Anomalous diffusion and ergodicity breaking in heterogeneous diffusion processes

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 $\{δ^2}$ 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 $\{δ^2}$. This model represents an alternative approach to non-ergodic, anomalous diffusion that might be particularly relevant for diffusion in heterogeneous media.

5 pages, 5 figures, Supplementary Material within source files

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