Diffusion Limit of the Low-Density Magnetic Lorentz Gas

TL;DR

This paper proves that in a low-density, magnetic Lorentz gas, the particle's distribution converges to a heat equation solution with a magnetic-field-dependent diffusion coefficient, under certain magnetic field strength conditions.

Contribution

It establishes the diffusion limit of the magnetic Lorentz gas, deriving the Green-Kubo formula for the diffusion coefficient from the generalized Boltzmann process.

Findings

Convergence to the heat equation in the low-density, diffusion limit.

Diffusion coefficient depends on magnetic field and is derived from the Green-Kubo formula.

Non-Markovian effects are negligible when magnetic field strength is below 8π/3.

Abstract

We consider the magnetic Lorentz gas proposed by Bobylev et al. [4], which describes a point particle moving in a random distribution of hard-disk obstacles in $\mathbb{R}^2$ under the influence of a constant magnetic field perpendicular to the plane. We show that, in the coupled low-density and diffusion limit, when the intensity of the magnetic field is smaller than $\frac{8\pi}{3}$, the non-Markovian effects induced by the magnetic field become sufficiently weak. Consequently, the particle's probability distribution converges to the solution of the heat equation with a diffusion coefficient dependent on the magnetic field and given by the Green-Kubo formula. This formula is derived from the generator of the generalized Boltzmann process associated with the generalized Boltzmann equation, as predicted in [4].