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We show that the spatially homogeneous Boltzmann equation evolves as the gradient flow of the entropy with respect to a suitable geometry on the space of probability measures which takes the collision process into account. This gradient flow structure allows to give a new proof for the convergence of Kac's random walk to the homogeneous Boltzmann equation, exploiting the stability of gradient flows. [ABSTRACT FROM AUTHOR]

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]

The Boltzmann equation is a typical example of partially dissipative equations, where the linearized collision operator is positive definite with respect to the microscopic part and the dissipation of the hydrodynamic part is discovered from the coupling structure between the transport operator and the linearized collision operator. Guo and Wang (Comm. Partial Differential Equations, 37, 2012) developed a general energy method for proving the optimal time decay rates of the solution to such type of equations in the whole space; however, the decay rate of the highest order spatial derivatives of the solution is not optimal. In this paper, by incorporating the high‐low frequency decomposition in the energy estimates, both linearly and nonlinearly, we prove the optimal decay rates of any high order spatial derivatives of the low frequency part of the solution to the Boltzmann equation and the almost exponential decay rate of the high frequency part, which imply in particular the optimal decay rate of the highest order spatial derivatives of the solution. Moreover, the velocity‐weighted assumption of the initial data required in Guo and Wang (Comm. Partial Differential Equations, 37, 2012) is removed by capturing the time‐weighted dissipation estimates via the time‐weighted energy method. The method can be applied to the compressible Navier–Stokes equations and many partially dissipative equations in kinetic theory and fluid dynamics. [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]

In this work, we study the formation of consensus in a hierarchical population. We derive the corresponding kinetic equations, and analyze the long time behavior of their solutions for the case of finite number of hierarchical levels, obtaining explicit formula for the consensus opinion. [ABSTRACT FROM AUTHOR]

The ellipsoidal BGK model (ES-BGK) is a generalized version of the BGK model where the local Maxwellian in the relaxation operator of the BGK model is extended to an ellipsoidal Gaussian with a parameter -1/2≤ν>1, so that the correct Prandtl number can be computed in the Navier--Stokes limit. In this work, we consider steady rarefied flows arising from the evaporation and condensation process between two parallel condensed phases, which is formulated in this paper as the existence problem of stationary solutions to the ES-BGK model in a bounded interval with the mixed boundary conditions. One of the key difficulties arises in the uniform control of the temperature tensor from below. In the noncritical case (-1/2≤ν>1), we utilize the property that the temperature tensor is equivalent to the temperature. In the critical case, (ν = 1/2), where such equivalence relation breaks down, we observe that the size of bulk velocity in the x direction can be controlled by the discrepancy of boundary flux, which enables one to bound the temperature tensor from below. [ABSTRACT FROM AUTHOR]

The Boltzmann equation with multiple-relaxation-time (MRT) collision operators has been widely employed in kinetic theory to describe the behavior of gases and liquids at the macro-level. Given the successful development of deep learning and the availability of data analytic tools, it is a feasible idea to try to solve the Boltzmann-MRT equation using a neural network-based method. Based on the canonical polyadic decomposition, a new physics-informed neural network describing the Boltzmann-MRT equation, named the network for MRT collision (NMRT), is proposed in this paper for solving the Boltzmann-MRT equation. The method of tensor decomposition in the Boltzmann-MRT equation is utilized to combine the collision matrix with discrete distribution functions within the moment space. Multiscale modeling is adopted to accelerate the convergence of high frequencies for the equations. The micro–macro decomposition method is applied to improve learning efficiency. The problem-dependent loss function is proposed to balance the weight of the function for different conditions at different velocities. These strategies will greatly improve the accuracy of the network. The numerical experiments are tested, including the advection–diffusion problem and the wave propagation problem. The results of the numerical simulation show that the network-based method can obtain a measure of accuracy at O 10 − 3 . [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]

Boundary effects play an important role in the study of hydrodynamic limits in the Boltzmann theory. We justify rigorously the validity of the hydrodynamic limit from the Boltzmann equation of soft potentials to the compressible Euler equations by the Hilbert expansion with multi-scales. Specifically, the Boltzmann solutions are expanded into three parts: interior part, viscous boundary layer and Knudsen boundary layer. Due to the weak effect of collision frequency of soft potentials, new difficulty arises when tackling the existence of Knudsen layer solutions with space decay rate, which has been overcome under some constraint conditions and losing velocity weight arguments. [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]

