Biased Brownian motion in extreme corrugated tubes

TL;DR

This paper analyzes biased Brownian motion in complex, smoothly varying tubes, deriving higher order corrections to the Fick-Jacobs approximation and demonstrating improved accuracy for extreme geometries through analytical and numerical methods.

Contribution

It introduces higher order correction terms to the Fick-Jacobs approximation for biased Brownian motion in corrugated tubes, enhancing accuracy for extreme geometries.

Findings

Higher order corrections improve the Fick-Jacobs approximation accuracy.

Analytical results are validated with finite element simulations.

The method outperforms traditional spatially dependent diffusion coefficient approaches.

Abstract

Biased Brownian motion of point-size particles in a three-dimensional tube with smoothly varying cross-section is investigated. In the fashion of our recent work [Martens et al., PRE 83,051135] we employ an asymptotic analysis to the stationary probability density in a geometric parameter of the tube geometry. We demonstrate that the leading order term is equivalent to the Fick-Jacobs approximation. Expression for the higher order corrections to the probability density are derived. Using this expansion orders we obtain that in the diffusion dominated regime the average particle current equals the zeroth-order Fick-Jacobs result corrected by a factor including the corrugation of the tube geometry. In particular we demonstrate that this estimate is more accurate for extreme corrugated geometries compared to the common applied method using the spatially dependent diffusion coefficient D(x,f). The analytic findings are corroborated with the finite element calculation of a sinusoidal-shaped tube.