Entropic particle transport: higher order corrections to the Fick-Jacobs diffusion equation

TL;DR

This paper derives higher order corrections to the Fick-Jacobs equation for particle transport in channels, providing a spatially dependent diffusion coefficient and validating results with numerical simulations.

Contribution

It introduces a method to include higher order corrections in the Fick-Jacobs approximation, accounting for spatially varying diffusion in channel transport.

Findings

Higher order corrections yield a spatially dependent diffusion coefficient D(x).

The average transport velocity is the product of the Fick-Jacobs result and <D(x)>.

Numerical simulations confirm the analytical predictions.

Abstract

Transport of point-size Brownian particles under the influence of a constant and uniform force field through a three-dimensional channel with smoothly varying periodic cross-section is investigated. Here, we employ an asymptotic analysis in the ratio between the difference of the widest and the most narrow constriction divided through the period length of the channel geometry. We demonstrate that the leading order term is equivalent to the Fick-Jacobs approximation. By use of the higher order corrections to the probability density we derive an expression for the spatially dependent diffusion coefficient D(x) which substitutes the constant diffusion coefficient present in the common Fick-Jacobs equation. In addition, we show that in the diffusion dominated regime the average transport velocity is obtained as the product of the zeroth-order Fick-Jacobs result and the expectation value of the spatially dependent diffusion coefficient <D(x)>. The analytic findings are corroborated with the precise numerical results of a finite element calculation of the Smoluchowski diffusive particle dynamics occurring in a reflection symmetric sinusoidal-shaped channel.