Hierarchical Lorentz Mirror Model: Normal Transport and a Universal $2/3$ Mean--Variance Law

TL;DR

This paper introduces a hierarchical Lorentz mirror model to analyze normal transport phenomena and uncovers a universal $2/3$ mean--variance law for conductance across dimensions, supported by theoretical proofs and numerical evidence.

Contribution

The paper presents a hierarchical version of the Lorentz mirror model with an exact recursion for crossings, proving normal transport in higher dimensions and predicting a universal variance-to-mean ratio.

Findings

Proves normal transport in dimensions $d extgreater=3$

Predicts the universal $2/3$ variance-to-mean ratio for conductance

Provides numerical evidence supporting the $2/3$ law in $d=3$

Abstract

The Lorentz mirror model provides a clean setting to study macroscopic transport generated solely by quenched environmental randomness. We introduce a hierarchical version whose distribution of left--right crossings satisfies an exact recursion. In dimensions $d\geq 3$, we prove normal transport: the mean conductance scales as (cross-section)/(length) on all length scales. A Gaussian closure, supported by numerics, predicts that the variance-to-mean ratio of the conductance converges to the universal value $2/3$ for all $d\geq 2$ (the ``$2/3$ law''). We provide numerical evidence for the $2/3$ law in the original (non-hierarchical) Lorentz mirror model in $d=3$, and conjecture that it is a universal signature of normal transport induced by random current matching. In the marginal case $d=2$, our hierarchical recursion reproduces the known scaling of the mean conductance and its variance. A YouTube video discussing the background and the main results of the paper is available: https://youtu.be/G1nqKd6MiXo