Fractality of the non-equilibrium stationary states of open volume-preserving systems: II. Galton boards

TL;DR

This paper investigates the fractal nature of non-equilibrium stationary states in open Galton board models, linking microscopic phase-space structures to macroscopic diffusion and entropy production.

Contribution

It provides analytical results connecting fractal phase-space structures with macroscopic diffusion and entropy production in Galton board models.

Findings

Fractal non-equilibrium stationary states are observed under flux boundary conditions.

Analytical statistics for multi-baker map models of diffusion are derived.

Fractality of invariant states explains positive entropy production.

Abstract

Galton boards are models of deterministic diffusion in a uniform external field, akin to driven periodic Lorentz gases, here considered in the absence of dissipation mechanism. Assuming 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. This is a purely diffusive process which admits fractal non-equilibrium stationary states under flux boundary conditions. Analytical results are obtained for the statistics of multi-baker maps modeling such a non-uniform diffusion process. A correspondence is established between the local phase-space statistics and their macroscopic counter-parts. The fractality of the invariant state is shown to be responsible for the positiveness of the entropy production rate.