Rank-dependent optimal resetting in multiparticle search

Abstract

In many soft-matter and biological systems, task completion relies on the cumulative arrival of multiple searchers rather than the speed of a single pioneer. The completion kinetics are therefore set not only by the first arrival, but by the full ordered sequence of first-passage times. Here, we determine how stochastic resetting optimizes these ordered arrivals for all arrival ranks. We construct an exact finite-$N$ reference for non-interacting Brownian searchers and obtain the mean ordered first-passage time $\langle T_{(k)} \rangle$ and its optimal resetting rate $r_k^*$. For searchers with identical initial conditions, $r_k^*$ increases monotonically with arrival rank and, with increasing population size, approaches the known large-$N$ quantile limit where a finite optimum appears only above a critical rank fraction $ϕ_c \simeq 0.412$. Spatial heterogeneity qualitatively reorganizes this sequence, shifting its maximum from late to early ranks even without particle interactions. We then compare this baseline with Brownian colloid experiments, interacting active Brownian particles, and a collective autochemotactic search with persistent environmental memory. Across these systems, sensitivity to resetting increases strongly with arrival rank, while deviations from appropriate non-interacting references reveal the influence of direct interactions, finite return overhead, and environmental memory. Our results show that optimal resetting in multiparticle search is governed by the required completion rank and must be evaluated relative to protocol- and geometry-matched baselines.