In this paper, we prove the compressible Euler limit from the Boltzmann equation with hard sphere collisional kernel and complete diffusive boundary condition in half-space by employing the Hilbert expansion which includes interior and Knudsen layers. This rigorously justifies the corresponding formal analysis in Sone's book [ Molecular gas dynamics , Birkhäuser Boston, Inc., Boston, MA, 2007] in the context of short time smooth solutions, and also generalizes the classic Caflisch's result [Comm. Pure Appl. Math. 33 (1980), pp. 651–666] to initial-boundary problem case. [ABSTRACT FROM AUTHOR]

In this work, we explore the Boltzmann equation with Fermi-Dirac statistics (briefly, BFD equation), which models the dilute gases when quantum effects are taking into consideration. Specifically, we firstly establish the uniform energy estimate and construct the global-in-time classical solutions of the scaled BFD equations under small assumption on the initial data. Then we use this uniform estimate to derive the acoustic limit from the scaled BFD equations within the context of classical solutions. Compared with our companion article Jiang and Zhou (2024) [23] , our uniform estimate in Knudsen number in this paper is crucial and hard to be obtained for the compressible Euler scaling and the acoustic limit is global-in-time, rather than almost global-in-time. [ABSTRACT FROM AUTHOR]

In this work, we generalize Kac's original many-particle binary stochastic model to derive a space homogeneous Boltzmann equation that includes a linear combination of higher-order collisional terms. First, we prove an abstract theorem about convergence from a finite hierarchy to an infinite hierarchy of coupled equations. We apply this convergence theorem on hierarchies for marginals corresponding to the generalized Kac model mentioned above. As a corollary, we prove propagation of chaos for the marginals associated to the generalized Kac model. In particular, the first marginal converges towards the solution of a Boltzmann equation including interactions up to a finite order and whose collision kernel is of Maxwell type with cut-off. [ABSTRACT FROM AUTHOR]

Benchmark calculations are reported for anisotropic scattering in Boltzmann equation solvers and Monte Carlo collisional models of electron swarms in gases. The work focuses on isotropic, forward, and screened Coulomb models for angular scattering in electron-neutral collisions. The impact of scattering on electron swarm parameters is demonstrated in both conservative and non-conservative model atoms. The practical implementation of anisotropic scattering in the kinetic models is discussed. [ABSTRACT FROM AUTHOR]

A semi-analytic method for quickly approximating the density-reduced critical electric field for arbitrary mixtures of gases is proposed and validated. Determination of this critical electric field is crucial for designing and testing alternatives to SF6 for insulating high voltage electrical equipment. We outline the theoretical basis of the approximation formula from electron fluid conservation equations, and demonstrate how for binary mixtures the critical electric field can be computed from the transport data of electrons in the pure gases. We demonstrate validity of the method in mixtures of N2 and O2, and SF6 and O2. We conclude with an application of the method to approximate the critical electric field for mixtures of SF6 and HFO1234ze(E), which is a high interest mixture being actively studied for high voltage insulation applications. [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]

Quite a many electron transport problems in condensed matter physics are analyzed with a quasiparticle Boltzmann equation. For sufficiently slowly varying weak external potentials it can be derived from the basic equations of quantum kinetics, provided quasiparticles can be defined and lead to a pole in the quantum‐mechanical propagators. The derivation breaks down, however, in the vicinity of an interface which constitutes an abrupt strong perturbation of the system. In this contribution we discuss in a tutorial manner a particular technique to systematically derive, for a planar, nonideal interface, matching conditions for the quasi‐particle Boltzmann equation. The technique is based on pseudizing the transport problem by two auxiliary interface‐free systems and matching Green functions at the interface. Provided quasiparticles exist in the auxiliary systems, the framework can be put onto the semiclassical level and the desired boundary conditions result. For ideal interfaces, the conditions can be guessed from flux conservation, but for complex interfaces this is no longer the case. The technique presented in this work is geared toward such interfaces. [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]

