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 mechanism. 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.