Papers › Critical number of walkers for diffusive search processes with resetting
Critical number of walkers for diffusive search processes with resetting
Marco Biroli, Satya N. Majumdar, Gregory Schehr
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We consider N Brownian motions diffusing independently on a line, starting at x₀>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 x₀ with rate r and (B) all walkers reset simultaneously to x₀ with rate r. We compute analytically the mean first-passage time to the origin and show that, as a function of r and for fixed x₀, 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. Continuing our results analytically to real N, we show that N_c=7.3264773… for protocol A and N_c=6.3555864… for protocol B, independently of x₀. Our theoretical predictions are verified in numerical Langevin simulations.
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