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We propose a kinetic model describing a polyatomic gas undergoing resonant collisions, in which the microscopic internal and kinetic energies are separately conserved during a collision process. This behaviour has been observed in some physical phenomena, for example in the collisions between selectively excited CO 2 molecules. We discuss the model itself, prove the related H-theorem and show that, at the equilibrium, two temperatures are expected. We eventually present a numerical illustration of the model and its main properties. [ABSTRACT FROM AUTHOR]

In this work, we derive a system of Boltzmann-type equations to describe the spread of contact-based infections, such as SARS-CoV-2 virus, at the microscopic scale, that is, by modeling the human-to-human mechanisms of transmission. To this end, we consider two populations, characterized by specific distribution functions, made up of individuals without symptoms (population 1) and infected people with symptoms (population 2). The Boltzmann operators model the interactions between individuals within the same population and among different populations with a probability of transition from one to the other due to contagion or, vice versa, to recovery. In addition, the influence of innate and adaptive immune systems is taken into account. Then, starting from the Boltzmann microscopic description we derive a set of evolution equations for the size and mean state of each population considered. Mathematical properties of such macroscopic equations, as equilibria and their stability, are investigated, and some numerical simulations are performed in order to analyze the ability of our model to reproduce the characteristic features of Covid-19 type pandemics. [ABSTRACT FROM AUTHOR]

At higher altitudes near space shuttles moving at hypersonic speed the air is excited to high temperatures. Then not only mechanical collisions are affecting the gas flow, but also chemical reactions have an impact on such hypersonic flows. In this work we insert chemical reactions, in form of dissociations and associations, in a model for a mixture of mono- and polyatomic (non-reacting) species. More general chemical reactions, e.g., bimolecular ones, can be obtained by instant combinations of the considered reactions. Polyatomicity is here modelled by a continuous internal energy variable and the evolution of the gas is described by a Boltzmann equation. In the Chapman-Enskog process—and related half-space problems—the linearized Boltzmann collision operator plays a central role. Here we extend some important properties of the linearized operator to the considered model with chemical reactions. A compactness result, that the linearized operator can be decomposed into a sum of a positive multiplication operator—the collision frequency—and a compact integral operator, is obtained. The terms of the integral operator are shown to be (at least) uniform limits of Hilbert-Schmidt integral operators and, thereby, compact operators. Self-adjointness of the linearized operator follows as a direct consequence. Also, bounds on—including coercivity of—the collision frequency is obtained for hard sphere, as well as hard potentials with cutoff, like models. As consequence, Fredholmness as well as the domain of the linearized operator are obtained. [ABSTRACT FROM AUTHOR]

A novel ES-BGK-based model of non-polytropic rarefied gases in the framework of kinetic theory is presented. Key features of this model are: an internal state density function depending only on the microscopic energy of internal modes (avoiding the dependence on temperature seen in previous reference studies); full compliance with the H-theorem; feasibility of the closure of the system of moment equations based on the maximum entropy principle, following the well-established procedure of rational extended thermodynamics. The structure of planar shock waves in carbon dioxide (CO 2 ) obtained with the present model is in general good agreement with that of previous results, except for the computed internal temperature profile, which is qualitatively different with respect to the results obtained in previous studies, showing here a consistently monotonic behavior across the shock structure, rather than the non monotonic behavior previously found. [ABSTRACT FROM AUTHOR]

A model polydisperse fluid represents many real fluids, such as colloidal suspensions and polymer solutions. In this study, we consider a concentrated size-polydisperse hard-sphere fluid with size derived from two different distribution functions, namely, uniform and Gaussian, and explore the effect of polydispersity and mass scaling on the transport properties in general. A simple analytical solution based on the Boltzmann transport equation is also presented (together with the solution using Chapman–Enskog (CE) method) using which various transport coefficients are obtained. The central idea of our approach is the realisation that, in polydisperse systems, the collision scattering cross-section is proportional to a random variable z which is equal to the sum of two random variables σ i and σ j (representing particle diameters), and the distribution of z can be written as the convolution of the two distributions P (σ i) and P (σ j) . In this work, we provide expressions for transport coefficients expressed as an explicit function of polydispersity index, δ , and their dependence on the nature of particle size distribution and mass scaling is explored. It is observed that in the low polydispersity limit, the transport coefficients are found to be insensitive to the type of size distribution functions considered. The analytical results (for diffusion coefficients and thermal conductivity) obtained using the CE method and our simple analytical approach agree well with the simulation. However, for shear viscosity, our analytical approach agrees for δ ≤ 20 % , while it agrees up to δ ≈ 40 % with the result obtained using the CE method (in the limit δ → 0 ). Interestingly, the effect of scaling mass (i.e., mass proportional to the particle size and thus a random variable) produces no significant qualitative difference. [ABSTRACT FROM AUTHOR]

