Linear Landau equation as a limit of a tagged particle in mean field interaction with a free gas

TL;DR

This paper proves that in high dimensions, the trajectory of a tagged particle interacting with a free gas converges to a diffusion process described by the linear Landau equation as the gas density increases, using stability and recollision controls.

Contribution

It establishes the convergence of a tagged particle's trajectory to a diffusion process governed by the linear Landau equation in dimensions four and higher, with new techniques for stability and recollision control.

Findings

Trajectory converges to diffusion process as gas density increases

Proof relies on long time stability of microscopic dynamics

Controls on particle recollisions are established

Abstract

We consider a tagged particle in mean field interaction with a free gas of density N at equilibrium. In dimensions $d\geq4$, we prove the convergence of its trajectory, as N goes to infinity, to the one of a diffusion process associated with the linear Landau equation. The proof of the convergence of the martingale problem relies on two key ingredients: long time stability results of the microscopic dynamics, and controls on the probability of particle recollisions.