Kinetic theory of diffusion in a channel of varying cross section

TL;DR

This paper derives a modified diffusion equation for particles in a channel with varying cross section, incorporating corrections for density inhomogeneity, based on kinetic theory and Chapman-Enskog expansion.

Contribution

It introduces a new derivation of a Ficks-Jacobs type equation with an effective diffusion coefficient accounting for density variations.

Findings

Derived a one-dimensional kinetic equation from the 2D Enskog-Boltzmann-Lorentz equation.

Obtained a macroscopic diffusion equation with correction terms for channel inhomogeneity.

Validated the modified diffusion coefficient's role in describing self-diffusion in variable cross section channels.

Abstract

Self-diffusion along the longitudinal coordinate in a channel of varying cross section is considered. The starting point is the two-dimensional Enskog-Boltzmann-Lorentz kinetic equation with appropriated boundary conditions. It is integrated over the transversal coordinate to get an approximated one-dimensional kinetic equation, keeping the relevant properties of the original one. Then, a macroscopic equation for the time evolution of the longitudinal density is derived, by means of a modified Chapman-Enskog expansion method, that takes into account the inhomogeneity of the equilibrium longitudinal density. This transport equation has the form of the phenomenological Ficks-Jacobs equation, but with an effective diffusion coefficient that contains corrections associated to the variation of the slope of the equilibrium longitudinal density profile.