Self-averaging and ergodicity of subdiffusion in quenched random media

We study the self-averaging properties and ergodicity of the mean square displacement m(t) of particles diffusing in d dimensional quenched random environments which give rise to subdiffusive average motion. These properties are investigated in terms of the sample to sample fluctuations as measured by the variance of m(t). We find that m(t) is not self-averaging for d<2 due to the inefficient disorder sampling by random motion in a single realization. For d≥2 in contrast, the efficient sampling of heterogeneity by the space random walk renders m(t) self-averaging and thus ergodic. This is remarkable because the average particle motion in d>2 obeys a CTRW, which by itself displays weak ergodicity breaking. This paradox is resolved by the observation that the CTRW as an average model does not reflect the disorder sampling by random motion in a single medium realization.