The probability distribution of Brownian motion in periodic potentials

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 $k_{B}T$.

arXiv comment6 pages, 2 figures, Accepted for publication in Phys. Rev. E

References in corpus (6)

• Brownian motors: noisy transport far from equilibrium
• Artificial Brownian motors: Controlling transport on the nanoscale
• Kinetic Equations for Diffusion in the Presence of Entropic Barriers
• A simple and effective Verlet-type algorithm for simulating Langevin dynamics
• Application of the G-JF Discrete-Time Thermostat for Fast and Accurate Molecular Simulations
• Controlling anomalous stresses in soft field-responsive systems

Cited by in corpus (5)

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• Infinite ergodic theory meets Boltzmann statistics
• Realizing Quantitative Quasiparticle Modeling of Skyrmion Dynamics in Arbitrary Potentials
• Brownian particles in periodic potentials: coarse-graining versus fine structure
• Nonsteady-state diffusion in two-dimensional periodic channels