Multiple scattering in random mechanical systems and diffusion approximation

TL;DR

This paper models multiple scattering in mechanical billiard systems using stochastic processes, demonstrating that as scattering weakens, the processes converge to diffusion described by a second order elliptic operator.

Contribution

It introduces a framework connecting deterministic billiard systems with stochastic processes and diffusion approximations, including explicit examples and numerical simulations.

Findings

Convergence of transition operators to a diffusion generator as scattering weakens

Identification of stationary measures as Maxwell-Boltzmann or Knudsen distributions

Numerical illustrations of the diffusion approximation in mechanical systems

Abstract

This paper is concerned with stochastic processes that model multiple (or iterated) scattering in classical mechanical systems of billiard type, defined below. From a given (deterministic) system of billiard type, a random process with transition probabilities operator P is introduced by assuming that some of the dynamical variables are random with prescribed probability distributions. Of particular interest are systems with weak scattering, which are associated to parametric families of operators P_h, depending on a geometric or mechanical parameter h, that approaches the identity as h goes to 0. It is shown that (P_h -I)/h converges for small h to a second order elliptic differential operator L on compactly supported functions and that the Markov chain process associated to P_h converges to a diffusion with infinitesimal generator L. Both P_h and L are selfadjoint (densely) defined on the space L2(H,{\eta}) of square-integrable functions over the (lower) half-space H in R^m, where {\eta} is a stationary measure. This measure's density is either (post-collision) Maxwell-Boltzmann distribution or Knudsen cosine law, and the random processes with infinitesimal generator L respectively correspond to what we call MB diffusion and (generalized) Legendre diffusion. Concrete examples of simple mechanical systems are given and illustrated by numerically simulating the random processes.