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The plasma heating by the α particle transport is a main self-heating source in inertial confinement fusion (ICF) that determines the capsule implosion performance. Due to the high energy of α particle and the high temperature in the ICF capsule hot spot, significant non-equilibrium effect exists and the continuum mechanics breaks down. For the numerical simulation of implosion and charged particle transport, the Boltzmann equation needs to be solved to capture the kinetic effects. However, the 7-dimensional Boltzmann equation, the highly frequent Coulomb collision, and the multi-folded integral stopping power formulation greatly limits the computational efficiency and challenges the computational power. To overcome the high computational cost of high-frequent Coulomb collisions, we propose a mixed collision model according to which the collisions are categorized into low-frequent large-angle collisions and high frequent small-angle grazing collisions. The large-angle collision process is precisely solved based on the Coulomb cross-section. For the highly frequent small-angle grazing, a statistic model is constructed with second-order accuracy in time. The mixed collision model reduces the computational cost of scattering calculation by two orders of magnitude. For the multi-folded integral stopping power formulation, a neural network is used to improve computational efficiency. Based on the proposed algorithm, we develop one-dimensional to three-dimensional module code to directly solve the α particle transport Boltzmann equation. The α transport module code is integrated into the multi-physics LARED-S program. The MC version ICF software is verified by a simulation study of the N210207 and N191110 experiment. [ABSTRACT FROM AUTHOR]

Numerical simulation results of basic exactly solvable fluid flows using the previously proposed by H. Chen Lattice Boltzmann Method (LBM) formulated on a general curvilinear coordinate system are presented. As was noted in the theoretical work of H. Chen, such curvilinear Lattice Boltzmann Method preserves a fundamental one-to-one exact advection feature in producing minimal numerical diffusion, as the Cartesian lattice Boltzmann model. As we numerically show, the new model converges to exact solutions of basic fluid flows with the increase of grid resolution in the presence of both natural curvilinear geometry and/or grid non-uniform contraction, both for near equilibrium and non-equilibrium LBM parameter choices.

A boundary value problem for the stationary Boltzmann equation with hard sphere in a half-space is considered for a binary mixture of gases, imposing slightly perturbed reflection boundary conditions for the both species. We assume that one of the species is dominant and close to equilibrium but the density of the other is small. Under this assumption, we prove the existence and uniqueness of the solution to the coupled system, and their accurate asymptotic behavior in the far field is obtained. Our result is based on the one species result. However, the system is not symmetric which leads us to the major difficulty in this paper. Thanks to the conservation laws for the collision operators, the nonlinear problem is solved. [ABSTRACT FROM AUTHOR]

In this paper, the authors study the 1D steady Boltzmann flow in a channel. The walls of the channel are assumed to have vanishing velocity and given temperatures θ0 and θ1. This problem was studied by Esposito-Lebowitz-Marra (1994, 1995) where they showed that the solution tends to a local Maxwellian with parameters satisfying the compressible Navier-Stokes equation with no-slip boundary condition. However, a lot of numerical experiments reveal that the fluid layer does not entirely stick to the boundary. In the regime where the Knudsen number is reasonably small, the slip phenomenon is significant near the boundary. Thus, they revisit this problem by taking into account the slip boundary conditions. Following the lines of [Coron, F., Derivation of slip boundary conditions for the Navier-Stokes system from the Boltzmann equation, J. Stat. Phys., 54(3–4), 1989, 829–857], the authors will first give a formal asymptotic analysis to see that the flow governed by the Boltzmann equation is accurately approximated by a superposition of a steady CNS equation with a temperature jump condition and two Knudsen layers located at end points. Then they will establish a uniform L∞ estimate on the remainder and derive the slip boundary condition for compressible Navier-Stokes equations rigorously. [ABSTRACT FROM AUTHOR]

In the present work, we adopt the idea of velocity averaging lemma to establish regularity for stationary linearized Boltzmann equations in a bounded convex domain. Considering the incoming data, with four iterations, we establish regularity in fractional Sobolev space in space variable up to order 1 - . [ABSTRACT FROM AUTHOR]

