Einstein relation in superdiffusive systems

TL;DR

This paper investigates the validity of the Einstein relation in superdiffusive systems modeled by Levy walks, finding it holds without current but fails when a mean drift is present, highlighting the role of current in Einstein relation breakdown.

Contribution

It demonstrates that the Einstein relation holds in superdiffusive Levy walk systems without current, but fails when a mean drift exists, emphasizing the impact of current on the relation.

Findings

Einstein relation holds in superdiffusive Levy walks without current.

The relation breaks down when a mean drift or current is present.

Presence of current, not anomalous diffusion, causes Einstein relation failure.

Abstract

We study the Einstein relation between diffusion and response to an external field in systems showing superdiffusion. In particular, we investigate a continuous time Levy walk where the velocity remains constant for a time \tau, with distribution P(\tau) \tau^{-g}. At varying g the diffusion can be standard or anomalous; in spite of this, if in the unperturbed system a current is absent, the Einstein relation holds. In the case where a current is present the scenario is more complicated and the usual Einstein relation fails. This suggests that the main ingredient for the breaking of the Einstein relation is not the anomalous diffusion but the presence of a mean drift (current).