Multiscale analysis of the conductivity in the Lorentz mirrors model

TL;DR

The paper tackles diffusion-like transport in the deterministic mirrors model by deploying a multiscale decomposition of slabs. It derives a recursion for the conductivity in 3D, showing the crossing probability scales as C_N ~ κ_∞/N with a finite κ_∞ ≈ 1.5403, closely matching the non-backtracking random walk value. A closure hypothesis controls correlations across scales, enabling a near-Markovian effective description and revealing the mechanism behind normal conductivity despite memory effects. The work suggests a general framework for analyzing transport in deterministic disordered systems and points to rigorous proofs via tightened bounds on correlation terms.

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 $κ/(κ+N)$ where $N$ is the width of the slab. We are able to compute $κ$ 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 $κ_n$ at scale $n$ : $κ_{n+1}=κ_n(1+\frac{κ_n}{2^{n}}α)$, up to $o(1/2^n)$ terms and with $α\simeq 0.0374$. This sequence has a finite limit.

Figures (7)

• Figure 1: Two-dimensional mirrors model on a portion of the lattice $\mathbb{Z}^2$. Mirrors (red) reflect the deterministic trajectory (blue). Periodic vertical boundary conditions are indicated by short arrows leaving and re-entering the domain.
• Figure 2: The slab $\Lambda_N$ as a subset of $\mathbb Z^d$ (here represented in two dimensions), together with the incoming sets $I_N^\pm$ (left) and outgoing sets $O_N^\pm$ (right) in the direction $\mathbf e_1$. The underlying lattice structure is shown explicitly as the edges of $\mathbb Z^2$, and the thick dots mark the phase-space positions where the velocities are attached.
• Figure 3: Multiscale composition in 3D: $2^n+2^n\!\to\!2^{n+1}$, then $2^{n+1}+2^{n+1}\!\to\!2^{n+2}$.
• Figure 4: Trajectories contributing to $R_11$ for $l=2$.
• Figure 5: The measured ratio $1+\Delta_n=c_{n+1}(2-c_n)/c_n$ is plotted in blue with a 95% confidence interval, while $1+c_n(2-c_n)(1-c_n)^2R_{11}$ is plotted in red. The contribution of $R_2$ lowers the red values significantly only for the two first values of $N$, the contribution of $R_{12}$ increases those values in a way that is almost not visible at this scale.