From single-particle stochastic kinetics to macroscopic reaction rates: fastest first-passage time of $N$ random walkers

TL;DR

This paper derives explicit formulas and analyzes the distribution of the fastest first-passage time for multiple diffusing particles reaching a target, revealing a strong initial condition dependence and implications for molecular reactions.

Contribution

It provides the first explicit formulas for the moments and distribution of the fastest first-passage time for multiple particles in a bounded domain, highlighting the impact of initial conditions.

Findings

Fastest first-passage time scales as 1/N and 1/N^2 for partial and perfect reactions.

Initial conditions significantly influence the scaling of the fastest first-passage time.

Analytic solutions, scaling arguments, and simulations validate the results.

Abstract

We consider the first-passage problem for $N$ identical independent particles that are initially released uniformly in a finite domain $\Omega$ and then diffuse toward a reactive area $\Gamma$, which can be part of the outer boundary of $\Omega$ or a reaction centre in the interior of $\Omega$. For both cases of perfect and partial reactions, we obtain the explicit formulas for the first two moments of the fastest first-passage time (fFPT), i.e., the time when the first out of the $N$ particles reacts with $\Gamma$. Moreover, we investigate the full probability density of the fFPT. We discuss a significant role of the initial condition in the scaling of the average fastest first-passage time with the particle number $N$, namely, a much stronger dependence ($1/N$ and $1/N^2$ for partially and perfectly reactive targets, respectively), in contrast to the well known inverse-logarithmic behaviour found when all particles are released from the same fixed point. We combine analytic solutions with scaling arguments and stochastic simulations to rationalise our results, which open new perspectives for studying the relevance of multiple searchers in various situations of molecular reactions, in particular, in living cells.