Probability distribution of Brownian motion in periodic potentials

Matan Sivan, Oded Farago

Research output: Contribution to journalArticlepeer-review

8 Scopus citations


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 ∼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 (1) the periodic length is much larger than the ballistic length of the dynamics, and (2) when the potential barrier ΔU=max[U(x)]-min[U(x)] is not much larger than the thermal energy kBT.

Original languageEnglish
Article number052117
JournalPhysical Review E
Issue number5
StatePublished - 15 Nov 2018

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Statistics and Probability
  • Condensed Matter Physics


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