The possibilities of simulation of viscous incompressible fluid flows with lattice Boltzmann method are considered. Unlike the traditional discretization approach based on the use of Navier–Stokes equations, the lattice Boltzmann method uses a mesoscopic model to simulate incompressible fluid flows. Macroscopic parameters of a fluid, such as density and velocity, are expressed through the moments of the discrete probability distribution function. Discretization of the lattice Boltzmann equation is carried out using schemes D2Q9 (two-dimensional case) and D3Q19 (three-dimensional case). To simulate collisions between pseudo-particles, the Bhatnaga r–Gross–Crooke approximation with one relaxation time is used. The specification of initial and boundary conditions (no penetration and no-slip conditions, outflow conditions, periodic conditions) is discussed. The patterns of formation and development of vortical flows in a square cavity and cubic cavities are computed. The results of calculations of flow characteristics in a square and cubic cavity at various Reynolds numbers are compared with data available in the literature and obtained based on the finite difference method and the finite volume method. The dependence of the numerical solution and location of critical points on faces of cubic cavity on the lattice size is studied. Computational time is compared with performance of fine difference and finite volume methods. The developed implementation of the lattice Boltzmann method is of interest for the transition to further modeling non-isothermal and high-speed compressible flows.

The existence and stability of collisional kinetic equations, especially non-cutoff Boltzmann equation, in a bounded domain with physical boundary condition is a longstanding open problem. This work proves the global stability of the Landau equation and non-cutoff Boltzmann equation in union of cubes with the specular reflection boundary condition when an initial datum is near Maxwellian. Moreover, the solution enjoys exponential large-time decay in the union of cubes. Our method is based on the fact that normal derivatives in cubes are also derivatives along the axis, which allows us to obtain high-order derivative estimates. [ABSTRACT FROM AUTHOR]

This article is dedicated to the derivation of a two-dimensional system of moment equations depending on the velocity of movement and the surface temperature of a body submerged in fluid, and macroscopic boundary conditions for the system of moment equations approximating the Maxwell microscopic boundary condition for the particle distribution function. The initial-boundary value problem for the Boltzmann equation with the Maxwell microscopic boundary condition is approximated by a corresponding problem for the system of moment equations with macroscopic boundary conditions. The number of moment equations and the number of macroscopic boundary conditions are interconnected and depend on the parity of the approximation of the system of moment equations. The setting of the initial-boundary value problem for a non-stationary, nonlinear two-dimensional system of moment equations in the first approximation with macroscopic boundary conditions is presented, and the solvability of the above-mentioned problem in the space of functions continuous in time and square-integrable in spatial variables is proven. [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]

It is challenging to perform a multiscale analysis of mesoscopic systems exhibiting singularities at the macroscopic scale. In this paper, we study the hydrodynamic limit of the Boltzmann equation ... toward the singular solutions of 2D incompressible Euler equations whose vorticity is unbounded: ∂t u + u · ∇x u + ∇x p = 0, div u = 0. We obtain a microscopic description of the singularity through the so-called kinetic vorticity and understand its behavior in the vicinity of the macroscopic singularity. As a consequence of our new analysis, we settle affirmatively an open problem of convergence toward Lagrangian solutions of the 2D incompressible Euler equation whose vorticity is unbounded (ω ∈ Lp for any fixed 1 ≤ p < ∞). Moreover, we prove the convergence of kinetic vorticities toward the vorticity of the Lagrangian solution of the Euler equation. In particular, we obtain the rate of convergence when the vorticity blows up moderately in Lp as p → ∞ (a localized Yudovich class). [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]

We consider the neural sparse representation to solve the Boltzmann equation with BGK and quadratic collision models, where a network-based ansatz that can approximate the distribution function with extremely high efficiency is proposed. Precisely, fully connected neural networks are employed in the time and physical space so as to avoid the discretization in space and time. Different low-rank representations are utilized in the microscopic velocity for the BGK and quadratic collision models, resulting in a significant reduction in the degree of freedom. We approximate the discrete velocity distribution in the BGK model using the canonical polyadic decomposition. For the quadratic collision model, a data-driven, SVD-based linear basis is built based on the BGK solution. All of these will significantly improve the efficiency of the network when solving the Boltzmann equation. Moreover, the specially designed adaptive-weight loss function is proposed with the strategies as multiscale input and Maxwellian splitting applied to further enhance the approximation efficiency and speed up the learning process. Several numerical experiments, including 1D wave and Sod tube problems and a 2D wave problem, demonstrate the effectiveness of these neural sparse representation methods. [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]

