid dynamics, in terms of an expansion around a single-particle distribution function, f^0 k, which incorporates (at least parts of) the momentum anisotropy via a suitable parametrization. We construct such an expansion in terms of polynomials in energy and momentum in the direction of the anisotropy and of irreducible tensors in the two-dimensional momentum subspace orthogonal to both the fluid velocity and the direction of the anisotropy. From the Boltzmann equation we then derive the set of equations of motion for the irreducible moments of the deviation of the single-particle distribution function from f^0 k. Truncating this set via the 14-moment approximation, we obtain the equations of motion of anisotropic dissipative fluid dynamics.
International Nuclear Information System (INIS)
We review our work on the application of the renormalization-group method to obtain first- and second-order relativistic hydrodynamics from the relativistic Boltzmann equation (RBE) as a dynamical system, with some corrections and new unpublished results. For the first-order equation, we explicitly obtain the distribution function in the asymptotic regime as the invariant manifold of the dynamical system, which turns out to be nothing but the matching condition defining the energy frame, i.e., the Landau-Lifshitz one. It is argued that the frame on which the flow of the relativistic hydrodynamic equation is defined must be the energy frame, if the dynamics should be consistent with the underlying RBE. A sketch is also given for derivation of the second-order hydrodynamic equation, i.e., extended thermodynamics, which is accomplished by extending the invariant manifold so that it is spanned by excited modes as well as the zero modes (hydrodynamic modes) of the linearized collision operator. On the basis of thus constructed resummed distribution function, we propose a novel ansatz for the functional form to be used in Grad moment method; it is shown that our theory gives the same expressions for the transport coefficients as those given in the Chapman-Enskog theory as well as the novel expressions for the relaxation times and lengths allowing natural interpretation. (orig.)
Energy Technology Data Exchange (ETDEWEB)
Tsumura, Kyosuke [Fujifilm Corporation, Analysis Technology Center, Kanagawa (Japan); Kunihiro, Teiji [Kyoto University, Department of Physics, Kyoto (Japan)
2012-11-15
We review our work on the application of the renormalization-group method to obtain first- and second-order relativistic hydrodynamics from the relativistic Boltzmann equation (RBE) as a dynamical system, with some corrections and new unpublished results. For the first-order equation, we explicitly obtain the distribution function in the asymptotic regime as the invariant manifold of the dynamical system, which turns out to be nothing but the matching condition defining the energy frame, i.e., the Landau-Lifshitz one. It is argued that the frame on which the flow of the relativistic hydrodynamic equation is defined must be the energy frame, if the dynamics should be consistent with the underlying RBE. A sketch is also given for derivation of the second-order hydrodynamic equation, i.e., extended thermodynamics, which is accomplished by extending the invariant manifold so that it is spanned by excited modes as well as the zero modes (hydrodynamic modes) of the linearized collision operator. On the basis of thus constructed resummed distribution function, we propose a novel ansatz for the functional form to be used in Grad moment method; it is shown that our theory gives the same expressions for the transport coefficients as those given in the Chapman-Enskog theory as well as the novel expressions for the relaxation times and lengths allowing natural interpretation. (orig.)
International Nuclear Information System (INIS)
The essential mathematical challenge in transport theory is based on the nonlinearity of the integro-differential equations governing classical thermodynamic systems on molecular kinetic level. It is the aim of this thesis to gain exact analytical solutions to the model Boltzmann equation suggested by Tjon and Wu. Such solutions afford a deeper insight into the dynamics of rarefied gases. Tjon and Wu have provided a stochastic model of a Boltzmann equation. Its transition probability depends only on the relative speed of the colliding particles. This assumption leads in the case of two translational degrees of freedom to an integro-differential equation of convolution type. According to this convolution structure the integro-differential equation is Laplace transformed. The result is a nonlinear partial differential equation. The investigation of the symmetries of this differential equation by means of Lie groups of transformation enables us to transform the originally nonlinear partial differential equation into ordinary differential equation into ordinary differential equations of Bernoulli type. (author)
Fermion particle production in semiclassical Boltzmann-Vlasov transport theory
International Nuclear Information System (INIS)
We present numerical solutions of the semiclassical Boltzmann-Vlasov equation for fermion particle-antiparticle production by strong electric fields in boost-invariant coordinates in (1+1) and (3+1) dimensional QED. We compare the Boltzmann-Vlasov results with those of recent quantum field theory calculations and find good agreement. We conclude that extending the Boltzmann-Vlasov approach to the case of QCD should allow us to do a thorough investigation of how backreaction affects recent results on the dependence of the transverse momentum distribution of quarks and antiquarks on a second Casimir invariant of color SU(3).
Hydrodynamic limit with geometric correction of stationary Boltzmann equation
Wu, Lei
2016-05-01
We consider the hydrodynamic limit of a stationary Boltzmann equation in a unit plate with in-flow boundary. The classical theory claims that the solution can be approximated by the sum of interior solution which satisfies steady incompressible Navier-Stokes-Fourier system, and boundary layer derived from Milne problem. In this paper, we construct counterexamples to disprove such formulation in L∞ both for its proof and result. Also, we show the hydrodynamic limit with a different boundary layer expansion with geometric correction.
Existence of the scattering matrix for the linearized Boltzmann equation
International Nuclear Information System (INIS)
Following Hejtmanek, we consider neutrons in infinite space obeying a linearized Boltzmann equation describing their interaction with matter in some compact set D. We prove existence of the S-matrix and subcriticality of the dynamics in the (weak-coupling) case where the mean free path is larger than the diameter of D uniform in the velocity. We prove existence of the S-matrix also for the case where D is convex and filled with uniformly absorbent material. In an appendix, we present an explicit example where the dynamics is not invertible on L+1, the cone of positive elements in L1. (orig.)
Pointwise Behavior of the Linearized Boltzmann Equation on Torus
Wu, Kung-Chien
2013-01-01
We study the pointwise behavior of the linearized Boltzmann equation on torus for non-smooth initial perturbation. The result reveals both the fluid and kinetic aspects of this model. The fluid-like waves are constructed as part of the long-wave expansion in the spectrum of the Fourier mode for the space variable, the time decay rate of the fluid-like waves depends on the size of the domain. We design a Picard-type iteration for constructing the increasingly regular kinetic-like waves, which ...
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
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.
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. PMID:27627421
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).
d'Eon, Eugene
2013-01-01
We derive new diffusion solutions to the monoenergetic generalized linear Boltzmann transport equation (GLBE) for the stationary collision density and scalar flux about an isotropic point source in an infinite $d$-dimensional absorbing medium with isotropic scattering. We consider both classical transport theory with exponentially-distributed free paths in arbitrary dimensions as well as a number of non-classical transport theories (non-exponential random flights) that describe a broader clas...
Reciprocal relations based on the non-stationary Boltzmann equation
Sharipov, Felix
2012-03-01
The reciprocal relations for open gaseous systems are obtained on the basis of main properties of the non-stationary Boltzmann equation and gas-surface interaction law. It is shown that the main principles to derive the kinetic coefficients satisfying the reciprocal relations remain the same as those used for time-independent gaseous systems [F. Sharipov, Onsager-Casimir reciprocal relations based on the Boltzmann equation and gas-surface interaction law single gas, Phys. Rev. 73 (2006) 026110]. First, the kinetic coefficients are obtained from the entropy production expression; then it is proved that the coefficient matrix calculated for time reversed source functions is symmetric. The proof is based on the reversibility of the gas-gas and gas-surface interactions. Three examples of applications of the present theory are given. None of these examples can be treated in the frame of the classical Onsager-Casimir reciprocal relations, which are valid only in a particular case, when the kinetic coefficients are odd or even with respect to the time reversion. The approach is generalized for gaseous mixtures.
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)
Boltzmann Equation Solver Adapted to Emergent Chemical Non-equilibrium
Birrell, Jeremiah
2014-01-01
We present a novel method to solve the spatially homogeneous and isotropic relativistic Boltzmann equation. We employ a basis set of orthogonal polynomials dynamically adapted to allow emergence of chemical non-equilibrium. Two time dependent parameters characterize the set of orthogonal polynomials, the effective temperature $T(t)$ and phase space occupation factor $\\Upsilon(t)$. In this first paper we address (effectively) massless fermions and derive dynamical equations for $T(t)$ and $\\Upsilon(t)$ such that the zeroth order term of the basis alone captures the number density and energy density of each particle distribution. We validate our method and illustrate the reduced computational cost and the ability to represent final state chemical non-equilibrium by studying a model problem that is motivated by the physics of the neutrino freeze-out processes in the early Universe, where the essential physical characteristics include reheating from another disappearing particle component ($e^\\pm$-annihilation).
Ma, John Z. G.; St.-Maurice, J.-P.
