On anomalous diffusion and the out of equilibrium response function in one-dimensional models

TL;DR

This paper investigates the validity of the Einstein relation in one-dimensional models with subdiffusive transport, showing that non-equilibrium currents, rather than anomalous diffusion itself, cause the breakdown of this fundamental relation.

Contribution

It demonstrates that non-equilibrium currents, not anomalous diffusion, are responsible for the failure of the Einstein relation in certain one-dimensional models.

Findings

The Einstein relation holds in the absence of currents.

Presence of stationary currents breaks the Einstein relation.

A generalized relation explains the breakdown under non-equilibrium conditions.

Abstract

We study how the Einstein relation between spontaneous fluctuations and the response to an external perturbation holds in the absence of currents, for the comb model and the elastic single-file, which are examples of systems with subdiffusive transport properties. The relevance of non-equilibrium conditions is investigated: when a stationary current (in the form of a drift or an energy flux) is present, the Einstein relation breaks down, as is known to happen in systems with standard diffusion. In the case of the comb model, a general relation, which has appeared in the recent literature, between the response function and an unperturbed suitable correlation function, allows us to explain the observed results. This suggests that a relevant ingredient in breaking the Einstein formula, for stationary regimes, is not the anomalous diffusion but the presence of currents driving the system out of equilibrium.