A minimal model for short-time diffusion in periodic potentials

TL;DR

This paper presents a minimal, analytically tractable model for understanding the short-time, non-diffusive dynamics of a colloidal particle in a periodic potential, capturing cage-like behavior and de-caging times.

Contribution

It introduces a simple two-state master equation model that accurately describes short-time diffusion plateaus and cage dynamics in periodic potentials.

Findings

Analytic expressions for plateau heights.

Estimate of de-caging time from non-Gaussian deviations.

Model matches observed short-time non-diffusive behavior.

Abstract

We investigate the dynamics of a single, overdamped colloidal particle, which is driven by a constant force through a one-dimensional periodic potential. We focus on systems with large barrier heights where the lowest-order cumulants of the density field, that is, average position and the mean-squared displacement, show nontrivial (non-diffusive) short-time behavior characterized by the appearance of plateaus. We demonstrate that this "cage-like" dynamics can be well described by a discretized master equation model involving two states (related to two positions) within each potential valley. Non-trivial predictions of our approach include analytic expressions for the plateau heights and an estimate of the "de-caging time" obtained from the study of deviations from Gaussian behaviour. The simplicity of our approach means that it offers a minimal model to describe the short-time behavior of systems with hindered dynamics.