2015-06-01
By applying a backward mapping technique, we solve the Boltzmann equation to investigate the effects of ion-neutral collisions on the ion velocity distribution and related transport properties in cylindrically symmetric, uniformly charged auroral ionosphere. Such a charge geometry introduces a radial electric field which increases linearly with distance from the axis of symmetry. In order to obtain complete analytical solutions for gaining physical insights into more complicated problems, we have substituted a relaxation collision model for the Boltzmann collision integral in the Boltzmann equation. Our calculations show that collisions drive the velocity distribution to a "horseshoe" shape after a few collision times. This feature extends to all radial positions as long as the electric field keeps increasing linearly versus radius. If the electric field is introduced suddenly, there is a transition from the collision-free pulsating Maxwellian distributions obtained in previous work (Ma and St.-Maurice, J. Geophys. Res., 113:A05312, 2008) to the "horseshoe" shapes on a time scale of within the few collision times. We also show how the transport properties evolve in a similar fashion, from oscillating to a non-oscillating features over the same time interval.
Tsumura, Kyosuke; Kunihiro, Teiji
2012-01-01
We review our work on the application of the renormalization-group method to obtain first- and second-order relativistic hydrodynamics of the relativistic Boltzmann equation (RBE) as a dynamical system, with some corrections and new unpublished results. For the first-order equation, we explicitly obtain the distribution function in the asymptotic regime as the invariant manifold of the dynamical system, which turns out to be nothing but the matching condition defining the energy frame, i.e., ...
International Nuclear Information System (INIS)
This paper derives an unbiased minimum variance estimator (UMVE) of a matrix exponential function of a normal wean. The result is then used to propose a reference scheme to solve Boltzmann/Bateman coupled equations, thanks to Monte Carlo transport codes. The last section will present numerical results on a simple example. (authors)
Neutron transport equation - indications on homogenization and neutron diffusion
International Nuclear Information System (INIS)
In PWR nuclear reactor, the practical study of the neutrons in the core uses diffusion equation to describe the problem. On the other hand, the most correct method to describe these neutrons is to use the Boltzmann equation, or neutron transport equation. In this paper, we give some theoretical indications to obtain a diffusion equation from the general transport equation, with some simplifying hypothesis. The work is organised as follows: (a) the most general formulations of the transport equation are presented: integro-differential equation and integral equation; (b) the theoretical approximation of this Boltzmann equation by a diffusion equation is introduced, by the way of asymptotic developments; (c) practical homogenization methods of transport equation is then presented. In particular, the relationships with some general and useful methods in neutronic are shown, and some homogenization methods in energy and space are indicated. A lot of other points of view or complements are detailed in the text or the remarks
Stamnes, K.; Lie-Svendsen, O.; Rees, M. H.
1991-01-01
The linear Boltzmann equation can be cast in a form mathematically identical to the radiation-transport equation. A multigroup procedure is used to reduce the energy (or velocity) dependence of the transport equation to a series of one-speed problems. Each of these one-speed problems is equivalent to the monochromatic radiative-transfer problem, and existing software is used to solve this problem in slab geometry. The numerical code conserves particles in elastic collisions. Generic examples are provided to illustrate the applicability of this approach. Although this formalism can, in principle, be applied to a variety of test particle or linearized gas dynamics problems, it is particularly well-suited to study the thermalization of suprathermal particles interacting with a background medium when the thermal motion of the background cannot be ignored. Extensions of the formalism to include external forces and spherical geometry are also feasible.
Tsumura, Kyosuke; Kunihiro, Teiji
2015-01-01
We derive the second-order hydrodynamic equation and the microscopic formulae of the relaxation times as well as the transport coefficients systematically from the relativistic Boltzmann equation. Our derivation is based on a novel development of the renormalization-group method, a powerful reduction theory of dynamical systems, which has been applied successfully to derive the non-relativistic second-order hydrodynamic equation Our theory nicely gives a compact expression of the deviation of the distribution function in terms of the linearized collision operator, which is different from those used as an ansatz in the conventional fourteen-moment method. It is confirmed that the resultant microscopic expressions of the transport coefficients coincide with those derived in the Chapman-Enskog expansion method. Furthermore, we show that the microscopic expressions of the relaxation times have natural and physically plausible forms. We prove that the propagating velocities of the fluctuations of the hydrodynamica...
Derivation of anisotropic dissipative fluid dynamics from the Boltzmann equation
Molnar, E; Rischke, D H
2016-01-01
Fluid-dynamical equations of motion can be derived from the Boltzmann equation in terms of an expansion around a single-particle distribution function which is in local thermodynamical equilibrium, i.e., isotropic in momentum space in the rest frame of a fluid element. To zeroth order this expansion yields ideal fluid dynamics, to first order Navier-Stokes theory, and to second order transient theories of dissipative fluid dynamics. However, in situations where the single-particle distribution function is highly anisotropic in momentum space, such as the initial stage of heavy-ion collisions at relativistic energies, such an expansion is bound to break down. Nevertheless, one can still derive a fluid-dynamical theory, so-called anisotropic fluid dynamics, in terms of an expansion around a single-particle distribution function which incorporates (at least parts of) the momentum anisotropy via a suitable parametrization. In this paper we derive, up to terms of second order in this expansion, the equations of mo...
High order numerical methods for the space non-homogeneous Boltzmann equation
International Nuclear Information System (INIS)
In this paper we present accurate methods for the numerical solution of the Boltzmann equation of rarefied gas. The methods are based on a time splitting technique. The transport is solved by a third order accurate (in space) positive and flux conservative (PFC) method. The collision step is treated by a Fourier approximation of the collision integral, which guarantees spectral accuracy in velocity, coupled with several high order integrators in time. Strang splitting is used to achieve second order accuracy in space and time. Several numerical tests illustrate the properties of the methods
International Nuclear Information System (INIS)
Purpose: The Linear Boltzmann Transport Equation (LBTE) solved through statistical Monte Carlo (MC) method provides the accurate dose calculation in radiotherapy. This work is to investigate the alternative way for accurately solving LBTE using deterministic numerical method due to its possible advantage in computational speed from MC. Methods: Instead of using traditional spherical harmonics to approximate angular scattering kernel, our deterministic numerical method directly computes angular scattering weights, based on a new angular discretization method that utilizes linear finite element method on the local triangulation of unit angular sphere. As a Result, our angular discretization method has the unique advantage in positivity, i.e., to maintain all scattering weights nonnegative all the time, which is physically correct. Moreover, our method is local in angular space, and therefore handles the anisotropic scattering well, such as the forward-peaking scattering. To be compatible with image-guided radiotherapy, the spatial variables are discretized on the structured grid with the standard diamond scheme. After discretization, the improved sourceiteration method is utilized for solving the linear system without saving the linear system to memory. The accuracy of our 3D solver is validated using analytic solutions and benchmarked with Geant4, a popular MC solver. Results: The differences between Geant4 solutions and our solutions were less than 1.5% for various testing cases that mimic the practical cases. More details are available in the supporting document. Conclusion: We have developed a 3D LBTE solver based on a new angular discretization method that guarantees the positivity of scattering weights for physical correctness, and it has been benchmarked with Geant4 for photon dose calculation
International Nuclear Information System (INIS)
This paper reports the development of an object-oriented programming methodology for particle simulations. It is established on the [m reductionist] view that many physical phenomena cana be reduced to many-body problems. By doing the reduction, many seemly unrelated physical phenomena can be simulated in a systematic way and a high-level programming system can be constructed to facilitate the programming and the solution of the simulations. In the object-oriented particle simulation methodology, a hierarchy of abstract particles is defined to represent a variety of characteristics in physical system simulations. A simulation program is constructed from particles derived from the abstract particles. The object- oriented particle simulation methodology provides a unifying modeling and simulation framework for a variety of simulation applications with the use of particle methods. It allows easy composition of simulation programs from predefined software modules and facilitates software reusability. It greatly increase the productivity of simulation program constructions. Boltzmann (after Ludwig Boltzmann, 1844-1906) is a prototype programming system in the object-oriented particle simulation methodology. Boltzmann is implemented in C++ and the X Window System. It contains a library of data types and functions that support simulations in particle methods. Moreover, it provides a visualization window to support friendly user-computer interaction. Examples of the application of the Boltzmann programming system are presented. The effectiveness of the object-oriented particle simulation methodology is demonstrated. A user's manual is included in the appendix
Incompressible Navier–Stokes equations from Boltzmann equations for reacting mixtures
International Nuclear Information System (INIS)
Incompressible Navier–Stokes equations for gas mixtures are derived from Boltzmann kinetic models in a suitable fluid dynamic limit. We consider polyatomic gases, each one endowed with a discrete set of internal energy levels. Specifically, we deal with a mixture of four polyatomic gases also undergoing chemical reactions. In the Maxwell molecule case, diffusion coefficients and contributions due to inelastic scattering and to chemical reactions may be explicitly computed. (paper)
Spherical harmonics and energy polynomial solution of the Boltzmann equation for neutrons, 1
International Nuclear Information System (INIS)
The approximate solution of the source-free energy-dependent Boltzmann transport equation for neutrons in plane geometry and isotropic scattering case was given by Leonard and Ferziger using a truncated development in a series of energy-polynomials for the energy dependent neutron flux and solving exactly for the angular dependence. The presence in the general solution of eigenfunctions belonging to a continuous spectrum gives rise to difficult analytical problems in the application of their method even to simple problems. To avoid such difficulties, the angular dependence is treated by a spherical harmonics method and a general solution of the energy-dependent transport equation in plane geometry and isotropic scattering is obtained, in spite of the appearance of matrices as argument of the angular polynomials
International Nuclear Information System (INIS)
A hybrid multigroup/continuous-energy Monte Carlo algorithm is developed for solving the Boltzmann-Fokker-Planck equation. This algorithm differs significantly from previous charged-particle Monte Carlo algorithms. Most importantly, it can be used to perform both forward and adjoint transport calculations, using the same basic multigroup cross-section data. The new algorithm is fully described, computationally tested, and compared with a standard condensed history algorithm for coupled electron-photon transport calculations
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.
