Transport properties of Brownian particles confined to a narrow channel by a periodic potential

TL;DR

This paper analyzes the transport behavior of Brownian particles in a narrow, periodically varying channel under external force, using asymptotic analysis and simulations to understand how confinement and potential shape influence particle movement.

Contribution

It extends the Fick-Jacobs approximation to include a periodic potential and compares analytical results with simulations, demonstrating excellent agreement.

Findings

Analytical expressions match Brownian dynamics simulations.

Confining potential influences transport similarly to solid channels.

Good agreement over a wide range of Peclet numbers.

Abstract

We investigate the transport of Brownian particles in a two-dimensional potential under the action of a uniform external force. The potential is periodic in one direction and confines the particle to a narrow channel of varying cross-section in the other direction. We apply the standard long-wave asymptotic analysis in the narrow dimension and show that the leading order term is equivalent to that obtained previously from a direct extension of the Fick-Jacobs approximation. We also show that the confining potential has similar effects on the transport of Brownian particles to those induced by a solid channel. Finally, we compare the analytical results with Brownian dynamics simulations in the case of a sinusoidal variation of the width of the parabolic potential in the cross-section. We obtain excellent agreement for the marginal probability distribution, the average velocity of the Brownian particles and the asymptotic dispersion coefficient, over a wide range of Peclet numbers.