Unlike the single species gases, the transport coefficients such as Fick, Soret, Dufour coefficients arise in the hydrodynamic limit of multi-species gas mixtures. To the best of the authors’ knowledge, no multi-component relaxational models is reported that produces all these values correctly. In this paper, we establish the existence of unique stationary mild solutions to the BGK models for gas mixtures which produces the correct Fick coefficients in the Navier–Stokes limit for inert gases (Brull in Eur J Mech B 33:74–86, 2012), and for reactive gases (Brull and Schneider in Commun Math Sci 12(7):1199–1223, 2014) in a unified manner. [ABSTRACT FROM AUTHOR]

We deal with McKean–Vlasov and Boltzmann-type jump equations. This means that the coefficients of the stochastic equation depend on the law of the solution, and the equation is driven by a Poisson point measure with intensity measure which depends on the law of the solution as well. Alfonsi and Bally (Construction of Boltzmann and McKean Vlasov type flows (the sewing lemma approach), 2021, arXiv:2105.12677) have proved that under some suitable conditions, the solution X t of such equation exists and is unique. One also proves that X t is the probabilistic interpretation of an analytical weak equation. Moreover, the Euler scheme X t P of this equation converges to X t in Wasserstein distance. In this paper, under more restrictive assumptions, we show that the Euler scheme X t P converges to X t in total variation distance and X t has a smooth density (which is a function solution of the analytical weak equation). On the other hand, in view of simulation, we use a truncated Euler scheme X t P , M which has a finite numbers of jumps in any compact interval. We prove that X t P , M also converges to X t in total variation distance. Finally, we give an algorithm based on a particle system associated with X t P , M in order to approximate the density of the law of X t . Complete estimates of the error are obtained. [ABSTRACT FROM AUTHOR]

A kinetic model of binary interactions is developed to describe an exchange market consisting of two groups (A and B) of manufacturing enterprises and two types of production factors (capital and labor). The relationship between production factors and output satisfies the Cobb-Douglas production function. The objective of the manufacturing enterprise is to obtain maximum profits. In the multi-agent market, manufacturing enterprises exchange surplus production factors for insufficient production factors. We assume that manufacturing enterprises in group A put all of the production factors into the market, while the manufacturing enterprises in group B have the priority of selecting a suitable percentage of production factors for exchange. A system of linear kinetic equations is developed for two groups of manufacturing enterprises to characterize the evolution of the quantity of exchanged production factors. The effect of various strategies on the price of production factors and the profit of manufacturing enterprises is explored. The findings reveal that the trading strategy of group B significantly impacts the price of production factors and effectively improves the profit under certain conditions. Furthermore, a system of Boltzmann equations for the probability distribution of production factors is presented to describe the price issues. Simulation experiments illustrate how the trading strategy and the elasticity coefficient of output influence the final price of production factors. [ABSTRACT FROM AUTHOR]

Unlike the case for classical particles, the literature on BGK type models for relativistic gas mixture is extremely limited. There are a few results in which such relativistic BGK models for gas mixture are employed to compute transport coefficients. However, to the best knowledge of authors, relativistic BGK models for gas mixtures with complete presentation of the relaxation operators are missing in the literature. In this paper, we fill this gap by suggesting a BGK model for relativistic gas mixtures for which the existence of each equilibrium coefficients in the relaxation operator is rigorously guaranteed in a way that all the essential physical properties are satisfied such as the conservation laws, the H-theorem, the capturing of the correct equilibrium state, the indifferentiability principle, and the recovery of the classical BGK model in the Newtonian limit. [ABSTRACT FROM AUTHOR]

