Biased diffusion in a piecewise linear random potential

TL;DR

This paper analyzes how particles diffuse under bias in a piecewise linear random potential, revealing conditions for normal and anomalous long-term behavior through Laplace transform techniques.

Contribution

It introduces a method to determine short- and long-time diffusion behavior in biased systems with piecewise linear potentials, including conditions for anomalous diffusion.

Findings

Short-time diffusion is always ballistic.

Long-time diffusion can be normal or anomalous.

Derived laws governing both diffusion regimes.

Abstract

We study the biased diffusion of particles moving in one direction under the action of a constant force in the presence of a piecewise linear random potential. Using the overdamped equation of motion, we represent the first and second moments of the particle position as inverse Laplace transforms. By applying to these transforms the ordinary and the modified Tauberian theorem, we determine the short- and long-time behavior of the mean-square displacement of particles. Our results show that while at short times the biased diffusion is always ballistic, at long times it can be either normal or anomalous. We formulate the conditions for normal and anomalous behavior and derive the laws of biased diffusion in both these cases.