Diffusion of a particle in the spatially correlated exponential random energy landscape: transition from normal to anomalous diffusion

TL;DR

This paper investigates how particle diffusion in a spatially correlated exponential energy landscape transitions from normal to anomalous behavior, revealing a two-stage process involving a singular diffusivity and the emergence of dispersive transport.

Contribution

It provides an exact calculation of diffusivity in correlated landscapes and identifies conditions leading to anomalous diffusion and dispersive regimes.

Findings

Diffusivity becomes singular at a specific temperature for slow decaying correlations.

A two-stage transition from normal to anomalous and then dispersive diffusion is identified.

The Einstein relation's validity is discussed in the context of existing diffusivity.

Abstract

Diffusive transport of a particle in spatially correlated random energy landscape having exponential density of states has been considered. We exactly calculate the diffusivity in the nondispersive quasi-equilibrium transport regime and found that for slow decaying correlation functions the diffusivity becomes singular at some particular temperature higher than the temperature of the transition to the true non-equilibrium dispersive transport regime. It means that the diffusion becomes anomalous and does not follow the usual $\propto t^{1/2}$ law. In such situation the fully developed non-equilibrium regime emerges in two stages: first, at some temperature there is the transition from the normal to anomalous diffusion, and then at lower temperature the average velocity for the infinite medium goes to zero, thus indicating the development of the true dispersive regime. Validity of the Einstein relation is discussed for the situation where the diffusivity does exist.