Anomalous diffusion and ergodicity breaking in heterogeneous diffusion processes

TL;DR

This paper investigates non-ergodic behavior in a simple Markovian diffusion process with space-dependent diffusion coefficients, revealing weak ergodicity breaking and anomalous diffusion in heterogeneous media.

Contribution

It introduces a model of space-dependent diffusion that exhibits anomalous diffusion and weak ergodicity breaking, providing insights into non-ergodic processes in heterogeneous environments.

Findings

Demonstrates non-ergodicity in space-dependent diffusion processes.

Shows anomalous diffusion with specific power-law dependence.

Identifies weak ergodicity breaking in both sub- and superdiffusive regimes.

Abstract

We demonstrate the non-ergodicity of a simple Markovian stochastic processes with space-dependent diffusion coefficient $D(x)$. For power-law forms $D(x) \simeq|x|^{\alpha}$, this process yield anomalous diffusion of the form $\ < x^2(t)\ > \simeq t^{2/(2-\alpha)}$. Interestingly, in both the sub- and superdiffusive regimes we observe weak ergodicity breaking: the scaling of the time averaged mean squared displacement $\{\delta^2}$ remains \emph{linear} and thus differs from the corresponding ensemble average $\ <x^2(t)\ >$. We analyze the non-ergodic behavior of this process in terms of the ergodicity breaking parameters and the distribution of amplitude scatter of $\{\delta^2}$. This model represents an alternative approach to non-ergodic, anomalous diffusion that might be particularly relevant for diffusion in heterogeneous media.