A deterministic particle method for the linearized Boltzmann equation
International Nuclear Information System (INIS)
We present a conservative particle method of approximation of integral operators which enables us to derive a numerical method of resolution of linear transport equations including integral terms. We prove the L∞ convergence of the method at the order ε2 + hm/εm for any bounded time as soon as the cut-off belongs to Wm1
Lattice Boltzmann Hydrodynamic and Transport Modeling of Everglades Mangrove Estuaries
Sukop, M. C.; Engel, V.
2010-12-01
Lattice Boltzmann methods are being developed and applied to simulate groundwater and surface water flows, and heat, solute, and particle transport. Their ability to solve Navier-Stokes, St. Venant, or Darcy equations with closely coupled solute transport and density-dependent flow effects in geometrically complex domains is attractive for inverse modeling of tracer release data and forward modeling of carbon transport in mangrove estuaries under various future conditions. Key physical processes to be simulated include tidal cycles, storm surge, sea level change, variable upstream stage, subsurface groundwater inputs, and precipitation/recharge and their effects on estuary salinity and carbon transport in the estuaries and groundwater beneath the mangroves. Carbon sources and storage in the aquifer and exchanges at the mangrove-estuary interface and carbon transformations in the water column also need to be simulated. Everglades tidal mangrove estuaries are characterized by relatively high velocity (approaching 1 m s-1) tidal flows. The channels are generally less than 2 m in depth. Tidal fluctuations approach 2 m leading to significant areas of periodic inundation and emergence of oyster beds, shell beaches, mangrove root masses, and sandy beaches. Initial models are two-dimensional, although a three-dimensional model explicitly incorporating bathymetry, density-dependent flow, and wind-driven circulation could be developed. Preliminary work highlights some of the abilities of early models. A satellite image of a 64-km2 area surrounding a CO2 flux tower is used to provide the model geometry. Model resolution is 15 m per grid node. A sinusoidal tidal stage variation and constant, high salinity are applied to the Gulf side of the model while a constant stage (corresponding to mean tide), zero salinity boundary is applied on the inland side. The Navier-Stokes equations coupled with the advection-diffusion equation are solved in the open channels. The mangrove areas
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.
On the solution of ion transport equation
International Nuclear Information System (INIS)
Two different techniques have been used to solve the Boltzmann-transport equation describing the penetration of light ions through solids in the continuous slowing down approximation (CSDA), namely maximum entropy and flux-limited.The solution- obtained for the scalar flux function θο(ξ,s) by using the flux-limited as well by means of the maximum entropy are agree with obtained by Chandrasekhar method
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...
Application of Littlewood-Paley decomposition to the regularity of Boltzmann type kinetic equations
International Nuclear Information System (INIS)
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∞ 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)
An Entropy Stable Discontinuous Galerkin Finite-Element Moment Method for the Boltzmann Equation
Abdelmalik, M R A
2016-01-01
This paper presents a numerical approximation technique for the Boltzmann equation based on a moment system approximation in velocity dependence and a discontinuous Galerkin finite-element approximation in position dependence. The closure relation for the moment systems derives from minimization of a suitable {\\phi}-divergence. This divergence-based closure yields a hierarchy of tractable symmetric hyperbolic moment systems that retain the fundamental structural properties of the Boltzmann equation. The resulting combined discontinuous Galerkin moment method corresponds to a Galerkin approximation of the Boltzmann equation in renormalized form. We present a new class of numerical flux functions, based on the underlying renormalized Boltzmann equation, that ensure entropy dissipation of the approximation scheme. Numerical results are presented for a one-dimensional test case.
Numerical solution of the linearized Boltzmann equation for an arbitrary intermolecular potential
International Nuclear Information System (INIS)
A numerical procedure to solve the linearized Boltzmann equation with an arbitrary intermolecular potential by the discrete velocity method is elaborated. The equation is written in terms of the kernel, which contains the differential cross section and represents a singularity. As an example, the Lennard-Jones potential is used and the corresponding differential cross section is calculated and tabulated. Then, the kernel is calculated so that to overcome its singularity. Once, the kernel is known and stored it can be used for many kinds of gas flows. In order to test the method, the transport coefficients, i.e. thermal conductivity and viscosity for all noble gases, are calculated and compared with those obtained by the variational method using the Sonine polynomials expansion. The fine agreement between the results obtained by the two different methods shows the feasibility of application of the proposed technique to calculate rarefied gas flows over the whole range of the Knudsen number.
Uniform in time lower bound for solutions to a quantum Boltzmann equation of bosons
Nguyen, Toan T.; Tran, Minh-Binh
2016-01-01
We consider the quantum Boltzmann equation, which describes the growth of the condensate, or in other words, models the interaction between excited atoms and a condensate. In this work, the full form of Bogoliubov dispersion law is considered, which leads to a detailed study of surface integrals inside the collision operator on energy manifolds. We prove that positive radial solutions of the quantum Boltzmann equation are bounded from below by a Gaussian, uniformly in time.
Tsumura, Kyosuke; Kikuchi, Yuta; Kunihiro, Teiji
2015-10-01
We derive the second-order hydrodynamic equation and the microscopic formulas of the relaxation times as well as the transport coefficients systematically from the relativistic Boltzmann equation. Our derivation is based on a novel development of the renormalization-group method, a powerful reduction theory of dynamical systems, which has been applied successfully to derive the nonrelativistic second-order hydrodynamic equation. Our theory nicely gives a compact expression of the deviation of the distribution function in terms of the linearized collision operator, which is different from those used as an ansatz in the conventional fourteen-moment method. It is confirmed that the resultant microscopic expressions of the transport coefficients coincide with those derived in the Chapman-Enskog expansion method. Furthermore, we show that the microscopic expressions of the relaxation times have natural and physically plausible forms. We prove that the propagating velocities of the fluctuations of the hydrodynamical variables do not exceed the light velocity, and hence our second-order equation ensures the desired causality. It is also confirmed that the equilibrium state is stable for any perturbation described by our equation.
International Nuclear Information System (INIS)
In this paper, we present a hybrid algorithm to solve the Linear Boltzmann Equation, specifically for application to problems containing regions of low scattering. The hybrid approach uses the Characteristics Method in low scattering regions, while the remaining regions are treated with the Discrete Ordinates Method (Sn). A new 3-D transport code (TITAN) has been developed based on the hybrid approach. In the TITAN code, a physical problem model is divided into a number of coarse meshes (blocks) in Cartesian geometry. TITAN allows different individual fine meshing schemes and angular quadrature sets for each coarse mesh. Either the characteristics solver or the Sn solver can be chosen to solve the Linear Boltzmann Equation within a coarse mesh. A shared scattering kernel allows an arbitrary order of anisotropic scattering in both block-oriented solvers. Angular and spatial projection techniques are developed to transfer angular fluxes on the interfaces of the coarse meshes. We have tested the performance and accuracy of the TITAN code on a number of benchmark problems. The results of a CT model are presented in this paper. The hybrid method shows higher computation efficiency than the regular Sn method. (authors)
The lattice Boltzmann model for the second-order Benjamin–Ono equations
International Nuclear Information System (INIS)
In this paper, in order to extend the lattice Boltzmann method to deal with more complicated nonlinear equations, we propose a 1D lattice Boltzmann scheme with an amending function for the second-order (1 + 1)-dimensional Benjamin–Ono equation. With the Taylor expansion and the Chapman–Enskog expansion, the governing evolution equation is recovered correctly from the continuous Boltzmann equation. The equilibrium distribution function and the amending function are obtained. Numerical simulations are carried out for the 'good' Boussinesq equation and the 'bad' one to validate the proposed model. It is found that the numerical results agree well with the analytical solutions. The present model can be used to solve more kinds of nonlinear partial differential equations
Simulation of a Natural Convection by the Hybrid Thermal Lattice Boltzmann Equation
Energy Technology Data Exchange (ETDEWEB)
Ryu, Seungyeob; Kang, Hanok; Seo, Jaekwang; Yun, Juhyeon; Zee, Sung-Quun [Korea Atomic Energy Research Institute, Taejon (Korea, Republic of)
2006-07-01
Recently, the lattice Boltzmann method(LBM) has gained much attention for its ability to simulate fluid flows, and for its potential advantages over conventional CFD method. The key advantages of LBM are, (1) suitability for parallel computations, (2) absence of the need to solve the time-consuming Poisson equation for a pressure, and (3) an ease with multiphase flows, complex geometries and interfacial dynamics may be treated. In spite of its success in solving various challenging problems involving athermal fluids, the LBM has not been able to handle realistic thermal fluids with a satisfaction. The difficulty encountered in the thermal LBM seems to be the numerical instabilities. The existing thermal lattice Boltzmann models may be classified into three categories based on their approach in solving the Boltzmann equation, namely, the multispeed, the passive scalar and the thermal energy distribution approach. For more details see Ref. In the present work, the hybrid thermal lattice Boltzmann scheme proposed by Lallemand and Luo is used for simulating a natural convection in a square cavity. They proposed a hybrid thermal lattice Boltzmann equation(HTLBE) in which the mass and momentum conservation equations are solved by using the multiple-relaxation-time(MRT) model, whereas the diffusion-advection equations for the temperature are solved separately by using finite-difference technique. The main objective of the present work is to establish the lattice Boltzmann method as a viable tool for the simulation of temperature fields at high Rayleigh numbers.