A half-space problem of a linear kinetic equation for gas molecules physisorbed close to a solid surface, relevant to a kinetic model of gas–surface interaction and derived by Aoki et al. (Phys. Rev. E 106:035306, 2022), is considered. The equation contains a confinement potential in the vicinity of the solid surface and an interaction term between gas molecules and phonons. It is proved that a unique solution exists when the incoming molecular flux is specified at infinity. This validates the natural observation that the half-space problem serves as the boundary condition for the Boltzmann equation. It is also proved that the sequence of approximate solutions used for the existence proof converges exponentially fast. In addition, numerical results showing the details of the solution to the half-space problem are presented. [ABSTRACT FROM AUTHOR]

We establish a connection between the relative Classical entropy and the relative Fermi–Dirac entropy, allowing to transpose, in the context of the Boltzmann or Landau equation, any entropy–entropy production inequality from one case to the other; therefore providing entropy–entropy production inequalities for the Boltzmann–Fermi–Dirac operator, similar to the ones of the Classical Boltzmann operator. We also provide a generalized version of the Csiszár–Kullback–Pinsker inequality to weighted L p norms, 1 ≤ p ≤ 2 and a wide class of entropies. [ABSTRACT FROM AUTHOR]

The main goal of this work is to check the hypothesis that the well-known N-particle statistical algorithm yields a solution estimate for the nonlinear Boltzmann equation with an error. For this purpose, practically important optimal relations between and the number of sample estimate values are determined. Numerical results for a problem with a known solution confirm that the formulated estimates and conclusions are satisfactory. [ABSTRACT FROM AUTHOR]

The present paper provides some arguments surrounding the controversies of boundary vorticity creation for incompressible viscous flow. Our discussion shows that boundary vorticity creation must be a viscous physical process. Importantly, it is emphasized that not only viscosity is responsible for spreading the generated vorticity out of the boundary, but also must be involved in the process of boundary vorticity creation to realize the no-slip boundary condition. Lyman flux is a part of the boundary vorticity flux under the Lighthill-Panton-Wu’s definition, which provides an alternative interpretation of boundary vorticity dynamics. Different from the existing inviscid interpretation, we insist that viscosity is fully indispensable for generating the Lyman flux through the tangential boundary acceleration and surface pressure gradient where the acceleration adherence is shown to be derived from the velocity adherence directly. Through a detailed discussion on interfacial vortex sheet and slip velocity, it is revealed that the velocity jump across the material interfacial vortex sheet (a thin viscous shear layer as the fluid viscosity approaches to zero) is physically different from that across the interface. In addition, it is shown that the formation of surface pressure distribution is an inviscid process while the subsequent boundary vorticity generation by the tangential pressure gradient must be a viscid process (contributed by the non-equilibrium particle relaxation effect). These two processes are separated by a non-zero time increment with the same order as the particle relaxation time. Then, the hydrodynamic limit of the Boltzmann equation is revisited to elaborate the crucial roles of viscosity for both the continuum and slip regimes. For continuum flows with a no-slip boundary, the physical carrier of the slip velocity in the inviscid Euler theory originates from the produced vorticity concentrated in the thin material vortex sheet. Interestingly, we find that Lyman flux must be a viscous boundary flux even for a slip boundary where the implicit viscid mechanism is attributed to the non-continuum effect hidden in the Knudsen layer. The present exploration suggests that a complete physical picture including the boundary vorticity creation and the formation of airfoil circulation should be built upon the viscous flow theory. [ABSTRACT FROM AUTHOR]

The linearized Boltzmann collision operator has a central role in many important applications of the Boltzmann equation. Recently some important classical properties of the linearized collision operator for monatomic single species were extended to multicomponent monatomic gases and polyatomic single species. For multicomponent polyatomic gases, the case where the polyatomicity is modelled by a discrete internal energy variable was considered lately. Here we consider the corresponding case for a continuous internal energy variable. Compactness results, stating that the linearized operator can be decomposed into a sum of a positive multiplication operator, the collision frequency, and a compact operator, bringing e.g., self-adjointness, is extended from the classical result for monatomic single species, under reasonable assumptions on the collision kernel. With a probabilistic formulation of the collision operator as a starting point, the compactness property is shown by a decomposition, such that the terms are, or at least are uniform limits of, Hilbert–Schmidt integral operators and therefore are compact operators. Moreover, bounds on—including coercivity of—the collision frequency are obtained for hard sphere like, as well as hard potentials with cutoff like, models, from which Fredholmness of the linearized collision operator follows, as well as its domain. [ABSTRACT FROM AUTHOR]

