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) |x|α, this process yield anomalous diffusion of the form \ < x²(t)\ > t2/(2-α). Interestingly, in both the sub- and superdiffusive regimes we observe weak ergodicity breaking: the scaling of the time averaged mean squared displacement \δ² remains linear and thus differs from the corresponding ensemble average \ <x²(t)\ >. We analyze the non-ergodic behavior of this process in terms of the ergodicity breaking parameters and the distribution of amplitude scatter of \δ². This model represents an alternative approach to non-ergodic, anomalous diffusion that might be particularly relevant for diffusion in heterogeneous media.