Electronic transport in VO2—Experimentally calibrated Boltzmann transport modeling
International Nuclear Information System (INIS)
Materials that undergo metal-insulator transitions (MITs) are under intense study, because the transition is scientifically fascinating and technologically promising for various applications. Among these materials, VO2 has served as a prototype due to its favorable transition temperature. While the physical underpinnings of the transition have been heavily investigated experimentally and computationally, quantitative modeling of electronic transport in the two phases has yet to be undertaken. In this work, we establish a density-functional-theory (DFT)-based approach with Hubbard U correction (DFT + U) to model electronic transport properties in VO2 in the semiconducting and metallic regimes, focusing on band transport using the Boltzmann transport equations. We synthesized high quality VO2 films and measured the transport quantities across the transition, in order to calibrate the free parameters in the model. We find that the experimental calibration of the Hubbard correction term can efficiently and adequately model the metallic and semiconducting phases, allowing for further computational design of MIT materials for desirable transport properties
Electronic transport in VO{sub 2}—Experimentally calibrated Boltzmann transport modeling
Energy Technology Data Exchange (ETDEWEB)
Kinaci, Alper; Rosenmann, Daniel; Chan, Maria K. Y., E-mail: debasish.banerjee@toyota.com, E-mail: mchan@anl.gov [Center for Nanoscale Materials, Argonne National Laboratory, Argonne, Illinois 60439 (United States); Kado, Motohisa [Higashifuji Technical Center, Toyota Motor Corporation, Susono, Shizuoka 410-1193 (Japan); Ling, Chen; Zhu, Gaohua; Banerjee, Debasish, E-mail: debasish.banerjee@toyota.com, E-mail: mchan@anl.gov [Materials Research Department, Toyota Motor Engineering and Manufacturing North America, Inc., Ann Arbor, Michigan 48105 (United States)
2015-12-28
Materials that undergo metal-insulator transitions (MITs) are under intense study, because the transition is scientifically fascinating and technologically promising for various applications. Among these materials, VO{sub 2} has served as a prototype due to its favorable transition temperature. While the physical underpinnings of the transition have been heavily investigated experimentally and computationally, quantitative modeling of electronic transport in the two phases has yet to be undertaken. In this work, we establish a density-functional-theory (DFT)-based approach with Hubbard U correction (DFT + U) to model electronic transport properties in VO{sub 2} in the semiconducting and metallic regimes, focusing on band transport using the Boltzmann transport equations. We synthesized high quality VO{sub 2} films and measured the transport quantities across the transition, in order to calibrate the free parameters in the model. We find that the experimental calibration of the Hubbard correction term can efficiently and adequately model the metallic and semiconducting phases, allowing for further computational design of MIT materials for desirable transport properties.
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
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.
Molnár, Etele; Rischke, Dirk H
2016-01-01
In Moln\\'ar et al. [Phys. Rev. D 93, 114025 (2016)] the equations of anisotropic dissipative fluid dynamics were obtained from the moments of the Boltzmann equation based on an expansion around an arbitrary anisotropic single-particle distribution function. In this paper we make a particular choice for this distribution function and consider the boost-invariant expansion of a fluid in one dimension. In order to close the conservation equations, we need to choose an additional moment of the Boltzmann equation. We discuss the influence of the choice of this moment on the time evolution of fluid-dynamical variables and identify the moment that provides the best match of anisotropic fluid dynamics to the solution of the Boltzmann equation in the relaxation-time approximation.
Normal and adjoint integral and integrodifferential neutron transport equations. Pt. 2
International Nuclear Information System (INIS)
Using the simplifying hypotheses of the integrodifferential Boltzmann equations of neutron transport, given in JEN 334 report, several integral equations, and theirs adjoint ones, are obtained. Relations between the different normal and adjoint eigenfunctions are established and, in particular, proceeding from the integrodifferential Boltzmann equation it's found out the relation between the solutions of the adjoint equation of its integral one, and the solutions of the integral equation of its adjoint one (author)
Weak and strong coupling limits of the Boltzmann equation in the relaxation-time approximation
Jaiswal, Amaresh; Redlich, Krzysztof
2016-01-01
We consider a momentum dependent relaxation time for the Boltzmann equation in the relaxation time approximation. We employ a power law parametrization for the momentum dependence of the relaxation time, and calculate the shear and bulk viscosity, as well as, the charge and heat conductivity. We show, that for the two popular parametrizations, referred to as the linear and quadratic ansatz, one can obtain transport coefficients which corresponds to the weak and strong coupling regimes, respectively. We also show that, for a system of massless particles with vanishing chemical potential, the off-equilibrium corrections to the phase-space distribution function calculated with the quadratic ansatz are identical with those of the Grad's 14-moment method.
Obliger, Amaël; Duvail, Magali; Jardat, Marie; Coelho, Daniel; Békri, Samir; Rotenberg, Benjamin
2013-07-01
We report the calculation of all the transfer coefficients which couple the solvent and ionic fluxes through a charged pore under the effect of pressure, electrostatic potential, and concentration gradients. We use a combination of analytical calculations at the Poisson-Nernst-Planck and Navier-Stokes levels of description and mesoscopic lattice simulations based on kinetic theory. In the absence of added salt, i.e., when the only ions present in the fluid are the counterions compensating the charge of the surface, exact analytical expressions for the fluxes in cylindrical pores allow us to validate a new lattice-Boltzmann electrokinetics (LBE) scheme which accounts for the osmotic contribution to the transport of all species. The influence of simulation parameters on the numerical accuracy is thoroughly investigated. In the presence of an added salt, we assess the range of validity of approximate expressions of the fluxes computed from the linearized Poisson-Boltzmann equation by a systematic comparison with LBE simulations. PMID:23944561
Jet propagation within a Linearized Boltzmann Transport model
International Nuclear Information System (INIS)
A Linearized Boltzmann Transport (LBT) model has been developed for the study of parton propagation inside quark–gluon plasma. Both leading and thermal recoiled partons are tracked in order to include the effect of jet-induced medium excitation. In this talk, we present a study within the LBT model in which we implement the complete set of elastic parton scattering processes. We investigate elastic parton energy loss and their energy and length dependence. We further investigate energy loss and transverse shape of reconstructed jets. Contributions from the recoiled thermal partons and jet-induced medium excitations are found to have significant influences on the jet energy loss and transverse profile
Jet propagation within a Linearized Boltzmann Transport model
Energy Technology Data Exchange (ETDEWEB)
Luo, Tan; He, Yayun [Key Laboratory of Quark and Lepton Physics (MOE) and Institute of Particle Physics, Central China Normal University, Wuhan 430079 (China); Wang, Xin-Nian [Key Laboratory of Quark and Lepton Physics (MOE) and Institute of Particle Physics, Central China Normal University, Wuhan 430079 (China); Nuclear Science Division, Mailstop 70R0319, Lawrence Berkeley National Laboratory, Berkeley, CA 94740 (United States); Zhu, Yan [Departamento de Física de Partículas and IGFAE, Universidade de Santiago de Compostela, E-15706 Santiago de Compostela, Galicia (Spain)
2014-12-15
A Linearized Boltzmann Transport (LBT) model has been developed for the study of parton propagation inside quark–gluon plasma. Both leading and thermal recoiled partons are tracked in order to include the effect of jet-induced medium excitation. In this talk, we present a study within the LBT model in which we implement the complete set of elastic parton scattering processes. We investigate elastic parton energy loss and their energy and length dependence. We further investigate energy loss and transverse shape of reconstructed jets. Contributions from the recoiled thermal partons and jet-induced medium excitations are found to have significant influences on the jet energy loss and transverse profile.
