Anomalous diffusion and ergodicity breaking in heterogeneous diffusion processes

Abstract

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