A diffusion limit for a test particle in a random distribution of scatterers

TL;DR

This paper demonstrates that a point particle in a random obstacle field exhibits diffusive behavior in a weak-coupling limit, with its probability distribution converging to the heat equation solution, characterized by a diffusion coefficient from the Landau equation.

Contribution

It establishes a diffusion limit for a particle in a random potential, connecting microscopic dynamics to macroscopic diffusion via the Landau equation.

Findings

Probability distribution converges to the heat equation solution.

Diffusion coefficient derived from the Green-Kubo formula.

Validates the diffusion approximation in the weak-coupling regime.

Abstract

We consider a point particle moving in a random distribution of obstacles described by a potential barrier. We show that, in a weak-coupling regime, under a diffusion limit suggested by the potential itself, the probability distribution of the particle converges to the solution of the heat equation. The diffusion coefficient is given by the Green-Kubo formula associated to the generator of the diffusion process dictated by the linear Landau equation.