In this study, the electron energy distribution function (EEDF), the electron swarm parameters, the effective ionization coefficients, and the critical field strength (dielectric strength) in binary He-H2 gas mixture which is used as cryogenic for high-temperature superconducting power applications, are evaluated using twoterm solution of the Boltzmann equation over the range of E/N ( the electric field to gas density) from 1 to 100 Td ( 1 Td=10-17 Vcm² ) at temperature 77 K and pressure 2MPa, taking into account elastic ( momentum transfer) and inelastic cross-sections. Using the electron energy distribution function (EEDF) electron swarm parameters (electron drift velocity, mean electron energy, diffusion coefficient, electron mobility, ionization and attachment coefficient) are calculated. At low reduced electric field strength E/N, the EEDF is close to Maxwellian distribution, at high E/N, due to vibrational excitation of H2, the calculated distribution function is nonMaxwellian. Besides, the Boltzmann equation analysis showed as the small mole fraction of H2 in the He-H2 mixture is increased, the electron energy distribution function EEDF shifts to lower energy region, the density-reduced ionization coefficient α/N and density-reduced effective ionization coefficient (α-η)/N decreases, whereas density-reduced attachment coefficient η/N, density-reduced critical electric field strength increases, (E/N)crt and critical electric field Ecrt increases. It is found that dielectric field strength depends on pressure and temperature. To confirm the validity of the two term solution of Boltzmann equation analysis, a set of elastic and inelastic cross-sections for each gas He and H2 are used to calculate the electron swarm parameters and dielectric field strength. Compared with previous experimental and theoretical literatures, the values obtained are generally in good agreement. [ABSTRACT FROM AUTHOR]

A new numerical method for solving the Boltzmann equation on a uniform mesh in velocity space is proposed. The asymptotic complexity of the method is , where is the total number of nodes on a three-dimensional mesh. The algorithm is efficient on relatively small meshes due to the simplicity of its operations and easy parallelization. The method preserves the most important properties of the solution, such as nonnegativity and conservation of total energy, momentum, and the number of particles. [ABSTRACT FROM AUTHOR]

Spectral methods, thanks to their high accuracy and the possibility of using fast algorithms, represent an effective way to approximate collisional kinetic equations in kinetic theory. On the other hand, the loss of some local invariants can lead to the wrong long time behavior of the numerical solution. We introduce in this paper a novel Fourier--Galerkin spectral method that improves the classical spectral method by making it conservative on the moments of the approximated distribution, without sacrificing its spectral accuracy or the possibility of using fast algorithms. The method is derived directly using a constrained best approximation in the space of trigonometric polynomials and can be applied to a wide class of problems where preservation of moments is essential. We then apply the new spectral method to the evaluation of the Boltzmann collision term and prove the spectral consistency and stability of the resulting Fourier--Galerkin approximation scheme. Various numerical experiments illustrate the theoretical findings. [ABSTRACT FROM AUTHOR]

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Possibility of the diverging gradient of the macroscopic quantity near the boundary is investigated by a mono-speed Lorentz-gas model, with a special attention to the regularizing effect of the grazing collision for the infinite-range potential on the velocity distribution function (VDF) and its influence on the macroscopic quantity. By careful numerical analyses of the steady one-dimensional boundary-value problem, it is confirmed that the grazing collision suppresses the occurrence of a jump discontinuity of the VDF on the boundary. However, as the price for that regularization, the collision integral becomes no longer finite in the direction of the molecular velocity parallel to the boundary. Consequently, the gradient of the macroscopic quantity diverges, even stronger than the case of the finite-range potential. A conjecture about the diverging rate in approaching the boundary is made as well for a wide range of the infinite-range potentials, accompanied by the numerical evidence. [ABSTRACT FROM AUTHOR]

