Effective one-dimensional description of confined diffusion biased by a transverse gravitational force

TL;DR

This paper derives an effective one-dimensional diffusion equation for particles in a 2D channel under transverse bias, incorporating a position-dependent diffusion coefficient influenced by the transverse force, verified through an exactly solvable model.

Contribution

The authors develop a recurrence mapping method to obtain an extended Fick-Jacobs equation with a corrected diffusion coefficient accounting for transverse bias.

Findings

Derived an approximate formula for the effective diffusion coefficient D(x)

Validated the approach with an exactly solvable linear cone model

Extended the Fick-Jacobs equation to include transverse force effects

Abstract

Diffusion of point-like non interacting particles in a two-dimensional (2D) channel of varying cross section is considered. The particles are biased by a constant force in the transverse direction. We apply our recurrence mapping procedure, which enables us to derive an effective one-dimensional (1D) evolution equation, governing the 1D density of the particles in the channel. In the limit of stationary flow, we arrive at an extended Fick-Jacobs equation, corrected by an effective diffusion coefficient D(x), depending on the longitudinal coordinate x. Our result is an approximate formula for D(x), involving also influence of the transverse force. Our calculations are verified on the stationary diffusion in a linear cone, which is exactly solvable.