We consider a Friedmann-Lemaître-Robertson-Walker physical metric g, an auxiliary metric q with a relativistic charged and colliding plasma in presence of a massive scalar field in Eddington-inspired-Born-Infeld theory of gravity. We first derive a governing system of second order nonlinear partial differential equations. By a judicious change of variables, we manage to build a system of partial differential equations of the first order equivalent to the system of the second order previously found. Then, by the method of characteristics applied to the Boltzmann equation which is a first order hyperbolic equation having as unknown the distribution function f, we construct an iterated sequence and prove the existence and uniqueness of a local solution on a positive time interval [0, T). Then, by the continuation criterion and under certain assumptions of smallness on the initial data, the Eddington parameter k and the dimensionless parameter λ , we show that this local solution is global in time. [ABSTRACT FROM AUTHOR]

For the Boltzmann equation with cutoff hard potentials, we construct the unique global solution converging with an exponential rate in large time to global Maxwellians not only for the specular reflection boundary condition with the bounded convex C 3 domain but also for a class of large amplitude initial data where the L ∞ norm with a suitable velocity weight can be arbitrarily large but the relative entropy needs to be small. A key point in the proof is to introduce a delicate triple nonlinear iterative process of estimating the gain terms of nonlinear collision operator. [ABSTRACT FROM AUTHOR]

The paper analyzes the uniform convergence in terms of the parameters of the scattering angle function of interacting molecules. This function is a key function from molecular kinetic theory and is used in computational technologies for determining the media transfer coefficients by the Chapman–Enskog method. The analysis will make it possible to reasonably determine the calculated values of transfer coefficients in gas mixtures in two-level numerical modeling of gas dynamics problems, which combine a description of macroprocesses based on macro-equations of heat and mass transfer with a micro description based on the Boltzmann equation. [ABSTRACT FROM AUTHOR]

The viscous and thermal velocity slip coefficients of various monatomic gases are computed via the linearized classical Boltzmann equation, with ab initio potential, subject to Maxwell and Cercignani–Lampis boundary conditions. Both classical and quantum interatomic interactions are considered. Comparisons with hard sphere and Lennard–Jones potentials, as well as the linearized Shakhov model are performed. The produced database is dense, covers the whole range of the accommodation coefficients and is of high accuracy. Using symbolic regression, very accurate closed form expressions of the slip coefficients, easily implemented in the future computational and experimental works, are deduced. The thermal slip coefficient depends, much more than the viscous one, on the intermolecular potential. For example, in the case of diffuse scattering, the relative differences in the viscous slip coefficient data between HS and AI potentials are less than 4%, whilst the corresponding ones in the thermal slip coefficient data are about 6% for He, reaching 15% for Xe. Quantum effects are considered for He, at temperatures 1–104 K to deduce that deviations from the classical behaviour are not important in the viscous slip coefficient, but they become important in the thermal slip coefficient, where the differences between the classical and quantum approaches reach 15% at 1 K. The computational effort of solving the linearized Boltzmann equation with ab initio and Lennard–Jones potentials is the same. Since ab initio potentials do not contain any adjustable parameters, it is recommended to use them at any temperature. [ABSTRACT FROM AUTHOR]

The ES-BGK model is a generalized version of the BGK model of the Boltzmann equation designed to reproduce the correct Prandtl number in the Navier–Stokes limit. We prove the existence and uniqueness of mild solutions to the ES-BGK model for polyatomic molecules when the total energy of the initial data is not necessarily finite. [ABSTRACT FROM AUTHOR]

In this article we study a kinetic model which describes the interaction between a gas and radiation. Specifically, we consider a scaling limit in which the interaction between the gas and the photons takes place much faster than the collisions between the gas molecules themselves. We prove in the homogeneous case that the solutions of the limit problem solve a kinetic equation for which a well-posedness theory is considered. The proof of the convergence to a new kinetic equation is obtained analyzing the dynamics of the gas–photon system near the slow manifold of steady states. [ABSTRACT FROM AUTHOR]