Energy Technology Data Exchange (ETDEWEB)
Riotto, A. [European Organization for Nuclear Research, Geneva (Switzerland). Theory Div.
1998-05-04
The closed time path (CTP) formalism is a powerful Green function formulation to describe non-equilibrium phenomena in field theory and it leads to a complete non-equilibrium quantum kinetic theory. In this paper we make use of the CTP formalism to write down a set of quantum Boltzmann equations describing the local number density asymmetries of the particles involved in supersymmetric electroweak baryogenesis. These diffusion equations automatically and self-consistently incorporate the CP-violating sources which fuel baryogenesis when transport properties allow the CP-violating charges to diffuse in front of the bubble wall separating the broken from the unbroken phase at the electroweak phase transition. This is a significant improvement with respect to recent approaches where the CP-violating sources are inserted by hand into the diffusion equations. Furthermore, the CP-violating sources and the particle number changing interactions manifest ````memory```` effects which are typical of the quantum transport theory and are not present in the classical approach. The slowdown of the relaxation processes may keep the system out of equilibrium for longer times and therefore enhance the final baryon asymmetry. We also stress that the classical approximation is not adequate to describe the quantum interference nature of CP violation and that a quantum approach should be adopted to compute the sources since they are most easily built up by the transmission of low momentum particles. (orig.). 26 refs.
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...
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.
Dynamics of Annihilation I : Linearized Boltzmann Equation and Hydrodynamics
de Soria, M. I. Garcia; Maynar, P.; Schehr, G.; Barrat, A.; Trizac, E.
2008-01-01
We study the non-equilibrium 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 {\\em ballistic annihilation} therefore constantly looses particles. The dynamics of perturbations around the free decay regime is investigated from the spectral properties of the linearized Boltzmann operator, that characterize linear excitations on all time scales. The linearized Boltzma...
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.
International Nuclear Information System (INIS)
We present a set of polynomial equations that provides models of the lattice Boltzmann theory for any required level of accuracy and for any dimensional space in a general form. We explicitly derive two- and three-dimensional models applicable to describe thermal compressible flows of the level of the Navier-Stokes equations.
Novel diagrammatic method for computing transport coefficients - beyond the Boltzmann approximation
International Nuclear Information System (INIS)
We propose a novel diagrammatic method for computing transport coefficients in relativistic quantum field theory. Our method is based on a reformulation and extension of the diagrammatic method by Eliashberg given in the imaginary-time formalism to the relativistic quantum field theory in the real-time formalism, in which the cumbersome analytical continuation problem can be avoided. The transport coefficients are obtained from a two-point function via Kubo formula. It is know that naive perturbation theory breaks down owing to a so called pinch singularity, and hence a resummation is required for getting a finite and sensible result. As a novel resummation method, we first decompose the two point function into the singular part and the regular part, and then reconstruct the diagrams. We find that a self-consistent equation for the two-point function has the same structure as the linearized Boltzmann equation. It is known that the two-point function at the leading order is equivalent to the linearized Boltzmann equation. We find the higher order corrections are nicely summarized as a renormalization of the vertex function, spectral function, and collision term. We also discuss the critical behavior of the transport coefficients near a phase transition, applying our method. (author)
Transport equation solving methods
International Nuclear Information System (INIS)
This work is mainly devoted to Csub(N) and Fsub(N) methods. CN method: starting from a lemma stated by Placzek, an equivalence is established between two problems: the first one is defined in a finite medium bounded by a surface S, the second one is defined in the whole space. In the first problem the angular flux on the surface S is shown to be the solution of an integral equation. This equation is solved by Galerkin's method. The Csub(N) method is applied here to one-velocity problems: in plane geometry, slab albedo and transmission with Rayleigh scattering, calculation of the extrapolation length; in cylindrical geometry, albedo and extrapolation length calculation with linear scattering. Fsub(N) method: the basic integral transport equation of the Csub(N) method is integrated on Case's elementary distributions; another integral transport equation is obtained: this equation is solved by a collocation method. The plane problems solved by the Csub(N) method are also solved by the Fsub(N) method. The Fsub(N) method is extended to any polynomial scattering law. Some simple spherical problems are also studied. Chandrasekhar's method, collision probability method, Case's method are presented for comparison with Csub(N) and Fsub(N) methods. This comparison shows the respective advantages of the two methods: a) fast convergence and possible extension to various geometries for Csub(N) method; b) easy calculations and easy extension to polynomial scattering for Fsub(N) method
Improved Multiple-Coarsening Methods for Sn Discretizations of the Boltzmann Equation
Energy Technology Data Exchange (ETDEWEB)
Lee, B
2008-12-01
In a recent series of articles, the author presented a multiple-coarsening multigrid method for solving S{sub n} discretizations of the Boltzmann transport equation. This algorithm is applied to an integral equation for the scalar flux or moments. Although this algorithm is very efficient over parameter regimes that describe realistic neutron/photon transport applications, improved methods that can reduce the computational cost are presented in this paper. These improved methods are derived through a careful examination of the frequencies, particularly the near-nullspace, of the integral equation. In the earlier articles, the near-nullspace components were shown to be smooth in angle in the sense that the angular fluxes generated by these components are smooth in angle. In this paper, we present a spatial description of these near-nullspace components. Using the angular description of the earlier papers together with the spatial description reveals the intrinsic space-angle dependence of the integral equation's frequencies. This space-angle dependence is used to determine the appropriate space-angle grids to represent and efficiently attenuate the near-nullspace error components on. It will be shown that these components can have multiple spatial scales. By using only the appropriate space-angle grids that can represent these spatial scales in the original multiple-coarsening algorithm, an improved algorithm is obtained. Moreover, particularly for anisotropic scattering, recognizing the strong angle dependence of the angular fluxes generated by the high frequencies of the integral equation, another improved multiple-coarsening scheme is derived. Restricting this scheme to the appropriate space-angle grids produces a very efficient method.
Diffusive limits for linear transport equations
International Nuclear Information System (INIS)
The authors show that the Hibert and Chapman-Enskog asymptotic treatments that reduce the nonlinear Boltzmann equation to the Euler and Navier-Stokes fluid equations have analogs in linear transport theory. In this linear setting, these fluid limits are described by diffusion equations, involving familiar and less familiar diffusion coefficients. Because of the linearity extant, one can carry out explicitly the initial and boundary layer analyses required to obtain asymptotically consistent initial and boundary conditions for the diffusion equations. In particular, the effects of boundary curvature and boundary condition variation along the surface can be included in the boundary layer analysis. A brief review of heuristic (nonasymptotic) diffusion description derivations is also included in our discussion
Entropy inequality and hydrodynamic limits for the Boltzmann equation.
Saint-Raymond, Laure
2013-12-28
Boltzmann brought a fundamental contribution to the understanding of the notion of entropy, by giving a microscopic formulation of the second principle of thermodynamics. His ingenious idea, motivated by the works of his contemporaries on the atomic nature of matter, consists of describing gases as huge systems of identical and indistinguishable elementary particles. The state of a gas can therefore be described in a statistical way. The evolution, which introduces couplings, loses part of the information, which is expressed by the decay of the so-called mathematical entropy (the opposite of physical entropy!). PMID:24249776
The scattering kernel of the nonlinear Boltzmann equation and its expansion into spherical harmonics
International Nuclear Information System (INIS)
The method of polynomial approximation i.e. the expansion of the particle density into spherical harmonics, well-known in the linear transport theory, has been generalized by F. Schuerrer 1984 to the nonlinear Boltzmann equation. For practical purposes, a truncation of the series expansion after a few terms requires a good convergence of the scattering kernel expansion into Legendre polynomials. In the present work a few simple examples are used to investigate the convergence. In the most simple case, the elastic collision on a fixed target, the PN-approximations of the scattering kernel in the linear theory serves as a standard of reference for the nonlinear case. A parameter study shows the different approximations as compared to the exact function. Then a more realistic model, a Maxwell gas of target particles, of different temperatures is investigated. The results of this parameter study are represented by three-dimensional computer graphics. In the last chapter the results are applied to the system of moment equations where a considerable simplification is justified for nearly isotropic velocity distributions. It is concluded that nonlinear transport not-too-far from equilibrium can be well described by a Pn approximation. 8 refs., 35 figs. (qui)
Peristaltic particle transport using the Lattice Boltzmann method
Energy Technology Data Exchange (ETDEWEB)
Connington, Kevin William [Los Alamos National Laboratory; Kang, Qinjun [Los Alamos National Laboratory; Viswanathan, Hari S [Los Alamos National Laboratory; Abdel-fattah, Amr [Los Alamos National Laboratory; Chen, Shiyi [JOHNS HOPKINS UNIV.
