Knudsen gas in a finite random tube: transport diffusion and first passage properties

TL;DR

This paper analyzes transport diffusion in a stochastic billiard model within a random tube, proving Fick's law, the equality of transport and self-diffusion coefficients, and examining crossing times and boundary effects.

Contribution

It establishes the validity of Fick's law and the equality of transport and self-diffusion coefficients in a random tube model, with rigorous proofs and analysis of boundary effects.

Findings

Linear density profile in the thermodynamic limit

Transport diffusion coefficient equals self-diffusion coefficient

Computed Milne extrapolation length depending on tube shape

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. We also study some properties of the crossing time and compute the Milne extrapolation length in dependence on the shape of the random tube.