Intrinsic randomness of transport coefficient in subdiffusion with static disorder

Abstract

Fluctuations in the time-averaged mean-square displacement for random walks on hypercubic lattices with static disorder are investigated. It is analytically shown that the diffusion coefficient becomes a random variable as a manifestation of weak ergodicity breaking. For two- and higher- dimensional systems, the distribution function of the diffusion coefficient is found to be the Mittag-Leffler distribution, which is the same as for the continuous-time random walk, whereas for one-dimensional systems a different distribution (a modified Mittag-Leffler distribution) arises. We also present a comparison of these two distributions in terms of an ergodicity-breaking parameter and show that the modified Mittag-Leffler distribution has a larger deviation from ergodicity. Some remarks on similarities between these results and observations in biological experiments are presented.