We study deterministic transport in a random medium using the mirrors model, a lattice Lorentz gas at unit density in which a particle moves deterministically in a frozen random configuration of mirror–scatterers. Despite the absence of chaos and the existence of finite trapping loops, numerical evidence suggests that the model exhibits normal conductivity in <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$d=3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo>=</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>. We develop a recursive multiscale expansion for the crossing probability of a slab of width N, showing that <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$C_N \sim \kappa /(\kappa +N)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>C</mi> <mi>N</mi> </msub> <mo>∼</mo> <mi>κ</mi> <mo stretchy="false">/</mo> <mrow> <mo stretchy="false">(</mo> <mi>κ</mi> <mo>+</mo> <mi>N</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> for large N and computing the conductivity constant <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\kappa $</EquationSource> <EquationSource Format="MATHML"><math> <mi>κ</mi> </math></EquationSource> </InlineEquation> through a renormalization procedure based on scale concatenation. The key idea is that a slab of width <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$2^{n+1}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mn>2</mn> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </math></EquationSource> </InlineEquation> may be decomposed into two independent slabs of width <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$2^n$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mn>2</mn> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation>, and that the crossing event can be expressed as a sum over trajectories that cross the left half, return a few times to the interface between the two halves, and finally cross the right half. This gives rise to a recursive relation in which the leading normal conduction scaling is propagated from scale <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$2^n$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mn>2</mn> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation> to scale <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$2^{n+1}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mn>2</mn> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </math></EquationSource> </InlineEquation>, while the subleading term yields the correction to the conductivity constant. For the mirrors model, it involves a correction factor with respect to a reference Markovian process that encodes correlations between crossings at scale <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$2^n$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mn>2</mn> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation>. These correlations are controlled through a closure assumption whose structure is shaped by the hard–core exclusion inherent to the reversible deterministic dynamics. The dominant contribution comes from trajectories with a single interface return, whose asymptotic contribution we identify explicitly. In <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$d=3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo>=</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation> the recursion takes the form <Equation ID="Equ65"> <EquationSource Format="TEX">$ \kappa _{n+1}=\kappa _n\Bigl (1+\alpha \,\frac{\kappa _n}{2^n}+o(2^{-n})\Bigr ), \qquad \alpha \simeq 0.0374, $</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <msub> <mi>κ</mi> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>κ</mi> <mi>n</mi> </msub> <mrow> <mo maxsize="1.623em" minsize="1.623em" stretchy="true">(</mo> </mrow> <mn>1</mn> <mo>+</mo> <mi>α</mi> <mspace width="0.166667em" /> <mfrac> <msub> <mi>κ</mi> <mi>n</mi> </msub> <msup> <mn>2</mn> <mi>n</mi> </msup> </mfrac> <mo>+</mo> <mi>o</mi> <mrow> <mo stretchy="false">(</mo> <msup> <mn>2</mn> <mrow> <mo>-</mo> <mi>n</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> <mrow> <mo maxsize="1.623em" minsize="1.623em" stretchy="true">)</mo> </mrow> <mo>,</mo> <mspace width="2em" /> <mi>α</mi> <mo>≃</mo> <mn>0.0374</mn> <mo>,</mo> </mrow> </math></EquationSource> </Equation>leading to a finite limit <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\kappa _\infty \simeq 1.5403$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>κ</mi> <mi>∞</mi> </msub> <mo>≃</mo> <mn>1.5403</mn> </mrow> </math></EquationSource> </InlineEquation>, in good agreement with numerical simulations performed in this paper. This value is close to the conductivity constant of a non–backtracking random walk, suggesting that the large–scale behavior of the mirrors model is effectively Markovian even though the microscopic dynamics is fully deterministic.

Multiscale Analysis of the Conductivity in the Lorentz Mirrors Model

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发表日期：

2026年5月26日

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Raphaël Lefevere

摘要

We study deterministic transport in a random medium using the mirrors model, a lattice Lorentz gas at unit density in which a particle moves deterministically in a frozen random configuration of mirror–scatterers. Despite the absence of chaos and the existence of finite trapping loops, numerical evidence suggests that the model exhibits normal conductivity in $d=3$ $d = 3$ . We develop a recursive multiscale expansion for the crossing probability of a slab of width N, showing that $C_N \sim \kappa /(\kappa +N)$ $C_{N} \sim κ / (κ + N)$ for large N and computing the conductivity constant $\kappa $ $κ$ through a renormalization procedure based on scale concatenation. The key idea is that a slab of width $2^{n+1}$ $2^{n + 1}$ may be decomposed into two independent slabs of width $2^n$ $2^{n}$ , and that the crossing event can be expressed as a sum over trajectories that cross the left half, return a few times to the interface between the two halves, and finally cross the right half. This gives rise to a recursive relation in which the leading normal conduction scaling is propagated from scale $2^n$ $2^{n}$ to scale $2^{n+1}$ $2^{n + 1}$ , while the subleading term yields the correction to the conductivity constant. For the mirrors model, it involves a correction factor with respect to a reference Markovian process that encodes correlations between crossings at scale $2^n$ $2^{n}$ . These correlations are controlled through a closure assumption whose structure is shaped by the hard–core exclusion inherent to the reversible deterministic dynamics. The dominant contribution comes from trajectories with a single interface return, whose asymptotic contribution we identify explicitly. In $d=3$ $d = 3$ the recursion takes the form $ \kappa _{n+1}=\kappa _n\Bigl (1+\alpha \,\frac{\kappa _n}{2^n}+o(2^{-n})\Bigr ), \qquad \alpha \simeq 0.0374, $ $κ_{n + 1} = κ_{n} (1 + α \frac{κ_{n}}{2^{n}} + o (2^{- n})), α ≃ 0.0374,$ leading to a finite limit $\kappa _\infty \simeq 1.5403$ $κ_{\infty} ≃ 1.5403$ , in good agreement with numerical simulations performed in this paper. This value is close to the conductivity constant of a non–backtracking random walk, suggesting that the large–scale behavior of the mirrors model is effectively Markovian even though the microscopic dynamics is fully deterministic.