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Anomalous diffusion and ergodicity breaking in heterogeneous diffusion processes

We demonstrate the non-ergodicity of a simple Markovian stochastic processes with space-dependent diffusion coefficient $D(x)$. For power-law forms $D(x) \simeq|x|^α$, this process yield anomalous diffusion of the form $\ < x^2(t)\ > \simeq t^{2/(2-α)}$. 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.