We consider the Lorentz gas in a distribution of scatterers which microscopically converges to a periodic distribution, and prove that the Lorentz gas in the low density limit satisfies a linear Boltzmann equation. This is in contrast with the periodic Lorentz gas, which does not satisfy the Boltzmann equation in the limit. [ABSTRACT FROM AUTHOR]

Numerically solving the Boltzmann equation is computationally expensive in part due to the number of variables the distribution function depends upon. Another contributor to the complexity of the Boltzmann Equation is the quadratic collision operator describing changes in the distribution function due to colliding particle pairs. Solving it as efficiently as possible has been a topic of recent research, e.g. Cai and Torrilhon (Phys Fluids 31(12):126105, 2019. ), Wang and Cai (J Comput Phys 397:108815, 2019. ), Cai et al. (Comput Fluids 200:104456, 2020. ). In this paper we exploit results from representation theory to find a very efficient algorithm both in terms of memory and computational time for the evaluation of the quadratic collision operator. With this novel approach we are also able to provide a meaningful interpretation of its structure. [ABSTRACT FROM AUTHOR]

In this work, a study is carried out on the solution of large linear systems of algebraic equations relevant to establish a general solution, based on a spectral formulation, to the discrete ordinates approximation of the two-dimensional particle transport equation in Cartesian geometry. The number of discrete ordinates (discrete directions of the particles) is determined by the order of the quadrature scheme on the unity sphere used to approximate the integral term of the linear Boltzmann equation (also called the transport equation). A nodal technique is applied to the discrete ordinates approximation of this equation, yielding to a system of first order ordinary differential equations for average unknowns along the directions x and y. The developed formulation is explicit for the spatial variables. The order of the linear system is defined by the number of discrete directions as well as the number of the spatial nodes. High-quality solutions are expected as both, the number of discrete directions and the refinement of the spatial mesh, increase. Here, the performance of direct and iterative methods, for the solution of the linear systems, are discussed, along with domain decomposition techniques and parallel implementation. Alternative arrangements in the configuration of the equations allowed solutions to higher order systems. A dependence on the type of the quadrature scheme as well as the class of problems to be solved (neutron or radiation problems, for instance) directly affect the final choice of the numerical algorithm. [ABSTRACT FROM AUTHOR]

In this note we study Boltzmann’s collision kernel for inverse power law interactions U s (r) = 1 / r s - 1 for s > 2 in dimension d = 3 . We prove the limit of the non-cutoff kernel to the hard-sphere kernel and give precise asymptotic formulas of the singular layer near θ ≃ 0 in the limit s → ∞ . Consequently, we show that solutions to the homogeneous Boltzmann equation converge to the respective solutions. [ABSTRACT FROM AUTHOR]

We prove the global existence of the non-negative unique mild solution for the Cauchy problem of the cutoff Boltzmann equation for soft potential model - 1 ≤ γ < 0 with the small initial data in three dimensional space. Thus our result fixes the gap for the case γ = - 1 in three dimensional space in the authors’ previous work (He and Jiang in J Stat Phys 168(2):470–481, 2017) where the estimate for the loss term was improperly used. The other gap in He and Jiang (2017) for the case γ = 0 in two dimensional space is recently fixed by Chen et al. (Arch Ration Mech Anal 240:327–381, 2021). The initial data f 0 is non-negative and satisfies that ‖ ⟨ v ⟩ ℓ γ f 0 (x , v) ‖ L x , v 3 ≪ 1 and ‖ ⟨ v ⟩ ℓ γ f 0 ‖ L x , v 15 / 8 < ∞ where ℓ γ = 0 when γ = - 1 and ℓ γ = (1 + γ) + when - 1 < γ < 0 . We also show that the solution scatters with respect to the kinetic transport operator. The novel contribution of this work lies in the exploration of the symmetric property of the gain term in terms of weighted estimate. It is the key ingredient for solving the model - 1 < γ < 0 when applying the Strichartz estimates. [ABSTRACT FROM AUTHOR]

We consider a modified Boltzmann equation which contains, together with the collision operator, an additional drift term which is characterized by a matrix A. Furthermore, we consider a Maxwell gas, where the collision kernel has an angular singularity. Such an equation is used in the study of homoenergetic solutions to the Boltzmann equation. Under smallness assumptions on the drift term, we prove that the longtime asymptotics is given by self-similar solutions. We work in the framework of measure-valued solutions with finite moments of order p > 2 and show existence, uniqueness and stability of these self-similar solutions for sufficiently small A. Furthermore, we prove that they have finite moments of arbitrary order if A is small enough. In addition, the singular collision operator allows to prove smoothness of these self-similar solutions. Finally, we study the asymptotics of particular homoenergetic solutions. This extends previous results from the cutoff case to non-cutoff Maxwell gases. [ABSTRACT FROM AUTHOR]

