Fick's law in a random lattice Lorentz gas

TL;DR

This paper proves that in a high-dimensional random lattice Lorentz gas, the macroscopic particle current obeys Fick's law, with the stationary current determined by the conductivity and reservoir chemical potentials, and shows exponential decay of deviations.

Contribution

It provides a rigorous proof of Fick's law for a high-dimensional Lorentz gas connected to reservoirs, highlighting the role of high dimension in simplifying particle interactions.

Findings

Stationary current satisfies Fick's law in high dimensions.

Probability of deviations from stationary current decays exponentially over time.

High dimension reduces particle loop and intersection probabilities.

Abstract

We provide a proof that the stationary macroscopic current of particles in a random lattice Lorentz gas satisfies Fick's law when connected to particles reservoirs. We consider a box on a d+1 dimensional lattice and when $d\geq7$, we show that under a diffusive rescaling of space and time, the probability to find a current different from its stationary value is exponentially small in time. Its stationary value is given by the conductivity times the difference of chemical potentials of the reservoirs. The proof is based on the fact that in high dimension, random walks have a small probability of making loops or intersecting each other when starting sufficiently far apart.