Self-gravitating fluid instabilities are analysed within the framework of a post-Newtonian Boltzmann equation coupled with the Poisson equations for the gravitational potentials of the post-Newtonian theory. The Poisson equations are determined from the knowledge of the energy–momentum tensor calculated from a post-Newtonian Maxwell–Jüttner distribution function. The one-particle distribution function and the gravitational potentials are perturbed from their background states, and the perturbations are represented by plane waves characterised by a wave number vector and time-dependent small amplitudes. The time-dependent amplitude of the one-particle distribution function is supposed to be a linear combination of the summational invariants of the post-Newtonian kinetic theory. From the coupled system of differential equations for the time-dependent amplitudes of the one-particle distribution function and gravitational potentials, an evolution equation for the mass density contrast is obtained. It is shown that for perturbation wavelengths smaller than the Jeans wavelength, the mass density contrast propagates as harmonic waves in time. For perturbation wavelengths greater than the Jeans wavelength, the mass density contrast grows in time, and the instability growth in the post-Newtonian theory is more accentuated than the one of the Newtonian theory. [ABSTRACT FROM AUTHOR]

A shock wave is a flow phenomenon that needs to be considered in the development of high-speed aircraft and engines. The traditional computational fluid dynamics (CFD) method describes it from the perspective of macroscopic variables, such as the Mach number, pressure, density, and temperature. The thickness of the shock wave is close to the level of the molecular free path, and molecular motion has a strong influence on the shock wave. According to the analysis of the Chapman-Enskog approach, the nonequilibrium effect is the source term that causes the fluid system to deviate from the equilibrium state. The nonequilibrium effect can be used to obtain a description of the physical characteristics of shock waves that are different from the macroscopic variables. The basic idea of the nonequilibrium effect approach is to obtain the nonequilibrium moment of the molecular velocity distribution function by solving the Boltzmann–Bhatnagar–Gross–Krook (Boltzmann BGK) equations or multiple relaxation times Boltzmann (MRT-Boltzmann) equations and to explore the nonequilibrium effect near the shock wave from the molecular motion level. This article introduces the theory and understanding of the nonequilibrium effect approach and reviews the research progress of nonequilibrium behavior in shock-related flow phenomena. The role of nonequilibrium moments played on the macroscopic governing equations of fluids is discussed, the physical meaning of nonequilibrium moments is given from the perspective of molecular motion, and the relationship between nonequilibrium moments and equilibrium moments is analyzed. Studies on the nonequilibrium effects of shock problems, such as the Riemann problem, shock reflection, shock wave/boundary layer interaction, and detonation wave, are introduced. It reveals the nonequilibrium behavior of the shock wave from the mesoscopic level, which is different from the traditional macro perspective and shows the application potential of the mesoscopic kinetic approach of the nonequilibrium effect in the shock problem. [ABSTRACT FROM AUTHOR]

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The phase space of a noncanonical Hamiltonian system is partially inaccessible due to dynamical constraints (Casimir invariants) arising from the kernel of the Poisson tensor. When an ensemble of noncanonical Hamiltonian systems is allowed to interact, dissipative processes eventually break the phase space constraints, resulting in a thermodynamic equilibrium described by a Maxwell–Boltzmann distribution. However, the time scale required to reach Maxwell–Boltzmann statistics is often much longer than the time scale over which a given system achieves a state of thermal equilibrium. Examples include diffusion in rigid mechanical systems, as well as collisionless relaxation in magnetized plasmas and stellar systems, where the interval between binary Coulomb or gravitational collisions can be longer than the time scale over which stable structures are self-organized. Here, we focus on self-organizing phenomena over spacetime scales such that particle interactions respect the noncanonical Hamiltonian structure, but yet act to create a state of thermodynamic equilibrium. We derive a collision operator for general noncanonical Hamiltonian systems, applicable to fast, localized interactions. This collision operator depends on the interaction exchanged by colliding particles and on the Poisson tensor encoding the noncanonical phase space structure, is consistent with entropy growth and conservation of particle number and energy, preserves the interior Casimir invariants, reduces to the Landau collision operator in the limit of grazing binary Coulomb collisions in canonical phase space, and exhibits a metriplectic structure. We further show how thermodynamic equilibria depart from Maxwell–Boltzmann statistics due to the noncanonical phase space structure, and how self-organization and collisionless relaxation in magnetized plasmas and stellar systems can be described through the derived collision operator.