We prove that exponential moments of a fluctuation of the pure transport equation decay pointwisely almost as fast as t-3 when the domain is any general strictly convex subset of R³ with the smooth boundary of the diffuse boundary condition. We prove the theorem by establishing a novel L¹-L∞ framework via stochastic cycles. [ABSTRACT FROM AUTHOR]

Studies in the collective motility of organisms use a range of analytical approaches to formulate continuous kinetic models of collective dynamics from rules or equations describing agent interactions. However, the derivation of these kinetic models often relies on Boltzmann's "molecular chaos" hypothesis, which assumes that correlations between individuals are short-lived. While this assumption is often the simplest way to derive tractable models, it is often not valid in practice due to the high levels of cooperation and self-organization present in biological systems. In this work, we illustrated this point by considering a general Boltzmann-type kinetic model for the alignment of self-propelled rods where rod reorientation occurs upon binary collisions. We examine the accuracy of the kinetic model by comparing numerical solutions of the continuous equations to an agent-based model that implements the underlying rules governing microscopic alignment. Even for the simplest case considered, our comparison demonstrates that the kinetic model fails to replicate the discrete dynamics due to the formation of rod clusters that violate statistical independence. Additionally, we show that introducing noise to limit cluster formation helps improve the agreement between the analytical model and agent simulations but does not restore the agreement completely. These results highlight the need to both develop and disseminate improved moment-closure methods for modeling biological and active matter systems., Competing Interests: Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper., (Published by Elsevier Inc.)

It is known that any laser has a pumping source, an amplifying medium and a resonator that oscillates the active mode representing the out-put wavelength of the laser. The suitable value for this ratio is 7:1 He/Ne in volume. These were to prepare under an ideal mixture of these gases as treated before by many experiment lists. The aim of this study to specify is to the basic portions of low power He-Ne gas lasers available in laboratories, and to test them in order to look for a probable factor of the efficiency as the whole. All calculations were performed, using calculators. The results were unique for all of them that the laser aperture also plays an important role in this criteria. The best fitting ratio between the radius of laser aperture and that of laser beam was determined and connected to the other parts of the efficiency, also the ratio of(ρ/ω) was fixed at 1.55.

The calculation of aerodynamic force and aerodynamic heating for flight vehicles round trip to Earth􀆳s atmosphere （over different flow regimes） is one of the challenges and hotspots in computational fluid dynamics. Since the flow scenarios faced by such vehicles are no longer purely continuous flow or rarefied flow， neither the Navier-Stokes equations’ solver nor the direct simulation Monte Carlo （DSMC） method can obtain accurate results. In recent years ，based on the Boltzmann model equation， which is independent of the continuity hypothesis， various numerical approaches have been developed to simulate fluid flow problems in all flow regimes by discretizing this equation in both physical space and velocity space. This paper reviews and analyzes the research progress of this kind of algorithm， including the gas kinetic unified algorithm （GKUA），the unified gas kinetic scheme （UGKS） and the improved discrete velocity method （IDVM）. It focuses on the basic ideas and the implementation of these methods and pays attention to their current progress and applications. At the same time， the IDVM is further extended to simulate the unsteady flows in all flow regimes. Finally， some problems existing in this kind of algorithm are discussed and the prospect for work in the future is proposed. [ABSTRACT FROM AUTHOR]

Departing from the weak solution, we prove the uniqueness and long-time behavior of the weak solution to the non-cutoff spatially homogeneous Boltzmann equation with moderate soft potentials. The ingredients of the proof lie in the development of the localized techniques in phase and frequency spaces and entropy methods. [ABSTRACT FROM AUTHOR]

The packing of hard-core particles in contact with their neighbors is considered as the simplest model of disordered particulate media. We formulate the statically determinate problem that allows analytic investigation of the statistical distribution of the contact force magnitude. A toy model of the Boltzmann-type equation for the contact force distribution probability is formulated and studied. An experimentally observed exponential distribution is derived. [ABSTRACT FROM AUTHOR]

