The Mirrors Model : Macroscopic Diffusion Without Noise or Chaos

TL;DR

This paper introduces the mirrors model, a deterministic and non-ergodic system, demonstrating that Fick's law of diffusion holds in high dimensions based on the behavior of crossing orbits, without relying on noise or chaos.

Contribution

It establishes a macroscopic diffusion law in a deterministic, non-ergodic system by linking orbit crossing behavior to Fick's law validity in high dimensions.

Findings

Fick's law holds in three or more dimensions for the mirrors model.

The behavior of crossing orbits determines macroscopic diffusion properties.

Negative correlations between crossing orbits are crucial for the diffusion law.

Abstract

We first clarify through classical examples the status of the laws of macroscopic physics as laws of large numbers. We next consider the mirrors model in a finite $d$-dimensional domain and connected to particles reservoirs at fixed chemical potentials. The dynamics is purely deterministic and non-ergodic. We study the macroscopic current of particles in the stationary regime. We show first that when the size of the system goes to infinity, the behaviour of the stationary current of particles is governed by the proportion of orbits crossing the system. This allows to formulate a necessary and sufficient condition on the distribution of the set of orbits that ensures the validity of Fick's law. Using this approach, we show that Fick's law relating the stationary macroscopic current of particles to the concentration difference holds in three dimensions and above. The negative correlations between crossing orbits play a key role in the argument.