In this review, we detail the commonality of mathematical intuitions that underlie three numerical methods used for the quantitative description of electron swarms propagating in a gas under the effect of externally applied electric and/or magnetic fields. These methods can be linked to the integral transport equation, following a common thread much better known in the theory of neutron transport than in the theory of electron transport. First, we discuss the exact solution of the electron transport problem using Monte Carlo (MC) simulations. In reality we will go even further, showing the interpretative role that the diagrams used in quantum theory and quantum field theory can play in the development of MC. Then, we present two methods, the Monte Carlo Flux and the Propagator method, which have been developed at this moment. The first one is based on a modified MC method, while the second shows the advantage of explicitly applying the mathematical idea of propagator to the transport problem.

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]

In the paper, we develop an L¹k∩Lpk approach to construct global solutions to the Cauchy problem on the non-cutoff Boltzmann equation near equilibrium in R³. In particular, only smallness of ∥Fxf0∥L¹∩Lp(R³k;L²(R³v)) with 3/2

The paper is concerned with the change of probability measures μ along non-random probability measure-valued trajectories ν t , t ∈ [ − 1 , 1 ]. Typically solutions to non-linear partial differential equations (PDEs), modeling spatial development as time progresses, generate such trajectories. Depending on in which direction the map ν ≡ ν 0 ↦ ν t does not exit the state space, for t ∈ [ − 1 , 0 ] or for t ∈ [ 0 , 1 ] , the Radon–Nikodym derivative d μ ∘ ν t / d μ is determined. It is also investigated how Fréchet differentiability of the solution map of the PDE can contribute to the existence of this Radon–Nikodym derivative. The first application is a certain Boltzmann type equation. Here, the Fréchet derivative of the solution map is calculated explicitly and quasi-invariance is established. The second application is a PDE related to the asymptotic behavior of a Fleming–Viot type particle system. Here, it is demonstrated how quasi-invariance can be used in order to derive a corresponding integration by parts formula. [ABSTRACT FROM AUTHOR]

The study proposes a new method called MTRMC to simulate flow in rarefied regimes, which are important in various industrial and engineering applications. This new method utilizes a modified collision function with smaller number of inter-molecular collisions, making it more computationally efficient than the widely used direct simulation Monte Carlo (DSMC) method. The MTRMC method is used to analyze the flow over a flat nano-plate at various free stream velocities, ranging from low to supersonic speeds. The results are compared with those from DSMC and time relaxed Monte Carlo (TRMC) schemes, and the findings show that the MTRMC method is in good agreement with the standard schemes, with a significant reduction in computational expense, up to 51% in some cases. [ABSTRACT FROM AUTHOR]

The antiparallel current‐induced spin polarization (CISP) in an inversion‐symmetric double‐quantum‐well structure with the antiparallel Rashba effective magnetic field is calculated by solving the Boltzmann equation which is derived for the distribution operator in layer‐pseudospin space in the relaxation‐time approximation. The antiparallel CISP in an analytical form, obtained for constant momentum relaxation time in the limit of large Fermi energy, exhibits factor‐two increase with increasing the interlayer‐tunneling strength. This increase originates from the correction (to the semiclassical CISP) created by the subband mixing due to the momentum dependence of the Rashba effective magnetic field. The present interlayer‐coupling dependence of the antiparallel CISP is successfully explained as the layer‐pseudospin Hanle effect in which the spin in the original Hanle effect is replaced by the layer pseudospin. The present finding suggests that the correction by band mixing can be significant in the transport properties of various systems with the momentum‐dependent effective magnetic field. [ABSTRACT FROM AUTHOR]