Universal Formula for Extreme First Passage Statistics of Diffusion

TL;DR

This paper derives a universal explicit formula for the moments of the fastest first passage times in diffusion processes, applicable across diverse conditions and geometries, unifying previous conjectures and heuristics.

Contribution

It provides the first universal, explicit formula for the moments of the fastest FPT in diffusion, applicable to complex, high-dimensional, and obstacle-laden environments.

Findings

The formula is valid for arbitrary dimensions and diffusion conditions.

It applies to Riemannian manifolds and environments with obstacles.

It confirms and generalizes previous conjectures about fastest FPTs.

Abstract

The timescales of many physical, chemical, and biological processes are determined by first passage times (FPTs) of diffusion. The overwhelming majority of FPT research studies the time it takes a single diffusive searcher to find a target. However, the more relevant quantity in many systems is the time it takes the fastest searcher to find a target from a large group of searchers. This fastest FPT depends on extremely rare events and has a drastically faster timescale than the FPT of a given single searcher. In this work, we prove a simple explicit formula for every moment of the fastest FPT. The formula is remarkably universal, as it holds for $d$-dimensional diffusion processes (i) with general space-dependent diffusivities and force fields, (ii) on Riemannian manifolds, (iii) in the presence of reflecting obstacles, and (iv) with partially absorbing targets. Our results rigorously confirm, generalize, correct, and unify various conjectures and heuristics about the fastest FPT.