Enhancement, slow relaxation, ergodicity and rejuvenation of diffusion in biased continuous-time random walks

TL;DR

This paper investigates how bias influences diffusion in random energy landscapes using continuous-time random walks, revealing superdiffusive behavior, ergodicity preservation, and a rejuvenation effect in non-equilibrium conditions.

Contribution

It introduces a novel mechanism showing bias-induced superdiffusion and explores ergodicity and rejuvenation phenomena in biased continuous-time random walks.

Findings

Bias enhances diffusion in random landscapes.

Superdiffusive behavior occurs when waiting time variance diverges.

Diffusivity shows rejuvenation with delayed measurement start.

Abstract

Bias plays an important role in the enhancement of diffusion in periodic potentials. Using the continuous-time random walk in the presence of a bias, we provide a novel mechanism for the enhancement of diffusion in a random energy landscape. When the variance of the waiting time diverges, in contrast to the bias-free case the dynamics with bias becomes superdiffusive. In the superdiffusive regime, we find a distinct initial ensemble dependence of the diffusivity. We show that the time-averaged variance converges to the corresponding ensemble-averaged variance, i.e., ergodicity is preserved. However, trajectory-to-trajectory fluctuations of the time-averaged variance decay slowly. Our finding suggests that in the superdiffusive regime the diffusivity for a non-equilibrium initial ensemble gradually increases to that for an equilibrium ensemble when the start of the measurement is delayed, corresponding to a rejuvenation of diffusivity.