Multiscale analysis of the conductivity in the Lorentz mirrors model
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866914454582591488 |
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| author | Lefevere, Raphael |
| author_facet | Lefevere, Raphael |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_24091 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Multiscale analysis of the conductivity in the Lorentz mirrors model Lefevere, Raphael Probability Statistical Mechanics Mathematical Physics 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. |
| title | Multiscale analysis of the conductivity in the Lorentz mirrors model |
| topic | Probability Statistical Mechanics Mathematical Physics |
| url | https://arxiv.org/abs/2510.24091 |