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We consider the first-passage problem for $N$ identical independent particlesthat are initially released uniformly in a finite domain $\Omega$ and thendiffuse toward a reactive area $\Gamma$, which can be part of the outerboundary of $\Omega$ or a reaction centre in the interior of $\Omega$. For bothcases of perfect and partial reactions, we obtain the explicit formulas for thefirst two moments of the fastest first-passage time (fFPT), i.e., the time whenthe first out of the $N$ particles reacts with $\Gamma$. Moreover, weinvestigate the full probability density of the fFPT. We discuss a significantrole of the initial condition in the scaling of the average fastestfirst-passage time with the particle number $N$, namely, a much strongerdependence ($1/N$ and $1/N^2$ for partially and perfectly reactive targets,respectively), in contrast to the well known inverse-logarithmic behaviourfound when all particles are released from the same fixed point. We combineanalytic solutions with scaling arguments and stochastic simulations torationalise our results, which open new perspectives for studying the relevanceof multiple searchers in various situations of molecular reactions, inparticular, in living cells.

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