Concentration statistics for transport in random media

Abstract

We study the ensemble statistics of the particle density in a random medium whose mean transport dynamics describes a continuous time random walk. Starting from a Langevin equation for the particle motion in a single disorder realization, we derive evolution equations for the n-point moments of concentration by coarse graining and ensemble averaging the microscale transport problem. The governing equations describe multidimensional continuous time random walks whose waiting time distribution is given in terms of the disorder distribution. We find that the concentration is not self-averaging even for normal mean behavior. The relative concentration variance for anomalous is larger than for normal mean behavior. These results may have some impact on risk and extreme value analysis in stochastic dynamic systems.