Flooding dynamics of diffusive dispersion in a random potential

TL;DR

This paper investigates how diffusion in a one-dimensional random potential affects first-passage times, revealing that typical times are governed by the highest barriers encountered, modeled through extreme value statistics.

Contribution

It introduces a flooding model to explain median first-passage times in diffusive systems within random potentials, emphasizing the role of extreme barriers.

Findings

Median first-passage time is determined by the highest encountered barriers.

Extreme value statistics effectively quantify the highest barriers.

Short-time dispersion is dominated by the flooding of the highest barriers.

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 $\bar T(x)$, which is an informative description of the typical course of the dispersion, with the expectation value $\langle T(x)\rangle$, which is dominated by rare events where there is an exceptionally high barrier to diffusion. We show that at relatively short times the median $\bar T(x)$ is explained by a 'flooding' model, where $T(x)$ is predominantly determined by the highest barriers which is encountered before reaching position $x$. These highest barriers are quantified using methods of extreme value statistics.