The influence of reactive cross sections upon the reaction rate of the simple chemical reaction A + A → p r o d u c t s — where the concentrations of the products are negligible in comparison with those of the reagents — is determined from the Boltzmann equation. This kind of reaction is known as “slow” reaction, since the time between the successive reactions is considered larger than the time between the elastic collisions. The internal degrees of freedom of the molecules of the gas are not taken into account. Two models for the reactive cross section are examined: (a) the line-of-centers energy model and (b) the modified line-of-centers energy model. Explicit expressions for the reaction rate as function of the activation energy for both models are given and it is shown how the reactant fields of particle number density and temperature evolve with time when the chemical reaction advances. [ABSTRACT FROM AUTHOR]

We study the self-similar solutions and related questions for the modified Boltzmann equation. This equation formally coincides with the classical spatially homogeneous Boltzmann equation in the presence of an artificial force term proportional to a matrix A. The modification is connected with applications to homoenergetic solutions of the spatially inhomogeneous Boltzmann equation. Our study is restricted to the case of Maxwell-type interactions. We investigate existence and uniqueness of self-similar solutions and their role as attractors for large values of time. Similar questions were studied recently under assumption of sufficient smallness of norm ‖ A ‖ without explicit estimates of that smallness. In this paper we fill this gap and prove, in particular, that all important facts related to self-similar solutions remain valid for moderately small values ‖ A ‖ = O (10 - 1) in appropriate dimensionless units. [ABSTRACT FROM AUTHOR]

In this paper we first deduce the estimates on the linearized Landau operator with Coulomb potential and then analyze its spectrum structure by using semigroup theory and linear operator perturbation theory. Based on these estimates, we give the precise time decay rate estimates on the semigroup generated by the linearized Landau operator so that the optimal time decay rates of the nonlinear Landau equation follow. In addition, we present a similar result for the non-angular cutoff Boltzmann equation with soft potentials. [ABSTRACT FROM AUTHOR]

The linearized Boltzmann collision operator appears in many important applications of the Boltzmann equation. Therefore, knowing its main properties is of great interest. This work extends some classical results for the linearized Boltzmann collision operator for monatomic single species to the case of polyatomic single species, while also reviewing corresponding results for multicomponent mixtures of monatomic species. The polyatomicity is modeled by a discrete internal energy variable, that can take a finite number of (given) different values. Results concerning the linearized Boltzmann collision operator being a nonnegative symmetric operator with a finite-dimensional kernel are reviewed. A compactness result, saying that the linearized operator can be decomposed into a sum of a positive multiplication operator, the collision frequency, and a compact operator, bringing e.g., self-adjointness, is extended from the classical result for monatomic single species, under reasonable assumptions on the collision kernel. With a probabilistic formulation of the collision operator as a starting point, the compactness property is shown by a splitting, such that the terms can be shown to be, or be the uniform limit of, Hilbert-Schmidt integral operators and as such being compact operators. Moreover, bounds on - including coercivity of - the collision frequency are obtained for a hard sphere like model, from which Fredholmness of the linearized collision operator follows, as well as its domain. [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]

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]

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]

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]

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]

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]

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]

In this paper, we study a class of kinetic equations describing radiative transfer in gases which include also the interaction of the molecules of the gas with themselves. We discuss several scaling limits and introduce some Euler-like systems coupled with radiation as an aftermath of specific scaling limits. We consider scaling limits in which local thermodynamic equilibrium (LTE) holds, as well as situations in which this assumption fails (non-LTE). The structure of the equations describing the gas-radiation system is very different in the LTE and non-LTE cases. We prove the existence of stationary solutions with zero velocities to the resulting limit models in the LTE case. We also prove the non-existence of a stationary state with zero velocities in a non-LTE case. [ABSTRACT FROM AUTHOR]

