A no-go theorem for ergodicity and an Einstein relation

TL;DR

This paper proves that for anomalous diffusion, either ergodicity or the Einstein relation must fail, contrasting with normal diffusion, and demonstrates this with the Lévy walk model.

Contribution

It establishes a no-go theorem linking ergodicity and the Einstein relation in anomalous diffusion, highlighting their incompatibility.

Findings

Either ergodicity or the Einstein relation is broken in anomalous diffusion.

A general relation between drift, MSD, and measurement time is derived.

The Lévy walk model exemplifies the no-go theorem.

Abstract

We provide a simple no-go theorem for ergodicity and the generalized Einstein relation for anomalous diffusion processes. The theorem states that either ergodicity in the sense of equal time and ensemble averaged mean squared displacements (MSD) is broken, and/or the generalized Einstein relation for time averaged diffusivity and mobility is invalid, which is in complete contrast to normal diffusion processes. We also give a general relation for the time averages of drift and MSD for \textit{ergodic} (in the MSD sense) anomalous diffusion processes, showing that the ratio of these quantities depends on the measurement time. The L\'evy walk model is used to exemplify the no-go theorem.