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