Universal Formula for Extreme First Passage Statistics of Diffusion

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.

11 pages, 3 figures

References in corpus (2)

• Mortality, Redundancy, and Diversity in Stochastic Search
• First Invader Dynamics in Diffusion-Controlled Absorption

Cited by in corpus (23)

• Distribution of extreme first passage times of diffusion
• From single-particle stochastic kinetics to macroscopic reaction rates: fastest first-passage time of $N$ random walkers
• Competition between slow and fast regimes for extreme first passage times of diffusion
• Extreme first passage times for random walks on networks
• Preface: New trends in first-passage methods and applications in the life sciences and engineering
• Depletion of Resources by a Population of Diffusing Species
• Extreme hitting probabilities for diffusion
• Being Heterogeneous Is Advantageous: Extreme Brownian Non-Gaussian Searches
• Extreme statistics of anomalous subdiffusion following a fractional Fokker-Planck equation: Subdiffusion is faster than normal diffusion
• First Passage Time for Many Particle Diffusion in Space-Time Random Environments
• Capture of a diffusive prey by multiple predators in confined space
• First passage time and change of entropy
• Target finding in fibrous biological environments
• Fastest first-passage time statistics for time-dependent particle injection
• Contact statistics in populations of noninteracting random walkers in two dimensions
• Target Searches of Interacting Brownian Particles
• Optimality of intercellular signaling: direct transport versus diffusion
• Optimal Searcher Distribution for Parallel Random Target Searches
• Fast decisions reflect biases, slow decisions do not
• Rare Events and Redundancy in Random Walkers Target Search in a Finite Domain
• First-passage properties of the jump process with a drift. The general case
• Concentration of Empirical First-Passage Times
• Encounter-based approach to target search problems: a review