Equilibration and diffusion for a dynamical Lorentz gas

TL;DR

This paper models a particle moving through a random array of scatterers with internal degrees of freedom, showing that its velocity distribution approaches a Maxwell-Boltzmann form with an effective temperature related to the scatterers' energy.

Contribution

It introduces a Markov chain model for the particle's energy evolution in a dynamical Lorentz gas with internal degrees of freedom, demonstrating thermalization to a Maxwell-Boltzmann distribution.

Findings

Velocity distribution approaches Maxwell-Boltzmann form

Effective temperature equals average scatterer energy

Model applies to high initial velocities

Abstract

We consider a model of a dynamical Lorentz gaz : a single particle is moving in $\mathbb{R}^d$ through an array of fixed an soft scatterers each possessing an internal degree of freedom coupled to the particle. Assuming the initial velocity is sufficiently high and modelling the parameters of the scatterers as random variables, we describe the evolution of the kinetic energy of the particle by a Markov chain for which each step corresponds to a collision. We show that the momentum distribution of the particle approaches a Maxwell-Boltzmann distribution with effective temperature $T$ such that $k_BT$ corresponds to an average of the scatterers' kinetic energy.