The Thermodynamic Limit of Extreme First-Passage Times
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2025
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866909974235447296 |
|---|---|
| author | Baravi, Talia Barkai, Eli |
| author_facet | Baravi, Talia Barkai, Eli |
| contents | The statistics of the slowest first-passage time among a large population of $N$ searchers is crucial for determining the completion time of many stochastic processes. Classical extreme-value theory predicts that for diffusing particles in a finite domain of size $L$, the slowest first passage time follows a Gumbel distribution, but a Fréchet distribution in an infinite domain. Here, we study the physically relevant thermodynamic limit where both $N$ and $L$ diverge while the density $ρ= N/L$ remains constant. We obtain an explicit solution for the extreme value in the thermodynamic limit, which recovers the Fréchet and Gumbel distributions in the low- and high-density limits, respectively, and reveals new, nontrivial behavior at intermediate densities. We then extend the framework to compact diffusion on fractal domains, showing that the walk dimension $d_w$ and fractal dimension $d_f$ control the extreme-value statistics via geometry-dependent scaling. The theory yields the full set of moments and finite-density corrections, providing a unified description of slowest-arrival times in confined Euclidean and fractal media. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_06098 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Thermodynamic Limit of Extreme First-Passage Times Baravi, Talia Barkai, Eli Statistical Mechanics The statistics of the slowest first-passage time among a large population of $N$ searchers is crucial for determining the completion time of many stochastic processes. Classical extreme-value theory predicts that for diffusing particles in a finite domain of size $L$, the slowest first passage time follows a Gumbel distribution, but a Fréchet distribution in an infinite domain. Here, we study the physically relevant thermodynamic limit where both $N$ and $L$ diverge while the density $ρ= N/L$ remains constant. We obtain an explicit solution for the extreme value in the thermodynamic limit, which recovers the Fréchet and Gumbel distributions in the low- and high-density limits, respectively, and reveals new, nontrivial behavior at intermediate densities. We then extend the framework to compact diffusion on fractal domains, showing that the walk dimension $d_w$ and fractal dimension $d_f$ control the extreme-value statistics via geometry-dependent scaling. The theory yields the full set of moments and finite-density corrections, providing a unified description of slowest-arrival times in confined Euclidean and fractal media. |
| title | The Thermodynamic Limit of Extreme First-Passage Times |
| topic | Statistical Mechanics |
| url | https://arxiv.org/abs/2509.06098 |