Lawley Sean D
Department of Mathematics, University of Utah, Salt Lake City, Utah 84112, USA.
Phys Rev E. 2020 Jan;101(1-1):012413. doi: 10.1103/PhysRevE.101.012413.
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.
许多物理、化学和生物过程的时间尺度由扩散的首次通过时间（FPT）决定。绝大多数FPT研究都聚焦于单个扩散搜索者找到目标所需的时间。然而，在许多系统中，更具相关性的量是最快的搜索者从一大群搜索者中找到目标所需的时间。这个最快的FPT取决于极其罕见的事件，并且其时间尺度比给定单个搜索者的FPT要快得多。在这项工作中，我们证明了一个关于最快FPT各阶矩的简单显式公式。该公式具有显著的通用性，因为它适用于以下d维扩散过程：（i）具有一般空间依赖扩散率和力场的过程；（ii）在黎曼流形上的过程；（iii）存在反射障碍物的过程；以及（iv）具有部分吸收目标的过程。我们的结果严格地证实、推广、修正并统一了关于最快FPT的各种猜想和启发式方法。
