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We consider transport diffusion in a stochastic billiard in a random tubewhich is elongated in the direction of the first coordinate (the tube axis).Inside the random tube, which is stationary and ergodic, non-interactingparticles move straight with constant speed. Upon hitting the tube walls, theyare reflected randomly, according to the cosine law: the density of theoutgoing direction is proportional to the cosine of the angle between thisdirection and the normal vector. Steady state transport is studied byintroducing an open tube segment as follows: We cut out a large finite segmentof the tube with segment boundaries perpendicular to the tube axis. Particleswhich leave this piece through the segment boundaries disappear from thesystem. Through stationary injection of particles at one boundary of thesegment a steady state with non-vanishing stationary particle current ismaintained. We prove (i) that in the thermodynamic limit of an infinite openpiece the coarse-grained density profile inside the segment is linear, and (ii)that the transport diffusion coefficient obtained from the ratio of stationarycurrent and effective boundary density gradient equals the diffusioncoefficient of a tagged particle in an infinite tube. Thus we prove Fick's lawand equality of transport diffusion and self-diffusion coefficients for quitegeneric rough (random) tubes. We also study some properties of the crossingtime and compute the Milne extrapolation length in dependence on the shape ofthe random tube.

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