Thomas GILBERT       Université Libre de Bruxelles

Equilibrium and non-equilibrium Galton boards

Galton boards are models of deterministic diffusion in a uniform
external field, akin to driven periodic Lorentz gases, which are here
considered in the absence of dissipation mechanisms. By considering a
cylindrical geometry with axis along the direction of the external field,
the two-dimensional board becomes a model for one-dimensional mass
transport along the direction of the external field. Equilibrium and
non-equilibrium stationary states arise, depending on the specific choice
of boundary conditions at the ends of the cylinder. While the former is
associated to a closed board and has a uniform invariant measure, the
latter is associated to an open board with the two ends in contact with
particle reservoirs and has a fractal invariant measure. Numerical results
are presented in support of this claim. A correspondence
is established between the local phase-space statistics and their
macroscopic counter-part. Analytical results are obtained for the statistics
of multi-baker maps associated to such a non-uniform diffusion process and
the fractality of the invariant state related to the positivity of the
entropy production rate.