Multiple Scattering in Random Mechanical Systems and Diffusion Approximation

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 ${\mathcal{L}}$ on compactly supported functions and that the Markov chain process associated to P _h converges to a diffusion with infinitesimal generator ${\mathcal{L}}$. Both P _h and ${\mathcal{L}}$ are self-adjoint (densely) defined on the space ${L^2(\mathbb{H},\eta)}$ of square-integrable functions over the (lower) half-space ${\mathbb{H}}$ in ${\mathbb{R}^m}$, where η 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 ${\mathcal{L}}$ 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.