Cells and organisms follow aligned structures in their environment, a process that can generate persistent migration paths. Kinetic transport equations are a popular modelling tool for describing biological movements at the mesoscopic level, yet their formulations usually assume a constant turning rate. Here we relax this simplification, extending to include a turning rate that varies according to the anisotropy of a heterogeneous environment. We extend known methods of parabolic and hyperbolic scaling and apply the results to cell movement on micropatterned domains. We show that inclusion of orientation dependence in the turning rate can lead to persistence of motion in an otherwise fully symmetric environment and generate enhanced diffusion in structured domains. [ABSTRACT FROM AUTHOR]

Efficient and accurate numerical approximation of the full Boltzmann equation has been a longstanding challenging problem in kinetic theory. This is mainly due to the high dimensionality of the problem and the complicated collision operator. In this work, we propose a highly efficient adaptive low rank method for the Boltzmann equation, concerning in particular the steady state computation. This method employs the fast Fourier spectral method (for the collision operator) and the dynamical low rank method to obtain computational efficiency. An adaptive strategy is introduced to incorporate the boundary information and control the computational rank in an appropriate way. Using a series of benchmark tests in 1D and 2D, we demonstrate the efficiency and accuracy of the proposed method in comparison to the full tensor grid approach. [ABSTRACT FROM AUTHOR]

This paper is concerned with the spatially homogeneous Boltzmann equation, with the assumption of Maxwellian interaction. We consider initial data that belong to a small neighborhood of the equilibrium, which is a Maxwellian distribution. We prove that the solution remains in another small neighborhood with the same center and converges to this equilibrium exponentially fast, with an explicit quantification. [ABSTRACT FROM AUTHOR]

The mainobject of this article is to revisit a certain double integral involving Kummer’s confluent hypergeometric function 1F1, which arose in the study of the collision terms of the celebrated Boltzmann equation in the kinetic theory of gases. Here, in this article, we propose to investigate some novel extensions and generalizations of this family of double integrals. We also point out some relevant connections of the results, which are presented here, with other related recent developments in the theory and applications of hypergeometric functions. 2020 Mathematics Subject Classifications: 33C15, 33C20, 33E12, 11M35, 33C60, 76P05, 82B40, 82C40 [ABSTRACT FROM AUTHOR]

In this paper we consider the Cauchy problem on the angular cutoff Boltzmann equation near global Maxwillians for soft potentials either in the whole space or in the torus. We establish the existence of global unique mild solutions in the space LpvL∞T Lx with polynomial velocity weights for suitably large p≤∞, whenever for the initial perturbation the weighted Lpv L∞x norm can be arbitrarily large but the L1xLv norm and the defect mass, energy, and entropy are sufficiently small. The proof is based on the local in time existence as well as the uniform a priori estimates via an interplay in Lpv L∞T L∞ and L∞ T L1v. [ABSTRACT FROM AUTHOR]

The dynamics of gases made of particles interacting dissipativelyâ€"known as granular gases â€"can be fully described by the translational and rotational motion of the individual particles; however, most of the results in the field refer to the limit of smooth particles, which implies that the rotational degrees of freedom are suppressed. Here we investigate the opposite limit: we consider a granular gas where the translational degrees of freedom are suppressed, and the key degrees of freedom are rotational. Our results indicate that for many-particle systems of pure rotators collective effects almost completely suppressed. This is in a sharp contrast to granular gases of smooth particles and other conventional matter where the translational degrees of freedom dominate the kinetics. [ABSTRACT FROM AUTHOR]

In this paper, we get the quantitative space-time behavior of the full Boltzmann equation with soft potentials (− 2 < γ < 0) in the close to equilibrium setting, under some velocity decay assumption, but without any Sobolev regularity assumption on the initial data. We find that both the large time and spatial behaviors depend on the velocity decay of the initial data and the exponent γ. The key step in our strategy is to obtain the L ∞ bound of a suitable weighted full Boltzmann equation directly, rather than using Green's function and Duhamel's principle to construct the pointwise structure of the solution as in [25]. This provides a new thinking in the related study. [ABSTRACT FROM AUTHOR]

