The distribution of free path lengths in the periodic Lorentz gas and related lattice point problems

TL;DR

This paper studies the distribution of free path lengths in the periodic Lorentz gas as scatterer size shrinks, establishing a limiting distribution and exploring related lattice point visibility problems.

Contribution

It proves the existence of a limiting distribution for free path lengths in the Boltzmann-Grad limit of the periodic Lorentz gas and discusses related lattice point problems.

Findings

Existence of a limiting distribution for free path lengths.

Characterization of the distribution in the Boltzmann-Grad limit.

Analysis of lattice point visibility from a fixed position.

Abstract

The periodic Lorentz gas describes the dynamics of a point particle in a periodic array of spherical scatterers, and is one of the fundamental models for chaotic diffusion. In the present paper we investigate the Boltzmann-Grad limit, where the radius of each scatterer tends to zero, and prove the existence of a limiting distribution for the free path length of the periodic Lorentz gas. We also discuss related problems, such as the statistical distribution of directions of lattice points that are visible from a fixed position.