  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.