We present a new accurate approach to solving a class of problems in the theory of rarefied–gas dynamics using a Physics-Informed Neural Networks framework, where the solution of the problem is approximated by the constrained expressions introduced by the Theory of Functional Connections. The constrained expressions are made by a sum of a free function and a functional that always analytically satisfies the equation constraints. The free function used in this work is a Chebyshev neural network trained via the extreme learning machine algorithm. The method is designed to accurately and efficiently solve the linear one-point boundary value problem that arises from the Bhatnagar–Gross–Krook model of the Poiseuille flow between two parallel plates for a wide range of Knudsen numbers. The accuracy of our results is validated via the comparison with the published benchmarks. [ABSTRACT FROM AUTHOR]

In this paper, we address the local well-posedness of the spatially inhomogeneous noncutoff Boltzmann equation when the initial data decays polynomially in the velocity variable. We consider the case of very soft potentials γ + 2s < 0. Our main result completes the picture for local well-posedness in this decay class by removing the restriction γ + 2s > - 3/2 of previous works. Our approach is entirely based on the Carleman decomposition of the collision operator into a lower order term and an integro-differential operator similar to the fractional Laplacian. Interestingly, this yields a very short proof of local well-posedness when γ ∈ (-3, 0] and s ∈ (0, 1/2) in a weighted C¹ space that we include as well. [ABSTRACT FROM AUTHOR]

The Boltzmann equation is a fundamental kinetic equation that describes the dynamics of dilute gas. In this paper we study the regularity of both dynamical and steady Boltzmann equations in a strictly convex domain with the Cercignani-Lampis (C-L) boundary condition. The C-L boundary condition describes the intermediate reflection law between diffuse reflection and specular reflection via two accommodation coefficients. We construct a local weighted C¹ dynamical solution using repeated interaction through the characteristic. When we assume small fluctuation to the wall temperature and accommodation coefficients, we construct weighted C¹ steady solution. [ABSTRACT FROM AUTHOR]

We study the steady Boltzmann equation in half-space, which arises in the Knudsen boundary layer problem, with diffusive reflection boundary conditions. Under certain admissible conditions, we establish the existence of a boundary layer solution for both linear and nonlinear Boltzmann equations in half-space with diffusive reflection boundary condition in L∞x,v when the far-field Mach number of the Maxwellian is zero. The continuity and the special decay of the solution are obtained. The uniqueness is established under some constraint conditions. [ABSTRACT FROM AUTHOR]

We consider neutral particle systems described by moments of a phase-space density and propose a realizability-preserving numerical method to evolve a spectral two-moment model for particles interacting with a background fluid moving with nonrelativistic velocities. The system of nonlinear moment equations, with special relativistic corrections to O (v / c) , expresses a balance between phase-space advection and collisions and includes velocity-dependent terms that account for spatial advection, Doppler shift, and angular aberration. The model is conservative for the correct O (v / c) Eulerian-frame number density and is consistent, to O (v / c) , with Eulerian-frame energy and momentum conservation. This model is closely related to the one promoted by Lowrie et al. [1] and similar to models currently used to study transport phenomena in large-scale simulations of astrophysical environments. The proposed numerical method is designed to preserve moment realizability, which guarantees that the moments correspond to a nonnegative phase-space density. The realizability-preserving scheme consists of the following key components: (i) a strong stability-preserving implicit-explicit (IMEX) time-integration method; (ii) a discontinuous Galerkin (DG) phase-space discretization with carefully constructed numerical fluxes; (iii) a realizability-preserving implicit collision update; and (iv) a realizability-enforcing limiter. In time integration, nonlinearity of the moment model necessitates solution of nonlinear equations, which we formulate as fixed-point problems and solve with tailored iterative solvers that preserve moment realizability with guaranteed global convergence. We also analyze the simultaneous Eulerian-frame number and energy conservation properties of the semi-discrete DG scheme and propose a "spectral redistribution" scheme that promotes Eulerian-frame energy conservation. Through numerical experiments, we demonstrate the accuracy and robustness of this DG-IMEX method and investigate its Eulerian-frame energy conservation properties. • A novel discontinuous Galerkin method is proposed for a radiation transport model. • The model is number conservative and evolves moments of a kinetic distribution. • Relativistic effects are included to account for interactions with a moving fluid. • The proposed method preserves physical realizability of the evolved moments. • Numerical experiments demonstrate the robustness of the proposed method. [ABSTRACT FROM AUTHOR]

