Kinetic transport in the two-dimensional periodic Lorentz gas

TL;DR

This paper investigates the kinetic transport properties of the two-dimensional periodic Lorentz gas, providing an explicit formula for the collision kernel and highlighting differences from disordered configurations.

Contribution

It derives an explicit, elementary formula for the collision kernel in the two-dimensional periodic Lorentz gas, advancing understanding of its kinetic transport behavior.

Findings

The macroscopic dynamics converge to a stochastic process in the Boltzmann-Grad limit.

The transport equation differs from the linear Boltzmann equation in the periodic case.

An explicit formula for the collision kernel is provided.

Abstract

The periodic Lorentz gas describes an ensemble of non-interacting point particles in a periodic array of spherical scatterers. We have recently shown that, in the limit of small scatterer density (Boltzmann-Grad limit), the macroscopic dynamics converges to a stochastic process, whose kinetic transport equation is not the linear Boltzmann equation--in contrast to the Lorentz gas with a disordered scatterer configuration. The present paper focuses on the two-dimensional set-up, and reports an explicit, elementary formula for the collision kernel of the transport equation.