2009-01-01
Peristaltic transport refers to a class of internal fluid flows where the periodic deformation of flexible containing walls elicits a non-negligible fluid motion. It is a mechanism used to transport fluid and immersed solid particles in a tube or channel when it is ineffective or impossible to impose a favorable pressure gradient or desirous to avoid contact between the transported mixture and mechanical moving parts. Peristaltic transport occurs in many physiological situations and has myriad industrial applications. We focus our study on the peristaltic transport of a macroscopic particle in a two-dimensional channel using the lattice Boltzmann method. We systematically investigate the effect of variation of the relevant dimensionless parameters of the system on the particle transport. We find, among other results, a case where an increase in Reynolds number can actually lead to a slight increase in particle transport, and a case where, as the wall deformation increases, the motion of the particle becomes non-negative only. We examine the particle behavior when the system exhibits the peculiar phenomenon of fluid trapping. Under these circumstances, the particle may itself become trapped where it is subsequently transported at the wave speed, which is the maximum possible transport in the absence of a favorable pressure gradient. Finally, we analyze how the particle presence affects stress, pressure, and dissipation in the fluid in hopes of determining preferred working conditions for peristaltic transport of shear-sensitive particles. We find that the levels of shear stress are most hazardous near the throat of the channel. We advise that shear-sensitive particles should be transported under conditions where trapping occurs as the particle is typically situated in a region of innocuous shear stress levels.
Fast Maximum Entropy Moment Closure Approach to Solving the Boltzmann Equation
Summy, Dustin; Pullin, Dale
2015-11-01
We describe a method for a moment-based solution of the Boltzmann Equation (BE). This is applicable to an arbitrary set of velocity moments whose transport is governed by partial-differential equations (PDEs) derived from the BE. The equations are unclosed, containing both higher-order moments and molecular-collision terms. These are evaluated using a maximum-entropy reconstruction of the velocity distribution function f (c , x , t) , from the known moments, within a finite-box domain of single-particle velocity (c) space. Use of a finite-domain alleviates known problems (Junk and Unterreiter, Continuum Mech. Thermodyn., 2002) concerning existence and uniqueness of the reconstruction. Unclosed moments are evaluated with quadrature while collision terms are calculated using any desired method. This allows integration of the moment PDEs in time. The high computational cost of the general method is greatly reduced by careful choice of the velocity moments, allowing the necessary integrals to be reduced from three- to one-dimensional in the case of strictly 1D flows. A method to extend this enhancement to fully 3D flows is discussed. Comparison with relaxation and shock-wave problems using the DSMC method will be presented. Partially supported by NSF grant DMS-1418903.
International Nuclear Information System (INIS)
A robust numerical solution of the nonlinear Poisson–Boltzmann equation for asymmetric polyelectrolyte solutions in discrete pore geometries is presented. Comparisons to the linearized approximation of the Poisson–Boltzmann equation reveal that the assumptions leading to linearization may not be appropriate for the electrochemical regime in many cementitious materials. Implications of the electric double layer on both partitioning of species and on diffusive release are discussed. The influence of the electric double layer on anion diffusion relative to cation diffusion is examined.
Numerical scheme for a spatially inhomogeneous matrix-valued quantum Boltzmann equation
Lu, Jianfeng; Mendl, Christian B.
2015-06-01
We develop an efficient algorithm for a spatially inhomogeneous matrix-valued quantum Boltzmann equation derived from the Hubbard model. The distribution functions are 2 × 2 matrix-valued to accommodate the spin degree of freedom, and the scalar quantum Boltzmann equation is recovered as a special case when all matrices are proportional to the identity. We use Fourier discretization and fast Fourier transform to efficiently evaluate the collision kernel with spectral accuracy, and numerically investigate periodic, Dirichlet and Maxwell boundary conditions. Model simulations quantify the convergence to local and global thermal equilibrium.
Numerical scheme for a spatially inhomogeneous matrix-valued quantum Boltzmann equation
Lu, Jianfeng
2014-01-01
We develop an efficient algorithm for a spatially inhomogeneous matrix-valued quantum Boltzmann equation derived from the Hubbard model. The distribution functions are 2 x 2 matrix-valued to accommodate the spin degree of freedom, and the scalar quantum Boltzmann equation is recovered as special case when all matrices are proportional to the identity. We use Fourier discretization and fast Fourier transform to efficiently evaluate the collision kernel with spectral accuracy, and numerically investigate periodic, Dirichlet and Maxwell boundary conditions. Model simulations quantify the convergence to local and global thermal equilibrium.
Well-Posedness of the Cauchy Problem for a Space-Dependent Anyon Boltzmann Equation
Arkeryd, Leif; Nouri, Anne
2015-01-01
A fully non-linear kinetic Boltzmann equation for anyons is studied in a periodic 1d setting with large initial data. Strong L 1 solutions are obtained for the Cauchy problem. The main results concern global existence, uniqueness and stabililty. We use the Bony functional, the two-dimensional velocity frame specific for anyons, and an initial layer analysis that moves the solution away from a critical value. 1 Anyons and the Boltzmann equation. Let us first recall the definition of anyon. Con...
A Fokker-Planck model of the Boltzmann equation with correct Prandtl number
Mathiaud, J
2015-01-01
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. This is obtained by replacing the diffusion coefficient (which is the equilibrium temperature) by a non diagonal temperature tensor, like the Ellipsoidal-Statistical model (ES) is obtained from the Bathnagar-Gross-Krook model (BGK) of the Boltzmann equation. Our model is proved to satisfy the properties of conservation and a H-theorem. A Chapman-Enskog analysis and two numerical tests show that a correct Prandtl number of 2/3 can be obtained.
Variational formulation of the steady Boltzmann equation for semiconductors and applications
International Nuclear Information System (INIS)
We present a variational formulation of the steady Boltzmann equation for semiconductors. In this formulation, the distribution function is replaced by a weighted distribution function, and the symmetry of the drift operator is obtained by using the parity operator. We show that the solutions of the Boltzmann equation for the weighted distribution function are stationary functions of a suitable functional, which takes into account realistic boundary conditions. After introducing a general numerical framework, the approach proposed is tested in the bulk case, by computing an approximate expression for carrier mobility in silicon.
Equations of motion of test particles for solving the spin-dependent Boltzmann-Vlasov equation
Xia, Yin; Xu, Jun; Li, Bao-An; Shen, Wen-Qing
2016-08-01
A consistent derivation of the equations of motion (EOMs) of test particles for solving the spin-dependent Boltzmann-Vlasov equation is presented. The resulting EOMs in phase space are similar to the canonical equations in Hamiltonian dynamics, and the EOM of spin is the same as that in the Heisenburg picture of quantum mechanics. Considering further the quantum nature of spin and choosing the direction of total angular momentum in heavy-ion reactions as a reference of measuring nucleon spin, the EOMs of spin-up and spin-down nucleons are given separately. The key elements affecting the spin dynamics in heavy-ion collisions are identified. The resulting EOMs provide a solid foundation for using the test-particle approach in studying spin dynamics in heavy-ion collisions at intermediate energies. Future comparisons of model simulations with experimental data will help to constrain the poorly known in-medium nucleon spin-orbit coupling relevant for understanding properties of rare isotopes and their astrophysical impacts.
Tsumura, Kyosuke
2012-01-01
We apply the renormalization-group method to obtain the first-order relativistic hydrodynamics of the relativistic Boltzmann equation (RBE) as a dynamical system: We explicitly obtain the distribution function in the asymptotic regime as the invariant manifold of the dynamical system, which turns out to be nothing but the matching condition defining the energy frame. It is argued that the frame on which the flow of relativistic hydrodynamic equation is defined must be the energy frame, i.e., the Landau-Lifshitz one, if the dynamics should be consistent with the underlying RBE. A sketch is also given for derivation of the second-order hydrodynamic equation, i.e., extended thermodynamics, which is accomplished by extending the invariant manifold so that it is spanned by excited modes as well as the zero modes (hydrodynamic modes) of the linearized collision operator. On the basis of thus constructed resummed distribution function, we propose a novel ansatz for the functional form to be used in Grad moment metho...
Regularized lattice Boltzmann model for a class of convection-diffusion equations.
Wang, Lei; Shi, Baochang; Chai, Zhenhua
2015-10-01
In this paper, a regularized lattice Boltzmann model for a class of nonlinear convection-diffusion equations with variable coefficients is proposed. The main idea of the present model is to introduce a set of precollision distribution functions that are defined only in terms of macroscopic moments. The Chapman-Enskog analysis shows that the nonlinear convection-diffusion equations can be recovered correctly. Numerical tests, including Fokker-Planck equations, Buckley-Leverett equation with discontinuous initial function, nonlinear convection-diffusion equation with anisotropic diffusion, are carried out to validate the present model, and the results show that the present model is more accurate than some available lattice Boltzmann models. It is also demonstrated that the present model is more stable than the traditional single-relaxation-time model for the nonlinear convection-diffusion equations. PMID:26565368
d'Eon, Eugene
2013-01-01
We derive new diffusion solutions to the monoenergetic generalized linear Boltzmann transport equation (GLBE) for the stationary collision density and scalar flux about an isotropic point source in an infinite $d$-dimensional absorbing medium with isotropic scattering. We consider both classical transport theory with exponentially-distributed free paths in arbitrary dimensions as well as a number of non-classical transport theories (non-exponential random flights) that describe a broader class of transport processes within partially-correlated random media. New rigorous asymptotic diffusion approximations are derived where possible. We also generalize Grosjean's moment-preserving approach of separating the first (or uncollided) distribution from the collided portion and approximating only the latter using diffusion. We find that for any spatial dimension and for many free-path distributions Grosjean's approach produces compact, analytic approximations that are, overall, more accurate for high absorption and f...
