Critical number of walkers for diffusive search processes with resetting.

Abstract

We consider N Brownian motions diffusing independently on a line, starting at x_{0}>0, in the presence of an absorbing target at the origin. The walkers undergo stochastic resetting under two protocols: (A) each walker resets independently to x_{0} with rate r and (B) all walkers reset simultaneously to x_{0} with rate r. We derive an explicit analytical expression for the mean first-passage time to the origin in terms of an integral which is evaluated numerically using Mathematica. We show that, as a function of r and for fixed x_{0}, it has a minimum at an optimal value r^{*}>0 as long as N<N_{c}. Thus resetting is beneficial for the search for N<N_{c}. When N>N_{c}, the optimal value occurs at r^{*}=0 indicating that resetting hinders search processes. We obtain different values of N_{c} for protocols A and B; indeed, for N≤7 resetting is beneficial in protocol A, while for N≤6 resetting is beneficial for protocol B. Our theoretical predictions are verified in numerical Langevin simulations.