• An approach to numerical simulation of shock waves in gas mixtures was proposed. • Software for simulating gas flows based on the Boltzmann equation was developed. • The structure of a shock wave in a binary gas mixture was computed. • The total accuracy of at least 1.4% was achieved. • Simulation of a shock wave propagation through a perforated surface was performed. The paper proposes an approach to numerical simulation of shock waves in rarefied gas mixtures on the basis of direct solution of the Boltzmann kinetic equation. Software for simulating the gas flows was developed. The structure of a shock wave in a binary gas mixture was computed with an accuracy controlled by the computational parameters. The computations were performed for various molecular masses ratios and Mach numbers. The total accuracy of at least 1.4% for the local values of the molecular densities and temperatures of the mixture components was achieved. Numerical simulation of a shock wave propagation through a periodically perforated surface was performed. The distributions of the macroscopic characteristics of the mixture components at various points in time were obtained. Unsteady areas of strong separation of the gas mixture components were discovered. [ABSTRACT FROM AUTHOR]

Motivated by the open problem of large-data global existence for the non-cutoff Boltzmann equation, we introduce a model equation that in some sense disregards the anisotropy of the Boltzmann collision kernel. We refer to this model equation as isotropic Boltzmann by analogy with the isotropic Landau equation introduced by Krieger and Strain (2012) [35]. The collision operator of our isotropic Boltzmann model converges to the isotropic Landau collision operator under a scaling limit that is analogous to the grazing collisions limit connecting (true) Boltzmann with (true) Landau. Our main result is global existence for the isotropic Boltzmann equation in the space homogeneous case, for certain parts of the "very soft potentials" regime in which global existence is unknown for the space homogeneous Boltzmann equation. The proof strategy is inspired by the work of Gualdani and Guillen (2022) [22] on isotropic Landau, and makes use of recent progress on weighted fractional Hardy inequalities. [ABSTRACT FROM AUTHOR]

The direct simulation Monte Carlo (DSMC) method is widely used for numerical solutions of the Boltzmann equation. However, the associated computational cost becomes prohibitive in the near-continuum regime. To address this limitation, the particle-based Fokker–Planck (FP) method has been extensively studied in the past decade. The FP equation, which describes Brownian motion, does not require resolution of the collisional time and length scales. While several monatomic FP models have been proposed, the modeling of diatomic gases within the FP framework has received limited attention. In this paper, we propose a new diatomic kinetic model, named the Fokker–Planck–Master (FPM) model, which can accurately describe energy exchanges between translational-rotational and translational-vibrational modes. The FPM model combines the FP equation to describe the evolution of translational and rotational modes, and the master equation to describe the evolution of the vibrational modes. The numerical test cases include relaxation problems, Couette flows, and hypersonic flows past a vertical flat plate. The results demonstrate that the FPM model shows good agreement with both analytical and DSMC solutions. • A new diatomic kinetic model based on the Fokker–Planck and equations. • The correct Prandtl number, accurate relaxation rates for internal energies, and the H-theorem are included in the model. • The development of a conservative scheme for robust numerical simulations. • The application of the model to rarefied and hypersonic flows. [ABSTRACT FROM AUTHOR]

The dynamics of a rarefied gas in a finite channel with the same temperatures and opposite velocities is a fundamental problem in kinetic theory. The relative motion of the planar boundaries can induce a non-equilibrium state which is referred to as the Couette flow. In this paper, we demonstrate that the unsteady Couette flow for the Boltzmann equation in 3D finite channel time asymptotically converges to the 1D steady state constructed in Duan et al. (2022), we also prove the exponential time decay rate as a byproduct. The validity of the analysis is established for all hard potentials. [ABSTRACT FROM AUTHOR]

