The periodic Lorentz gas in the Boltzmann-Grad limit

Abstract

Consider the motion of a single point particle bouncing in a fixed system of spherical obstacles. It is assumed that collisions are perfectly elastic, and that the particle is subject to no external force between collisions, so that the particle moves at constant speed. This type of dynamical system belongs to the class of dispersing billiards, and is referred to as a “Lorentz gas”. A Lorentz gas is called periodic when the obstacle centers form a lattice. Assuming that the initial position and direction of the particle are distributed under some smooth density with respect to the uniform measure, one seeks the evolution of that density under the dynamics defined by the particle motion in some large scale limit for which the number of collisions per unit of time is of the order of unity. This scaling limit is known as “the Boltzmann-Grad limit”, and is the regime of validity for the Boltzmann equation in the kinetic theory of gases. Whether this evolution is governed in such a limit by a PDE analogous to the Boltzmann equation is a natural question, and the topic of this paper.