The probability distribution of Brownian motion in periodic potentials

TL;DR

This paper derives an analytical expression for the probability distribution of an overdamped Brownian particle in a periodic potential, validating it with simulations and recovering known formulas for effective diffusion.

Contribution

It provides a general solution for the probability distribution function in periodic potentials, including asymptotic accuracy and recovery of the Lifson-Jackson formula.

Findings

Analytical PDF matches Langevin simulations under specified conditions.

Solution is valid for any periodic even potential.

Recovers the Lifson-Jackson formula for effective diffusion.

Abstract

We calculate the probability distribution function (PDF) of an overdamped Brownian particle moving in a periodic potential energy landscape $U(x)$. The PDF is found by solving the corresponding Smoluchowski diffusion equation. We derive the solution for any periodic even function $U(x)$, and demonstrate that it is asymptotically (at large time $t$) correct up to terms decaying faster than $\sim t^{-3/2}$. As part of the derivation, we also recover the Lifson-Jackson formula for the effective diffusion coefficient of the dynamics. The derived solution exhibits agreement with Langevin dynamics simulations when (i) the periodic length is much larger than the ballistic length of the dynamics, and (ii) when the potential barrier $\Delta U=\max(U(x))-\min(U(x))$ is not much larger than the thermal energy $k_BT$.