Fluctuations and self-averaging in random trapping transport: The diffusion coefficient.

Abstract

On the basis of a self-consistent cluster effective-medium approximation for random trapping transport, we study the problem of self-averaging of the diffusion coefficient in a nonstationary formulation. In the long-time domain, we investigate different cases that correspond to the increasing degree of disorder. In the regular and subregular cases the diffusion coefficient is found to be a self-averaging quantity-its relative fluctuations (relative standard deviation) decay in time in a power-law fashion. In the subdispersive case the diffusion coefficient is self-averaging in three dimensions (3D) and weakly self-averaging in two dimensions (2D) and one dimension (1D), when its relative fluctuations decay anomalously slowly logarithmically. In the dispersive case, the diffusion coefficient is self-averaging in 3D, weakly self-averaging in 2D, and non-self-averaging in 1D. When non-self-averaging, its fluctuations remain of the same order as, or larger than, its average value. In the irreversible case, the diffusion coefficient is non-self-averaging in any dimension. In general, with the decreasing dimension and/or increasing disorder, the self-averaging worsens and eventually disappears. In the cases of weak self-averaging and, especially, non-self-averaging, the reliable reproducible experimental measurements are highly problematic. In all the cases under consideration, asymptotics with prefactors are obtained beyond the scaling laws. Transition between all cases is analyzed as the disorder increases.