Solutions of first passage times problems: a biscaling approach
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arXiv
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| Format: | Preprint |
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2025
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| author | Baravi, Talia Kessler, David A. Barkai, Eli |
| author_facet | Baravi, Talia Kessler, David A. Barkai, Eli |
| contents | We study the first-passage time (FPT) problem for widespread recurrent processes in confined though large systems and present a comprehensive framework for characterizing the FPT distribution over many time scales. We find that the FPT statistics can be described by two scaling functions: one corresponds to the solution for an infinite system, and the other describes a scaling that depends on system size. We find a universal scaling relationship for the FPT moments $\langle t^q \rangle$ with respect to the domain size and the source-target distance. This scaling exhibits a transition at $q_c=θ$, where $θ$ is the persistence exponent. For low-order moments with $q<q_c$, convergence occurs towards the moments of an infinite system. In contrast, the high-order moments, $q>q_c$, can be derived from an infinite density function. The presented uniform approximation, connecting the two scaling functions, provides a description of the first-passage time statistics across all time scales. We extend the results to include diffusion in a confining potential in the high-temperature limit, where the potential strength takes the place of the system's size as the relevant scale. This study has been applied to various mediums, including a particle in a box, two-dimensional wedge, fractal geometries, non-Markovian processes and the non-equilibrium process of resetting. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_15956 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Solutions of first passage times problems: a biscaling approach Baravi, Talia Kessler, David A. Barkai, Eli Statistical Mechanics We study the first-passage time (FPT) problem for widespread recurrent processes in confined though large systems and present a comprehensive framework for characterizing the FPT distribution over many time scales. We find that the FPT statistics can be described by two scaling functions: one corresponds to the solution for an infinite system, and the other describes a scaling that depends on system size. We find a universal scaling relationship for the FPT moments $\langle t^q \rangle$ with respect to the domain size and the source-target distance. This scaling exhibits a transition at $q_c=θ$, where $θ$ is the persistence exponent. For low-order moments with $q<q_c$, convergence occurs towards the moments of an infinite system. In contrast, the high-order moments, $q>q_c$, can be derived from an infinite density function. The presented uniform approximation, connecting the two scaling functions, provides a description of the first-passage time statistics across all time scales. We extend the results to include diffusion in a confining potential in the high-temperature limit, where the potential strength takes the place of the system's size as the relevant scale. This study has been applied to various mediums, including a particle in a box, two-dimensional wedge, fractal geometries, non-Markovian processes and the non-equilibrium process of resetting. |
| title | Solutions of first passage times problems: a biscaling approach |
| topic | Statistical Mechanics |
| url | https://arxiv.org/abs/2503.15956 |