Diffusion of the random Lorentz process in a magnetic field

TL;DR

This paper proves that a charged particle moving in a 2D plane with randomly placed scatterers under a magnetic field behaves like a Brownian motion in the low-density and long-time limit, extending previous coupling methods.

Contribution

It extends the coupling method to show the invariance principle for the magnetic Lorentz gas in an intermediate scaling limit.

Findings

Rescaled trajectories converge to Brownian motion

Validates the invariance principle in a magnetic field setting

Extends previous methods to magnetic Lorentz gases

Abstract

Consider the motion of a charged, point particle moving in the complement of a Poisson distribution of hard sphere scatterers in two dimensions under the effect of a fixed magnetic field. Building on, and extending a coupling method established by the authors, we show that this 'magnetic Lorentz gas' satisfies an invariance principle in an intermediate scaling limit. That is, we apply the low-density (Boltzmann-Grad) limit and simultaneously take the limit as time goes to infinity, then prove convergence of the rescaled trajectory to a Brownian motion in this limit.