Fick's Law for the Lorentz Model in a weak coupling regime

TL;DR

This paper proves that in a weak coupling regime, the Lorentz gas exhibits diffusion described by Fick's law, with the diffusion coefficient derived from the Green-Kubo formula linked to the Landau equation.

Contribution

It establishes a rigorous connection between the Lorentz gas in a weak coupling regime and macroscopic diffusion laws, extending previous results.

Findings

The macroscopic current follows Fick's law in the stationary state.

The diffusion coefficient is characterized by the Green-Kubo formula.

The results connect microscopic dynamics to macroscopic diffusion in the Lorentz model.

Abstract

In this paper we deal with further recent developments, strictly connected to the recent result obtained by Basile, Nota, Pezzotti and Pulvirenti. We consider the Lorentz gas out of equilibrium in a weak coupling regime. Each obstacle of the Lorentz gas generates a smooth radially symmetric potential with compact support. We prove that the macroscopic current in the stationary state is given by the Fick's law of diffusion. The diffusion coefficient is given by the Green-Kubo formula associated to the generator of the diffusion process dictated by the linear Landau equation.