Multiscale analysis of the conductivity in the Lorentz mirrors model

TL;DR

This paper analyzes the conductivity in the Lorentz mirrors model using a multiscale approach, deriving a recursive relation for conductivity and computing crossing probabilities in a deterministic system within a random environment.

Contribution

It introduces a perturbative multiscale method to compute conductivity and crossing probabilities in the Lorentz mirrors model, providing a recursive relation for conductivity in three dimensions.

Findings

Crossing probability scales as κ/(κ+N) with computed κ

Recursive relation for conductivity in 3D: κ_{n+1} = κ_n(1 + (κ_n/2^n)α)

Conductivity sequence converges to a finite limit

Abstract

We consider the mirrors model in $d$ dimensions on an infinite slab and with unit density. This is a deterministic dynamics in a random environment. We argue that the crossing probability of the slab goes like $\kappa/(\kappa+N)$ where $N$ is the width of the slab. We are able to compute $\kappa$ perturbatively by using a multiscale approach. The only small parameter involved in the expansion is the inverse of the size of the system. This approach rests on an inductive process and a closure assumption adapted to the mirrors model. For $d=3$, we propose the recursive relation for the conductivity $\kappa_n$ at scale $n$ : $\kappa_{n+1}=\kappa_n(1+\frac{\kappa_n}{2^{n}}\alpha)$, up to $o(1/2^n)$ terms and with $\alpha\simeq 0.0374$. This sequence has a finite limit.