The telegraph equation in charged particle transport
Gombosi, T. I.; Jokipii, J. R.; Kota, J.; Lorencz, K.; Williams, L. L.
1993-01-01
We present a new derivation of the telegraph equation which modifies its coefficients. First, an infinite order partial differential equation is obtained for the velocity space solid angle-averaged phase-space distribution of particles which underwent at least a few collisions. It is shown that, in the lowest order asymptotic expansion, this equation simplifies to the well-known diffusion equation. The second-order asymptotic expansion for isotropic small-angle scattering results in a modified telegraph equation with a signal propagation speed of v(5/11) exp 1/2 instead of the usual v/3 exp 1/2. Our derivation of a modified telegraph equation follows from an expansion of the Boltzmann equation in the relevant smallness parameters and not from a truncation of an eigenfunction expansion. This equation is consistent with causality. It is shown that, under steady state conditions in a convecting plasma, the telegraph equation may be regarded as a diffusion equation with a modified transport coefficient, which describes a combination of diffusion and cosmic-ray inertia.
Numerical investigations of low-density nozzle flow by solving the Boltzmann equation
Deng, Zheng-Tao; Liaw, Goang-Shin; Chou, Lynn Chen
1995-01-01
A two-dimensional finite-difference code to solve the BGK-Boltzmann equation has been developed. The solution procedure consists of three steps: (1) transforming the BGK-Boltzmann equation into two simultaneous partial differential equations by taking moments of the distribution function with respect to the molecular velocity u(sub z), with weighting factors 1 and u(sub z)(sup 2); (2) solving the transformed equations in the physical space based on the time-marching technique and the four-stage Runge-Kutta time integration, for a given discrete-ordinate. The Roe's second-order upwind difference scheme is used to discretize the convective terms and the collision terms are treated as source terms; and (3) using the newly calculated distribution functions at each point in the physical space to calculate the macroscopic flow parameters by the modified Gaussian quadrature formula. Repeating steps 2 and 3, the time-marching procedure stops when the convergent criteria is reached. A low-density nozzle flow field has been calculated by this newly developed code. The BGK Boltzmann solution and experimental data show excellent agreement. It demonstrated that numerical solutions of the BGK-Boltzmann equation are ready to be experimentally validated.
Suriyawichitseranee, A.; Grigoriev, Yu. N.; Meleshko, S. V.
2014-01-01
The paper is devoted to group analysis of the spatially homogeneous and isotropic Boltzmann equation with a source term. In fact, the Fourier transform of the Boltzmann equation with respect to the molecular velocity variable is considered. Using a particular class of solutions, the determining equation for the admitted Lie group is reduced to a partial differential equation for the source function. The latter equation is analyzed by an algebraic method. A complete group classification of the...
Generalized Boltzmann equations for on-shell particle production in a hot plasma
Jakovác, A
2002-01-01
A novel refinement of the conventional treatment of Kadanoff--Baym equations is suggested. Besides the Boltzmann equation another differential equation is used for calculating the evolution of the non-equilibrium two-point function. Although it was usually interpreted as a constraint on the solution of the Boltzmann equation, we argue that its dynamics is relevant to the determination and resummation of the particle production cut contributions. The differential equation for this new contribution is illustrated in the example of the cubic scalar model. The analogue of the relaxation time approximation is suggested. It results in the shift of the threshold location and in smearing out of the non-analytic threshold behaviour of the spectral function. Possible consequences for the dilepton production are discussed.
Monitoring derivation of the quantum linear Boltzmann equation
Hornberger, Klaus; Vacchini, Bassano
2007-01-01
We show how the effective equation of motion for a distinguished quantum particle in an ideal gas environment can be obtained by means of the monitoring approach introduced in [EPL 77, 50007 (2007)]. The resulting Lindblad master equation accounts for the quantum effects of the scattering dynamics in a non-perturbative fashion and it describes decoherence and dissipation in a unified framework. It incorporates various established equations as limiting cases and reduces to the classical linear...
The Green's function for the three-dimensional linear Boltzmann equation via Fourier transform
Machida, Manabu
2016-04-01
The linear Boltzmann equation with constant coefficients in the three-dimensional infinite space is revisited. It is known that the Green's function can be calculated via the Fourier transform in the case of isotropic scattering. In this paper, we show that the three-dimensional Green's function can be computed with the Fourier transform even in the case of arbitrary anisotropic scattering.
A Revisiting of the -Stability Theory of the Boltzmann Equation Near Global Maxwellians
Ha, Seung-Yeal; Xiao, Qinghua
2015-07-01
We study the -stability theory of the Boltzmann equation near a global Maxwellian. When an initial datum is a perturbation of a global Maxwellian, we show that the -distance between two classical solutions can be controlled by the initial data in a Lipschitz manner, which illustrates the Lipschitz continuity of the solution operator for the Boltzmann equation in -topology. Our local-in-time -stability results cover cutoff very soft potentials as well as non-cutoff hard and soft potentials. These cases were not treated in the previous work (Ha et al. in Arch Ration Mech Anal 197:657-688, 2010). Thus, our results together with the results in Ha et al. (2010) complete the -stability theory for the Boltzmann equation near a global Maxwellian. For this -stability estimate, we use the coercivity estimate of the linearized collision operator, the smallness of perturbation in a mixed Lebesgue norm, and Strichartz-type estimates of perturbation. We also show that for all classical solutions available in the literature, the Lipschitz constant can be chosen as independent of time to obtain the uniform -stability of the Boltzmann equation.
General particle transport equation. Final report
International Nuclear Information System (INIS)
The general objectives of this research are as follows: (1) To develop fundamental models for fluid particle coalescence and breakage rates for incorporation into statistically based (Population Balance Approach or Monte Carlo Approach) two-phase thermal hydraulics codes. (2) To develop fundamental models for flow structure transitions based on stability theory and fluid particle interaction rates. This report details the derivation of the mass, momentum and energy conservation equations for a distribution of spherical, chemically non-reacting fluid particles of variable size and velocity. To study the effects of fluid particle interactions on interfacial transfer and flow structure requires detailed particulate flow conservation equations. The equations are derived using a particle continuity equation analogous to Boltzmann's transport equation. When coupled with the appropriate closure equations, the conservation equations can be used to model nonequilibrium, two-phase, dispersed, fluid flow behavior. Unlike the Eulerian volume and time averaged conservation equations, the statistically averaged conservation equations contain additional terms that take into account the change due to fluid particle interfacial acceleration and fluid particle dynamics. Two types of particle dynamics are considered; coalescence and breakage. Therefore, the rate of change due to particle dynamics will consider the gain and loss involved in these processes and implement phenomenological models for fluid particle breakage and coalescence
Zhang, Jianying; Yan, Guangwu
2016-04-01
A lattice Boltzmann model for solving the (2+1) dimensional cubic-quintic complex Ginzburg-Landau equation (CQCGLE) is proposed. Different from the classic lattice Boltzmann models, this lattice Boltzmann model is based on uniformly distributed lattice points in a two-dimensional space, and the evolution of the model is about a spatial axis rather than time. The algorithm provides advantages similar to the lattice Boltzmann method in that it is easily adapted to complex Ginzburg-Landau equations. Numerical results reproduce the phenomena of the fusion of necklace-ring pattern and the effect of non-linearity on the soliton in the CQCGLE.
Stability of Global Solution to Boltzmann-Enskog Equation with External Force
Institute of Scientific and Technical Information of China (English)
JIANG ZHENG-LU; MA LI-JUN; YAO ZHENG-AN
2012-01-01
In the presence of external forces depending only on the time and space variables,the Boltzmann-Enskog equation formally conserves only the mass of the system,and its entropy functional is also nonincreasing.Corresponding to this type of equation,we first give some hypotheses of its bicharacteristic equations and then get some results about the stablity of its global solution with the help of two new Lyapunov functionals:one is to describe interactions between particles with different velocities and the other is to measure the L1 distance between two mild solutions.The former Lyapunov functional yields the time-asymptotic convergence of global classical solutions to the collision free motion while the latter is applied into the verification of the L1 stability of global mild solutions to the Boltzmann-Enskog equation for a moderately or highly dense gas in the influence of external forces.
From Newton's law to the linear Boltzmann equation without cut-off
Ayi, Nathalie
2016-01-01
We provide a rigorous derivation of the linear Boltzmann equation without cutoff 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. 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 combin...