In this article we study the Gevrey regularization effect for the spatially inhomogeneous Boltzmann equation without angular cutoff. This equation is partially elliptic in the velocity direction and degenerates in the spatial variable. We consider the nonlinear Cauchy problem for the fluctuation around the Maxwellian distribution and prove that any solution with mild regularity will become smooth in the Gevrey class at positive time with the Gevrey index depending on the angular singularity. Our proof relies on the symbolic calculus for the collision operator and the global subelliptic estimate for the Cauchy problem of the linearized Boltzmann operator. [ABSTRACT FROM AUTHOR]

This work proves the global existence to Boltzmann equation in the whole space with very soft potential γ ∈ [ 0 , d) and angular cutoff, in the framework of small perturbation of equilibrium state. In this article, we generalize the estimate on linearized collision operator L to the case of very soft potential and obtain the spectrum structure of the linearized Boltzmann operator correspondingly. The global classic solution can be derived by the method of strongly continuous semigroup. For soft potential, the linearized Boltzmann operator could not give spectral gap; hence, we have to consider a weighted velocity space in order to obtain algebraic decay in time. [ABSTRACT FROM AUTHOR]

The goal of this work is to build up the capability of quasi-particle simulation (QuiPS), a novel flow solver, such that it can adequately model the rarefied portion of an atmospheric reentry trajectory. Direct simulation Monte Carlo (DSMC) is the conventional solver for such conditions, but struggles to resolve transient flows, trace species, and high-level internal energy states due to stochastic noise. Quasi-particle simulation (QuiPS) is a novel Boltzmann solver that describes a system with a discretized, truncated velocity distribution function. The resulting fixed-velocity, variable weight quasi-particles enable smooth variation of macroscopic properties. The distribution function description enables the use of a variance-reduced collision model, greatly minimizing expense near equilibrium. This work presents the addition of a neutral air chemistry model to QuiPS and some demonstrative 0D simulations. The explicit representation of internal distributions in QuiPS reveals some of the flaws in existing physics models. Variance reduction, a key feature of QuiPS, can greatly reduce expense of multi-dimensional calculations, but is only cheaper when the gas composition is near chemical equilibrium. [ABSTRACT FROM AUTHOR]

We discuss the numerical aspects of the Boltzmann transport equation (BE) for electrons in semiconductor devices, which is stabilized by Godunov's scheme. The k-space is discretized with a grid based on the total energy to suppress spurious diffusion in the stationary case. Band structures of arbitrary shape can be handled. In the stationary case, the discrete BE yields always nonnegative distribution functions and the corresponding system matrix has only eigenvalues with positive real parts (diagonally dominant matrix) resulting in an excellent numerical stability. In the transient case, this property yields an upper limit for the time step ensuring the stability of the CPU-efficient forward Euler scheme and a positive distribution function. Similar to the Monte-Carlo (MC) method, the discrete BE can be solved in time together with the Poisson equation (PE), where the time steps for the PE are split into shorter time steps for the BE, which can be performed at minor additional computational cost. Thus, similar to the MC method, the transient approach is matrix-free and the solution of memory and CPU intensive large systems of linear equations is avoided. The numerical properties of the approach are demonstrated for a silicon nanowire NMOSFET, for which the stationary I–V characteristics, small-signal admittance parameters and the switching behavior are simulated with and without strong scattering. The spurious damping introduced by Godunov's (upwind) scheme is found to be negligible in the technically relevant frequency range. The inherent asymmetry of the upwind scheme results in an error for very strong scattering that can be alleviated by a finer grid in transport direction. [ABSTRACT FROM AUTHOR]

Can Brownian motion arise from a deterministic system of particles? This paper addresses this question by analysing the derivation of Brownian motion as the limit of a deterministic hard-spheres gas with Lanford’s theorem. In particular, we examine the role of the Boltzmann-Grad limit in the loss of memory of the deterministic system and compare this derivation and the derivation of Brownian motion with the Langevin equation. [ABSTRACT FROM AUTHOR]

