Approach to asymptotically diffusive behavior for Brownian particles in periodic potentials : extracting information from transients

TL;DR

This paper investigates how to accurately extract the long-term diffusion constant of Brownian particles in periodic potentials from transient MSD data, providing explicit formulas and analysis of the constant term in the linear fit.

Contribution

It introduces a method to compute the non-zero constant term in the MSD linear fit, revealing its dependence on potential curvature and temperature for one-dimensional systems.

Findings

The constant term in the MSD linear fit is always positive for Brownian particles in periodic potentials.

At low temperatures, the constant depends on the curvature of the potential's minima.

For a symmetric continuous time random walk, the constant can be positive or negative, relating to the variance of hopping times.

Abstract

A Langevin process diffusing in a periodic potential landscape has a time dependent diffusion constant which means that its average mean squared displacement (MSD) only becomes linear at late times. The long time, or effective diffusion constant, can be estimated from the slope of a linear fit of the MSD at late times. Due to the cross over between a short time microscopic diffusion constant, which is independent of the potential, to the effective late time diffusion constant, a linear fit of the MSD will not in general pass through the origin and will have a non-zero constant term. Here we address how to compute the constant term and provide explicit results for Brownian particles in one dimension in periodic potentials. We show that the constant is always positive and that at low temperatures it depends on the curvature of the minimum of the potential. For comparison we also consider the same question for the simpler problem of a symmetric continuous time random walk in discrete space. Here the constant can be positive or negative and can be used to determine the variance of the hopping time distribution.