Brownian diffusion in a dilute field of traps is Fickean but non-Gaussian

Abstract

Motivated by experimental observations, we look for the diffusion of Brownian particles in a medium where they can be either trapped in randomly disposed deep traps or where they diffuse by the regular Fick's law outside of the traps. This process can be represented by two coupled equations - one valid inside the traps and another one outside - yielding the probability distribution of the distance run as a function of time. This probability depends on a unique dimensionless parameter λ which is proportional to the product of the (small) density of the traps times the long time of staying in the traps. The mean-square displacement is proportional to the lag time, and for finite or large λ the probability density is no longer Gaussian but exponential on intermediate time scales.