The Elo rating system, which was originally proposed by Arpad Elo for chess, has become one of the most important rating systems in sports, economics and gaming. Its original formulation is based on two-player zero-sum games, but it has been adapted for team sports and other settings. In 2015, Junca and Jabin proposed a kinetic version of the Elo model, and showed that under certain assumptions the ratings do converge towards the players' strength. In this paper, we generalize their model to account for variable performance of individual players or teams. We discuss the underlying modelling assumptions, derive the respective formal mean-field model and illustrate the dynamics with computational results. This article is part of the theme issue 'Kinetic exchange models of societies and economies'. [ABSTRACT FROM AUTHOR]

Motivated by recent successes in model-based pre-election polling, we propose a kinetic model for opinion formation which includes voter demographics and socio-economic factors like age, sex, ethnicity, education level, income and other measurable factors like behaviour in previous elections or referenda as a key driver in the opinion formation dynamics. The model is based on Toscani's kinetic opinion formation model (Toscani G. 2006 Kinetic models of opinion formation. Commun. Math. Sci.4, 481–496.) and the leader–follower model of Düring et al. (Düring B. et al. 2009 Boltzmann and Fokker–Planck equations modelling opinion formation in the presence of strong leaders. Proc. R. Soc. A465, 3687–3708.), and leads to a system of coupled Boltzmann-type equations and associated, approximate Fokker–Planck-type systems. Numerical examples using data from general elections in the UK show the effect different demographics have on the opinion formation process and the outcome of elections. This article is part of the theme issue 'Kinetic exchange models of societies and economies'. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new semi-Lagrangian scheme for the polyatomic ellipsoidal BGK model. In order to avoid time step restrictions coming from convection term and small Knudsen number, we combine a semi-Lagrangian approach for the convection term with an implicit treatment for the relaxation term. We show how to explicitly solve the implicit step, thus obtaining an efficient and stable scheme for any Knudsen number. We also derive an explicit error estimate on the convergence of the proposed scheme for every fixed value of the Knudsen number. [ABSTRACT FROM AUTHOR]

The Berry curvature in Weyl semimetal (WSM) induces chiral surface states through Fermi arcs, giving rise to an anomalous intrinsic Hall response. The Weyl semimetals are conducting in the bulk; we study the surface-bulk scattering mechanism and build an original Boltzmann equation for the surface electron transport in Weyl semimetals. We derive a general formula for the anomalous Hall conductivity (AHC) as a function of chemical potential, surface-bulk coupling strength and sample size, which is valid for both diffusive and ballistic limits. The ballistic limit is beyond the Green's function as the ladder approximation deteriorates at small chemical potential, showing unique advantages of our Boltzmann method. Moreover, we investigate the energy dependence of anomalous Hall conductivity and find it non-monotonic with temperature. [ABSTRACT FROM AUTHOR]

Jeans instability is analyzed in an expanding universe within the framework of BGK model of the Boltzmann equation and Poisson equations. The background is characterized by a comoving Maxwellian distribution function and a spacetime Newtonian gravitational potential which satisfy the BGK model of the Boltzmann–Poisson equations without the necessity to invoke "Jeans swindle." The perturbations of the distribution function and Newtonian gravitational potentials from their background states are represented by plane waves of small amplitudes and a differential equation for the density contrast is determined. The density contrast differential equation was solved numerically and it is shown: (i) Jeans instability is characterized by perturbation wavelengths larger than Jeans wavelength where the density contrast grows with time. The growth of the density contrast is less accentuated for the case where the particle collisions are considered due to an energy dissipation; (ii) for perturbation wavelengths smaller than Jeans wavelength the density contrast has an oscillatory behavior in time and the oscillations for the case where the collisions are taken into account fade away in time due to the energy dissipation. [ABSTRACT FROM AUTHOR]