Flooding dynamics of diffusive dispersion in a random potential

Abstract

We discuss the combined effects of overdamped motion in a quenched random potential and diffusion, in one dimension, in the limit where the diffusion coefficient is small. Our analysis considers the statistics of the mean first-passage time T(x) to reach position x, arising from different realisations of the random potential. Specifically, we contrast the median $${\bar{T}}(x)$$ T ¯ ( x ) , which is an informative description of the typical course of the motion, with the expectation value $$\langle T(x)\rangle $$ ⟨ T ( x ) ⟩ , which is dominated by rare events where there is an exceptionally high barrier to diffusion. We show that at relatively short times the median $${\bar{T}}(x)$$ T ¯ ( x ) is explained by a ‘flooding’ model, where T(x) is predominantly determined by the highest barriers which are encountered before reaching position x. These highest barriers are quantified using methods of extreme value statistics.