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 ofN non-communicating, randomly moving searchers (numbered by the indexk,k = 1, 2,..., N), which all start their random motion simultaneously at the same point in space. For eachrealization of the search process, we record the unordered set of time moments{τk}, whereτk is the time of thefirst passage of the kth searcher to the location of the target. Clearly,τks are independent, identically distributed random variables with the same distribution functionΨ(τ). We evaluate thenthe distribution P(ω) of the random variable , where is the ensemble-averaged realization-dependent first passage time. We show thatP(ω) exhibits quite a non-trivial and sometimes a counterintuitive behavior. Wedemonstrate that in some well-studied cases (e.g. Brownian motion in finited-dimensionaldomains) the mean first passage time is not a robust measure of the search efficiency, despite the factthat 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 extractedfrom a single-particle tracking should be regarded with appropriate caution because of thesignificant sample-to-sample fluctuations.