Stochastic Dynamics and Hierarchy for the Boltzmann Equation with Arbitrary Differential Scattering Cross Section

The stochastic dynamics for point particles that corresponds to the Boltzmann equation with arbitrary differential scattering cross section is constructed. We derive the stochastic Boltzmann hierarchy the solutions of which outside the hyperplanes of lower dimension where the point particles interact are equal to the product of one-particle correlation functions, provided that the initial correlation functions are products of one-particle correlation functions. A one-particle correlation function satisfies the Boltzmann equation. The Kac dynamics in the momentum space is obtained.__________Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 56, No. 12, pp. 1629–1653, December, 2004.     

Stochastic dynamics and Boltzmann hierarchy. II

Stochastic dynamics corresponding to the Boltzmann hierarchy is constructed. The Liouville-Itô equations are obtained, from which we derive the Boltzmann hierarchy regarded as an abstract evolution equation. We construct the semigroup of evolution operators and prove the existence of solutions of the Boltzmann hierarchy in the space of sequences of integrable and bounded functions. On the basis of these results, we prove the existence of global solutions of the Boltzmann equation and the existence of the Boltzmann-Grad limit for an arbitrary time interval.     

Stochastic dynamics and Boltzmann hierarchy. III

Stochastic dynamics corresponding to the Boltzmann hierarchy is constructed. The Liouville-Itô equations are obtained, from which we derive the Boltzmann hierarchy regarded as an abstract evolution equation. We construct the semigroup of evolution operators and prove the existence of solutions of the Boltzmann hierarchy in the space of sequences of integrable and bounded functions. On the basis of these results, we prove the existence of global solutions of the Boltzmann equation and the existence of the Boltzmann-Grad limit for an arbitrary time interval.     

Stochastic dynamics and Boltzmann hierarchy. I

Stochastic dynamics corresponding to the Boltzmann hierarchy is constructed. The Liouville-Itô equations are obtained, from which we derive the Boltzmann hierarchy regarded as an abstract evolution equation. We construct the semigroup of evolution operators and prove the existence of solutions of the Boltzmann hierarchy in the space of sequences of integrable and bounded functions. On the basis of these results, we prove the existence of global solutions of the Boltzmann equation and the existence of the Boltzmann-Grad limit for an arbitrary time interval.     

Spatially-Homogeneous Boltzmann Hierarchy as Averaged Spatially-Inhomogeneous Stochastic Boltzmann Hierarchy

We introduce the stochastic dynamics in the phase space that corresponds to the Boltzmann equation and hierarchy and is the Boltzmann–Grad limit of the Hamiltonian dynamics of systems of hard spheres. By the method of averaging over the space of positions, we derive from it the stochastic dynamics in the momentum space that corresponds to the space-homogeneous Boltzmann equation and hierarchy. Analogous dynamics in the mean-field approximation was postulated by Kac for the explanation of the phenomenon of propagation of chaos and derivation of the Boltzmann equation.     

Conservative numerical method for solving the averaged Boltzmann equation

A method is proposed for averaging the Boltzmann kinetic equation with respect to transverse velocities. A system of two integro-differential equations for two desired functions depending only on the longitudinal velocity is derived. The gas particles are assumed to interact as absolutely hard spheres. The integrals in the equations are double. The reduction in the number of variables in the desired functions and the low multiplicity of the integrals ensure the computational efficiency of the averaged equations. A numerical method of discrete ordinates is developed that effectively solves the gas relaxation problem based on the averaged equations. The method is conservative, and the number of particles, momentum, and energy are conserved automatically at every time step.     

On large deviations for particle systems associated with spatially homogeneous Boltzmann type equations

Summary We consider a dynamical interacting particle system whose empirical distribution tends to the solution of a spatially homogeneous Boltzmann type equation, as the number of particles tends to infinity. These laws of large numbers were proved for the Maxwellian molecules by H. Tanaka [Tal] and for the hard spheres by A.S. Sznitman [Szl]. In the present paper we investigate the corresponding large deviations: the large deviation upper bound is obtained and, using convex analysis, a non-variational formulation of the rate function is given. Our results hold for Maxwellian molecules with a cutoff potential and for hard spheres.     

About Kacʼs program in kinetic theory

In this Note we present the main results from the recent work of Mischler and Mouhot (2011) [15], which answers several conjectures raised fifty years ago by Kac (1956) [10]. There Kac introduced a many-particle stochastic process (now denoted as Kac?s master equation) which, for chaotic data, converges to the spatially homogeneous Boltzmann equation. We answer the three following questions raised in Kac (1956) [10]: (1) prove the propagation of chaos for realistic microscopic interactions (i.e. in our results: hard spheres and true Maxwell molecules); (2) relate the time scales of relaxation of the stochastic process and of the limit equation by obtaining rates independent of the number of particles; (3) prove the convergence of the many-particle entropy towards the Boltzmann entropy of the solution to the limit equation (microscopic justification of the H-theorem of Boltzmann in this context). These results crucially rely on a new theory of quantitative uniform in time estimates of propagation of chaos.     

Analog of the Liouville Equation and BBGKY Hierarchy for a System of Hard Spheres with Inelastic Collisions

Dynamics of a system of hard spheres with inelastic collisions is investigated. This system is a model for granular flow. The map induced by a shift along the trajectory does not preserve the volume of the phase space, and the corresponding Jacobian is different from one. A special distribution function is defined as the product of the usual distribution function and the squared Jacobian. For this distribution function, the Liouville equation with boundary condition is derived. A sequence of correlation functions is defined for canonical and grand canonical ensemble. The generalized BBGKY hierarchy and boundary condition are deduced for correlation functions. __________ Published in Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 6, pp. 818–839, June, 2005.     

