Two-dimensional Lorentz process for magnetotransport: Boltzmann-Grad limit

TL;DR

This paper analyzes the behavior of charged particles in a two-dimensional system with scatterers under a magnetic field, demonstrating that their distribution follows a generalized Boltzmann equation with non-Markovian features in the low-density limit.

Contribution

It proves the convergence of the particle distribution to a generalized Boltzmann equation with memory effects in the Boltzmann-Grad limit.

Findings

Distribution follows a generalized linear Boltzmann equation

Boltzmann's chaos fails in this model

Kinetic equation includes non-Markovian terms

Abstract

We study a system of charged, noninteracting classical particles moving in a Poisson distribution of hard-disk scatterers in two dimensions, under the effect of a magnetic field perpendicular to the plane. We prove that, in the low-density (Boltzmann-Grad) limit, the particle distribution evolves according to a generalized linear Boltzmann equation, previously derived and solved by Bobylev et al. [4, 5, 6]. In this model, Boltzmann's chaos fails, and the kinetic equation includes non-Markovian terms. The ideas of [13] can be however adapted to prove convergence of the process with memory.