The periodic Lorentz gas in the Boltzmann-Grad limit: asymptotic estimates

TL;DR

This paper studies the asymptotic behavior of a particle in a periodic array of scatterers, showing it converges to a Markov process with specific transition probabilities, differing from the random case.

Contribution

It provides new asymptotic estimates for transition probabilities in the periodic Lorentz gas, refining previous bounds on free path length distributions.

Findings

Sharpened bounds on free path length distribution

Asymptotic estimates for Markov process transition probabilities

Distinct transport equation from the random scatterer case

Abstract

The dynamics of a point particle in a periodic array of spherical scatterers converges, in the limit of small scatterer size, to a random flight process, whose paths are piecewise linear curves generated by a Markov process with memory two. The corresponding transport equation is distinctly different from the linear Boltzmann equation observed in the case of a random configuration of scatterers. In the present paper we provide asymptotic estimates for the transition probabilities of this Markov process. Our results in particular sharpen previous upper and lower bounds on the distribution of free path lengths obtained by Bourgain, Golse and Wennberg.