Non-equilibrium version of the Einstein relation

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 $v\mathrm{/}D$ 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.