Hydrodynamic limits with shock waves of the Boltzmann equation

We show that piecewise smooth solutions with shocks of the Euler equations in gas dynamics can be obtained as the zero Knudsen number limit of solutions of the Boltzmann equation for hard sphere collision model. The construction of the Boltzmann solutions is done in two steps. First we introduce a generalized Hilbert expansion with shock layer correction to construct approximations to the solutions of the Boltzmann equations with small Knudsen numbers. We then apply the recently developed macro‐micro decomposition and energy method for Boltzmann shock layers to construct the exact Boltzmann solutions through the stability analysis. © 2004 Wiley Periodicals, Inc.     

Stationary solutions of the linearized Boltzmann equation in a half-space

Stationary half-space solutions of the linearized Boltzmann equation are studied by energy estimates methods. We extend the results of Bardos, Caflisch and Nicolaenko for a gas of hard spheres to a general potential. Asymptotic behaviour is obtained for hard as well as soft potentials and compared to the case of hard spheres.     

The grad limit for a system of soft spheres

A model describing the dynamics of a system of soft spheres is introduced. The model contains a random mechanism which mollifies the deterministic model of hard sphere molecules. It allows a rigorous deduction of the corresponding Boltzmann equation for a wide class of initial data and arbitrarily long times.     

Solutions of Bogolyubov equations for one-dimensional system of hard spheres

For a one-dimensional system of particles that interact as hard elastic spheres the existence of global solutions to the Cauchy problem for the Bogolyubov equations is proved for initial data in spaces of sequences of summable and bounded functions.Institute of Mathematics, Uzb. SSR Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 91, No. 1, pp. 120–128, April, 1992.     

Convergence of a Stochastic Particle Approximation for Measure Solutions of the 2D Keller-Segel System

The analysis of a stochastic interacting particle scheme for the approximation of measure solutions of the parabolic-elliptic Keller–Segel system in 2D is continued. In previous work it has been shown that solutions of a regularized scheme converge to solutions of the regularized Keller–Segel system, when the number of particles tends to infinity. In the present work, the regularization is eliminated in the particle model, which requires an application of the framework of time dependent measures with diagonal defects, developed by Poupaud. The subsequent many particle limit of the BBGKY hierarchy can be solved using measure solutions of the Keller–Segel system and the molecular chaos assumption. However, a uniqueness result for the limiting hierarchy and therefore a proof of propagation of chaos is missing. Finally, the dynamics of strong measure solutions, i.e., sums of smooth distributions and Delta measures, of the particle model is discussed formally for the cases of 2 and 3 particles. The blow-up behavior for more than 2 particles is not completely understood.     

Spherical averages in the space of marked lattices

A marked lattice is a d-dimensional Euclidean lattice, where each lattice point is assigned a mark via a given random field on \({\mathbb {Z}}^d\). We prove that, if the field is strongly mixing with a faster-than-logarithmic rate, then for every given lattice and almost every marking, large spheres become equidistributed in the space of marked lattices. A key aspect of our study is that the space of marked lattices is not a homogeneous space, but rather a non-trivial fiber bundle over such a space. As an application, we prove that the free path length in a crystal with random defects has a limiting distribution in the Boltzmann-Grad limit.     

First passage Monte Carlo algorithms for solving coupled systems of diffusion–reaction equations

We suggest in this letter a new Random Walk on Spheres (RWS) stochastic algorithm for solving systems of coupled diffusion–reaction equations where the random walk is living both on the randomly walking spheres and inside the relevant balls. The method is mesh free both in space and time, and is well applied to solve high-dimensional problems with complicated domains. The algorithms are based on tracking the trajectories of the diffusing particles exactly in accordance with the probabilistic distributions derived from the explicit representation of the relevant Green functions for balls and spheres. They can be conveniently used not only for the solutions, but also for a direct calculation of fluxes to any part of the boundary without calculating the whole solution in the domain. Some applications to exciton flux calculations in the diffusion imaging method in semiconductors are discussed.     

A probabilistic model associated with the pressureless gas dynamics

Using a method of stochastic perturbation of a Langevin system associated with the non-viscous Burgers equation we introduce a system of PDE that can be considered as a regularization of the pressureless gas dynamics describing sticky particles. By means of this regularization we describe how starting from smooth data a δ-singularity arises in the component of density. Namely, we find the asymptotics of solution at the point of the singularity formation as the parameter of stochastic perturbation tends to zero. Then we introduce a generalized solution in the sense of free particles (FP-solution) as a special limit of the solution to the regularized system. This solution corresponds to a medium consisting of non-interacting particles. The FP-solution is a bridging step to constructing solutions to the Riemann problem for the pressureless gas dynamics describing sticky particles. We analyze the difference in the behavior of discontinuous solutions for these two models and the relations between them. In our framework we obtain a unique entropy solution to the Riemann problem in 1D case.     

Derivation of the Boltzmann equation and entropy production in functional mechanics

A derivation of the Boltzmann equation from the Liouville equation by the use of the Grad limiting procedure in a finite volume is proposed. We introduce two scales of space-time: macro- and microscale and use the BBGKY hierarchy and the functional formulation of classical mechanics. According to the functional approach to mechanics, a state of a system of particles is formed from the measurements, which are rational numbers. Hence, one can speak about the accuracy of the initial probability density function in the Liouville equation. We assume that the initial data for the microscopic density functions are assigned by the macroscopic one (so one can say about a kind of hierarchy and subordination of the microscale to the macroscale) and derive the Boltzmann equation, which leads to the entropy production.