Normal and anomalous diffusion of Brownian particles on disordered potentials

TL;DR

This paper investigates how Brownian particles transition from normal to anomalous diffusion on disordered potentials, deriving exact formulas for diffusion coefficients and exponents, and validating results with simulations.

Contribution

It provides analytical expressions for diffusion behavior on disordered potentials with arbitrary distributions, highlighting conditions for normal and anomalous diffusion.

Findings

Gaussian-distributed potentials lead to always normal diffusion.

Exponential-distributed potentials cause vanishing diffusion below a critical temperature.

Analytical calculation of the anomalous diffusion exponent using the random trap model.

Abstract

In this work we study the transition from normal to anomalous diffusion of Brownian particles on disordered potentials. The potential model consists of a series of "potential hills" (defined on unit cell of constant length) whose heights are chosen randomly from a given distribution. We calculate the exact expression for the diffusion coefficient in the case of uncorrelated potentials for arbitrary distributions. We particularly show that when the potential heights have a Gaussian distribution (with zero mean and a finite variance) the diffusion of the particles is always normal. In contrast when the distribution of the potential heights are exponentially distributed we show that the diffusion coefficient vanishes when system is placed below a critical temperature. We calculate analytically the diffusion exponent for the anomalous (subdiffusive) phase by using the so-called "random trap model". We test our predictions by means of Langevin simulations obtaining good agreement within the accuracy of our numerical calculations.