Transport diffusion coefficient for a Knudsen gas in a random tube

TL;DR

This paper proves Fick's law and the equality of transport and self-diffusion coefficients for particles in a random tube, showing linear density profiles and consistent diffusion behavior in the thermodynamic limit.

Contribution

It establishes the equivalence of transport and self-diffusion coefficients and confirms Fick's law for a broad class of random, rough tubes using stochastic billiard models.

Findings

Linear density profile in the thermodynamic limit

Transport diffusion coefficient equals self-diffusion coefficient

Validation of Fick's law in random tube models

Abstract

We consider transport diffusion in a stochastic billiard in a random tube which is elongated in the direction of the first coordinate (the tube axis). Inside the random tube, which is stationary and ergodic, non-interacting particles move straight with constant speed. Upon hitting the tube walls, they are reflected randomly, according to the cosine law: the density of the outgoing direction is proportional to the cosine of the angle between this direction and the normal vector. Steady state transport is studied by introducing an open tube segment as follows: We cut out a large finite segment of the tube with segment boundaries perpendicular to the tube axis. Particles which leave this piece through the segment boundaries disappear from the system. Through stationary injection of particles at one boundary of the segment a steady state with non-vanishing stationary particle current is maintained. We prove (i) that in the thermodynamic limit of an infinite open piece the coarse-grained density profile inside the segment is linear, and (ii) that the transport diffusion coefficient obtained from the ratio of stationary current and effective boundary density gradient equals the diffusion coefficient of a tagged particle in an infinite tube. Thus we prove Fick's law and equality of transport diffusion and self-diffusion coefficients for quite generic rough (random) tubes.