Solutions of first passage times problems: a biscaling approach

Talia Baravi; David A. Kessler; Eli Barkai

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arXiv · 2503.15956

Solutions of first passage times problems: a biscaling approach

Talia Baravi·David A. Kessler·Eli Barkai

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Abstract

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 $q_c=\theta$, where $\theta$ is the persistence exponent. For low-order moments with $q q_c$, 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.

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Talia Baravi, David A. Kessler, Eli Barkai. 2025-03-20. Solutions of first passage times problems: a biscaling approach. https://arxiv.org/abs/2503.15956

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