Non-ergodicity, fluctuations, and criticality in heterogeneous diffusion   processes
  We study the stochastic behavior of heterogeneous diffusion processes with the power-law dependence $D(x)\sim|x|^{\alpha}$ of the generalized diffusion coefficient encompassing sub- and superdiffusive anomalous diffusion. Based on statistical measures such as the amplitude scatter of the time averaged mean squared displacement of individual realizations, the ergodicity breaking and non-Gaussianity parameters, as well as the probability density function $P(x,t)$ we analyze the weakly non-ergodic character of the heterogeneous diffusion process and, particularly, the degree of irreproducibility of individual realization. As we show, the fluctuations between individual realizations increase with growing modulus $|\alpha|$ of the scaling exponent. The fluctuations appear to diverge when the critical value $\alpha=2$ is approached, while for even larger $\alpha$ the fluctuations decrease, again. At criticality, the power-law behavior of the mean squared displacement changes to an exponentially fast growth, and the fluctuations of the time averaged mean squared displacement do not seem to converge for increasing number of realizations. From a systematic comparison we observe some striking similarities of the heterogeneous diffusion process with the familiar subdiffusive continuous time random walk process with power-law waiting time distribution and diverging characteristic waiting time. 
Andrey G. Cherstvy and Ralf Metzler
cond-mat.stat-mech