A hybrid kinetic-fluid model for solving the gas dynamics Boltzmann-BGK equation
Crouseilles, Nicolas; Degond, Pierre; Lemou, Mohammed
2004-01-01
International audience Our purpose s toderive a hybrid model for particles systems which combines a kinetic description of the fast particles with a fluid description of the thermal ones. Fats particles will be described through a collisional kinetic equation of Boltzmann-BGK type while thermal particles will be modeled by means of a system of a Euler type equations. A conservative numerical scheme is constructed and enables us to validate the approach on various numerical tests.
The Incompressible Navier-Stokes Limit of the Boltzmann Equation for Hard Cutoff Potentials
Golse, François; Saint-Raymond, Laure
2008-01-01
The present paper proves that all limit points of sequences of renormalized solutions of the Boltzmann equation in the limit of small, asymptotically equivalent Mach and Knudsen numbers are governed by Leray solutions of the Navier-Stokes equations. This convergence result holds for hard cutoff potentials in the sense of H. Grad, and therefore completes earlier results by the same authors [Invent. Math. 155, 81-161 (2004)] for Maxwell molecules.
Steady detonation waves via the Boltzmann equation for a reacting mixture
International Nuclear Information System (INIS)
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
Application of Boltzmann equation to electron transmission and seconary electron emission
International Nuclear Information System (INIS)
A method is presented for numerical treatment of integro-differential equation, based upon finite difference techniques. This method allows to formulate in a satisfactory manner the Boltzmann's equation applied to backscattering, transmission and secondary emission of metallic targets, avoiding must of the restrictive hypothesis, used until now in these models. For aluminium, the calculated energy spectra, angular distribution, transmission and backscattering coefficients, and secondary emission yield, are found to be in good agreement with experiment
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.
Spherical Harmonic Expansion Method for Coupled Electron-Phonon Boltzmann Transport
Santia, Marco; Albrecht, John
2014-03-01
Thermoelectric transport modeling often relies on independent Boltzmann transport equations (BTEs) for electrons and phonons which work best near equilibrium (linearized) and steady-state. Device design relies heavily on this baseline approximation. Monte Carlo methods can allow for complex physical interactions (e.g., anharmonicity) but their stochastic nature has practical limits. Distribution functions with wide disparities in population (e.g., ratios >108 between majority and minority carriers.[1]) are a computational challenge. We present a coupled BTE solver based on a k-space spherical harmonic expansion (SHE) of the distribution functions and eigenstates of electrons and phonons. The method is deterministic and allows for detailed treatments of scattering processes, yet ameliorates the issues with population disparity within phase space. We set the formalism and examine the accuracy of the SHE for phonon band structures, calculate scattering rates determined within that representation, and compare our preliminary results for distribution statistics in control examples such as thermal conductivity and drift velocity.
Hammou, H.; Ginzburg, I.; Boulerhcha, M.
2011-01-01
We develop two-relaxation-times Lattice Boltzmann schemes (TRT) with two relaxation functions Kð~r; tÞ for solving highly non-linear equations for groundwater modeling in d-dimensions, namely, the Richards equation for water content distribution hð~r; tÞ in unsaturated flow and the associated transport equation for solute concentration Cð~r; tÞ, advected by the local Darcian water flux. The method is verified against the analytical solutions and the HYDRUS code where the TRT schemes behave mo...
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.
International Nuclear Information System (INIS)
Handling the highly anisotropic scattering of fast neutrons with conventional methods usually means that high-order Legendre expansions are necessary to obtain correct angular fluxes. This drawback in standard transport calculations is avoided by applying the Boltzmann-Fokker-Planck (BEP) equation approach which has been used in both neutral and charged-particle transport problems. Previously, Caro and Ligou, and Morel have introduced Fokker-Planck decomposition methods, which decompose elastic scattering cross section into forward-peaked and smooth components. A new method for decomposing scattering cross sections for Boltzmann-Fokker-Planck equation is presented. We start from the basic data σs(μ)(given by ENDF/B-VI) to get more correctly determined BFP data. In this method, we use Legendre expansion for smooth component and exponential function, which Caro and Ligou used in their paper, for forward-peaked component. In addition, by using RMS errors and an extra degree of freedom (Y), we conserve both moment and scattering cross section
Liu, Zhe; Jiang, Liwei; Zheng, Yisong
2016-01-01
The diagonal and Hall conductivities of graphene arising from the spin-orbit coupling impurity scattering are theoretically studied. Based on the continuous model, i.e. the massless Dirac equation, we derive analytical expressions of the conductivity tensor from both the Kubo and Boltzmann transport theories. By performing numerical calculations, we find that the Kubo quantum transport result of the diagonal conductivity within the self-consistent Born approximation exhibits an insulating gap around the Dirac point. And in this gap a well-defined quantized spin Hall plateau occurs. This indicates the realization of the quantum spin Hall state of graphene driven by the spin-orbit coupling impurities. In contrast, the semi-classical Boltzmann theory fails to predict such a topological insulating phase. The Boltzmann diagonal conductivity is nonzero even in the insulating gap, in which the Boltzmann spin Hall conductivity does not exhibit any quantized plateau. PMID:27029398
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...
Goal-Oriented Adaptivity and Multilevel Preconditioning for the Poisson-Boltzmann Equation
Aksoylu, Burak; Cyr, Eric; Holst, Michael
2011-01-01
In this article, we develop goal-oriented error indicators to drive adaptive refinement algorithms for the Poisson-Boltzmann equation. Empirical results for the solvation free energy linear functional demonstrate that goal-oriented indicators are not sufficient on their own to lead to a superior refinement algorithm. To remedy this, we propose a problem-specific marking strategy using the solvation free energy computed from the solution of the linear regularized Poisson-Boltzmann equation. The convergence of the solvation free energy using this marking strategy, combined with goal-oriented refinement, compares favorably to adaptive methods using an energy-based error indicator. Due to the use of adaptive mesh refinement, it is critical to use multilevel preconditioning in order to maintain optimal computational complexity. We use variants of the classical multigrid method, which can be viewed as generalizations of the hierarchical basis multigrid and Bramble-Pasciak-Xu (BPX) preconditioners.
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)
Sliding periodic boundary conditions for lattice Boltzmann and lattice kinetic equations
Adhikari, R.; Desplat, J. -C.; Stratford, K.
2005-01-01
We present a method to impose linear shear flow in discrete-velocity kinetic models of hydrodynamics through the use of sliding periodic boundary conditions. Our method is derived by an explicit coarse-graining of the Lees-Edwards boundary conditions for Couette flow in molecular dynamics, followed by a projection of the resulting equations onto the subspace spanned by the discrete velocities of the lattice Boltzmann method. The boundary conditions are obtained without resort to perturbative ...
A Combined MPI-CUDA Parallel Solution of Linear and Nonlinear Poisson-Boltzmann Equation
José Colmenares; Antonella Galizia; Jesús Ortiz; Andrea Clematis; Walter Rocchia
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 ...
Spherical-harmonic type expansion for the Boltzmann equation in semiconductor devices
Directory of Open Access Journals (Sweden)
Armando Majorana
1998-10-01
Full Text Available The Boltzmann equation for an electron gas in a semiconductor is considered. The electron energy is assumed to have a very general form, so that, for instance, parabolic or non parabolic band approximations can be treated. A technique, which recalls the classical moment method due to Grad, to deduce an approximate quasi-hydrodynamical model is shown and compared with the spherical harmonic expansion. Some characteristics of the model, as entropy inequality, are explicitly presented.
A Spectral Study of the Linearized Boltzmann Equation for Diffusively Excited Granular Media
Rey, Thomas
2013-01-01
In this work, we are interested in the spectrum of the diffusively excited granular gases equation, in a space inhomogeneous setting, linearized around an homogeneous equilibrium. We perform a study which generalizes to a non-hilbertian setting and to the inelastic case the seminal work of Ellis and Pinsky about the spectrum of the linearized Boltzmann operator. We first give a precise localization of the spectrum, which consists in an essential part lying on the left of the imaginary axis an...
Dilaton and off-shell (non-critical string) effects in Boltzmann equation for species abundances
Lahanas, A B; Nanopoulos, Dimitri V
2006-01-01
In this work we derive the modifications to the Boltzmann equation governing the cosmic evolution of relic abundances induced by dilaton dissipative-source and non-critical-string terms in dilaton-driven non-equilibrium string Cosmologies. We also discuss briefly the most important phenomenological consequences, including modifications of the constraints on the available parameter space of cosmologically appealing particle physics models, imposed by recent precision data of astrophysical measurements.
Dilaton and off-shell (non-critical string) effects in Boltzmann equation for species abundances
Lahanas, Ab; Mavromatos, Ne; Nanopoulos, Dv
In this work we derive the modifications to the Boltzmann equation governing the cosmic evolution of relic abundances induced by dilaton dissipative-source and non-critical-string terms in dilaton-driven non-equilibrium string Cosmologies. We also discuss briefly the most important phenomenological consequences, including modifications of the constraints on the available parameter space of cosmologically appealing particle physics models, imposed by recent precision data of astrophysical measurements.