Lorentz Gas at a Positive Temperature

TL;DR

This paper studies a Lorentz gas with moving scatterers, revealing universal scaling laws for particle velocity and displacement growth, and deriving distribution forms in various dimensions.

Contribution

It introduces a model of a Lorentz gas with dynamic scatterers and derives universal scaling behaviors for particle velocity and displacement.

Findings

Particle speed grows as t^{lambda/(4+lambda)} in 3D.

Displacement grows linearly with time universally.

Velocity and position distributions approach universal scaling forms.

Abstract

We investigate the evolution of a particle in a Lorentz gas where the background scatters move and collide with each other. As in the standard Lorentz gas, we assume that the particle is negligibly light in comparison with scatters. We show that the average particle speed grows in time as t^{lambda/(4+lambda)} in three dimensions when the particle-scatter potential diverges as r^{-lambda} in the small separation limit. The typical displacement of the particle exhibits a universal linear growth in time independently on the density of the background gas and the particle-scatter interaction. The velocity and position distributions approach universal scaling forms. We determine the former, while for the position distribution we establish conjecturally exact scaling forms for the one and two-dimensional Lorentz gas.