The Boltzmann-Grad limit of the periodic Lorentz gas

TL;DR

This paper investigates the behavior of a particle in a periodic array of scatterers under low-density conditions, deriving a stochastic process that describes its dynamics in the Boltzmann-Grad limit, revealing a Markov process with memory two.

Contribution

It constructs a novel stochastic process model for the Lorentz gas in the Boltzmann-Grad limit, capturing the particle's dynamics with a Markov process with memory two.

Findings

The limiting process is a piecewise linear curve.

Consecutive segments are generated by a Markov process with memory two.

The model accurately describes the low-density limit dynamics.

Abstract

We study the dynamics of a point particle in a periodic array of spherical scatterers, and construct a stochastic process that governs the time evolution for random initial data in the limit of low scatterer density (Boltzmann-Grad limit). A generic path of the limiting process is a piecewise linear curve whose consecutive segments are generated by a Markov process with memory two.