The kinetics of formation of silver nanoparticles using trisodium citrate as the reducing agent was studied in order to evaluate the rate constants and the rate expression. The inability to measure the concentration of the reactant (precursor silver salt) at millimolar concentrations using conventional spectrophotometric techniques renders the studies of kinetics cumbersome. An attempt was made to study the kinetics of this reaction by measuring the concentration of silver nanoparticles instead of the silver ions as a function of time. The initial concentration of the reducing agent (trisodium citrate) was taken to be nearly 20 times that of the initial concentration of the silver ions. Hence, the reaction could be modelled as pseudo-first-order kinetics, considering the bimolecular nature of the reaction. The final second-order rate constant was evaluated using integral method of analysis as 0.254 l (g min)–1. The key steps in the formation of silver nanoparticles (i.e., reduction, nucleation, growth and saturation) were modelled as a sigmoidal plot using Boltzmann equation. A very good fit of experimental data (R2 ≈ 0.99) was observed with the model. [ABSTRACT FROM AUTHOR]

A technique is considered for modeling the electrophysical characteristics of silicon ultrathin field-effect nanotransistors with a fully enclosed gate, the channel lengths of which are in the sub-20 nm range. The technique is a combination of the 3D Poisson, 2D Schrödinger, and 1D Boltzmann equations, provided that their solutions are self-consistent. To achieve this aim, the scattering model is simplified using the relaxation time approximation. Relaxation time linearization allows us to obtain a direct solution to the 1D transport Boltzmann equation in a very efficient way. The prototypes of nanotransistors with topological parameters close to the scaling limit in the range of control voltages up to 0.4 V are analyzed. The simulation results fully illustrate the unique electrophysical properties of the considered transistor architecture. This is a full-fledged operation at a low supply voltage and ideal sub-threshold characteristics in a wide range of topological parameters. [ABSTRACT FROM AUTHOR]

We propose a hybrid method for the multi-scale kinetic equations in the framework of the hyperbolic moment method (Cai and Li in SIAM J Sci Comput 32(5):2875–2907, 2010). In this method, the fourth order moment system is chosen as the governing equations in the fluid region, while the hyperbolic moment system with arbitrary order is chosen as the governing equations in the kinetic region. When transiting from the fluid regime to the kinetic regime, the maximum entropy principle is adopted to reconstruct the kinetic distribution function, so that the information in the fluid region can be utilized thoroughly. Moreover, only one uniform set of numerical scheme is needed for both the fluid and kinetic regions. Numerical tests validate this new hybrid method. [ABSTRACT FROM AUTHOR]

We complete the investigation of the feasibility in principle to close the moment system for the kinetic equation via invariant relations obtained by group analysis methods. For the one-dimensional equation, we obtain an affirmative answer, find an invariant closure of the moment system, and reveal several new phenomena. [ABSTRACT FROM AUTHOR]

A physically clear probabilistic model of a gas from hard spheres is considered using the theory of random processes and in terms of the classical kinetic theory for the densities of distribution functions in the phase space: from the system of nonlinear stochastic differential equations (SDEs), first the generalized and then the random and nonrandom Boltzmann integro-differential equations are derived, taking into account the correlations and fluctuations. The main feature of the original model is the random nature of the intensity of the jump measure and its dependence on the process itself. For the sake of completeness, we briefly recall the transition to increasingly rough approximations in accordance with a decrease in the dimensionless parameter, the Knudsen number. As a result, stochastic and nonrandom macroscopic equations are obtained that differ from the system of Navier–Stokes equations or systems of quasi-gas dynamics. The key difference of this derivation is a more accurate averaging over the velocity due to the analytical solution of the SDE with respect to the Wiener measure, in the form of which the intermediate meso-model in the phase space is presented. This approach differs significantly from the traditional one, which uses not the random process itself, but its distribution function. The emphasis is on the transparency of the assumptions when moving from one level of detail to another, rather than on numerical experiments, which contain additional approximation errors. [ABSTRACT FROM AUTHOR]

We study the boundary value problem of two stationary BGK-type models—the BGK model for fast chemical reaction and the BGK model for slow chemical reaction—and provide a unified argument to establish the existence and uniqueness of stationary flows of reactive BGK models in a slab. For both models, the main difficulty arise in the uniform control of the auxiliary parameters from above and below, since, unlike the BGK models for non-reactive gases, the auxiliary parameters for the reactive BGK models are defined through highly nonlinear relations. To overcome this difficulty, we introduce several nonlinear functionals that capture essential structures of such nonlinear relations such as the monotonicity in specific variables, that enable one to derive necessary estimates for the auxiliary parameters. [ABSTRACT FROM AUTHOR]