First passages for a search by a swarm of independent random searchers

Abstract

In this paper we study some aspects of search for an immobile target by a swarm of N non-communicating, randomly moving searchers (numbered by the index k, k = 1, 2, . . . ,N), which all start their random motion simultaneously at the same point in space. For each realization of the search process, we record the unordered set of time moments {τk}, where τk is the time of the first passage of the kth searcher to the location of the target. Clearly, τks are independent, identically distributed random variables with the same distribution function Ψ(τ ). We evaluate then the distribution P(ω) of the random variable ω ∼ τ1/¯τ , where ¯τ = N−1N k=1 τk is the ensemble-averaged realization-dependent first passage time. We show that P(ω) exhibits quite a non-trivial and sometimes a counterintuitive behavior. We demonstrate that in some well-studied cases (e.g. Brownian motion in finite d-dimensional domains) the mean first passage time is not a robust measure of the search efficiency, despite the fact that Ψ(τ ) has moments of arbitrary order. This implies, in particular, that even in this simplest case (not to mention complex systems and/or anomalous diffusion) first passage data extracted from a single-particle tracking should be regarded with appropriate caution because of the significant sample-to-sample fluctuations.