Nonequilibrium version 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.

Negative mobility, sliding and delocalization for stochastic networks

“…The weighted drift velocity can be calculated from the spectrum [23], and the result is obviously in agreement with Eq. ( 4).…”

Relaxation rate of a stochastic spreading process in a closed ring

“…(12) can be used in order to predict Γ. For this purpose v and D are independently calculated using a standard procedure that is outlined in Appendix A of [17]. Indeed we observe in Fig.1 a nice agreement between this prediction (solid and dashed green lines) and the previously calculated relaxation rate (blue symbols).…”

“…A straightforward analysis of the time-dependent spreading process, see e.g. [17], shows that the drift velocity and the diffusion coefficient are given by the following expressions:…”

“…In the range s 1/2 < s < s 2 the diffusion coefficient is large but finite and becomes N dependent. In [17] a heuristic approach has been attempted in order to figure out this N dependence. In the present work we would like to adopt a more rigorous approach.…”

“…In the presence of disorder, the forward and backward rates are random numbers. Here we summarize known analytical expressions for v and D based on [2], and notations as in [16,17]. Taking the infinite chain limit, and using units such that the lattice spacing is a = 1, the expression for the drift velocity is…”