Diffusive Transport in Periodic Potentials: Underdamped Dynamics

TL;DR

This paper introduces a comprehensive method for calculating the diffusion tensor of Brownian particles in periodic potentials, applicable across dimensions and dissipation regimes, with explicit formulas and numerical techniques.

Contribution

It provides a systematic, rigorous approach for diffusion tensor calculation, including explicit formulas in 1D and higher-order corrections, applicable in underdamped and overdamped limits.

Findings

Explicit diffusion coefficient formula in 1D for underdamped limit

Higher order corrections to Lifson-Jackson formula in overdamped limit

Numerical method performs well across all dissipation values

Abstract

In this paper we present a systematic and rigorous method for calculating the diffusion tensor for a Brownian particle moving in a periodic potential which is valid in arbitrary dimensions and for all values of the dissipation. We use this method to obtain an explicit formula for the diffusion coefficient in one dimension which is valid in the underdamped limit, and we also obtain higher order corrections to the Lifson-Jackson formula for the diffusion coefficient in the overdamped limit. A numerical method for calculating the diffusion coefficient is also developed and is shown to perform extremely well for all values of the dissipation.