Steady-state conduction in self-similar billiards

TL;DR

This paper investigates steady-state conduction in a self-similar Lorentz billiard channel, revealing fractal invariant measures and a connection between phase-space contraction and entropy production.

Contribution

It introduces a self-similar billiard model with a nonequilibrium steady state and demonstrates the fractal nature of its invariant measure and thermodynamic consistency.

Findings

Invariant measure exhibits fractal properties.

Phase-space contraction rate matches entropy production near equilibrium.

Steady particle flow from small to large scales.

Abstract

The self-similar Lorentz billiard channel is a spatially extended deterministic dynamical system which consists of an infinite one-dimensional sequence of cells whose sizes increase monotonically according to their indices. This special geometry induces a nonequilibrium stationary state with particles flowing steadily from the small to the large scales. The corresponding invariant measure has fractal properties reflected by the phase-space contraction rate of the dynamics restricted to a single cell with appropriate boundary conditions. In the near-equilibrium limit, we find numerical agreement between this quantity and the entropy production rate as specified by thermodynamics.