Non-equilibrium version of the Einstein relation

TL;DR

This paper investigates how the classical Einstein relation between diffusion and drift is violated in non-equilibrium systems, specifically analyzing a lattice-based Brownian motion with asymmetric transition rates.

Contribution

It introduces a non-equilibrium framework showing the $v/D$ ratio as a non-linear function of affinity, highlighting the dependence on microscopic details.

Findings

The Einstein relation does not hold in non-equilibrium conditions.

The $v/D$ ratio varies non-linearly with affinity.

The microscopic transition rates influence the violation of the Einstein relation.

Abstract

The celebrated Einstein relation between the diffusion coefficient $D$ and the drift velocity $v$ is violated in non-equilibrium circumstances. We analyze how this violation emerges for the simplest example of a Brownian motion on a lattice, taking into account the interplay between the periodicity, the randomness and the asymmetry of the transition rates. Based on the non-equilibrium fluctuation theorem the $v/D$ ratio is found to be a non-linear function of the affinity. Hence it depends in a non-trivial way on the microscopics of the sample.