Fluctuations and first-passage properties of systems of Brownian particles with reset.

Abstract

We study, analytically and numerically, stationary fluctuations in two models involving N Brownian particles undergoing stochastic resetting in one dimension. We start with the well-known reset model where the particles reset to the origin independently (model A). Then we introduce nonlocal interparticle correlations by postulating that only the particle farthest from the origin can be reset to the origin (model B). At long times, models A and B approach nonequilibrium steady states. In the limit of N→∞, the steady-state particle density in model A has an infinite support, whereas in model B, it has a compact support, like the recently studied Brownian bees model. A finite system radius, which scales at large N as lnN, appears in model A when N is finite. In both models, we study stationary fluctuations of the center of mass of the system and of the radius of the system due to the random character of the Brownian motion and of the resetting events. In model A, we determine exact distributions of these two quantities. The variance of the center of mass for both models scales as 1/N. The variance of the radius is independent of N in model A and exhibits an unusual scaling (lnN)/N in model B. The latter scaling is intimately related to the 1/f noise in the radius autocorrelation. Finally, we evaluate the mean first-passage time (MFPT) to a distant target in model A, model B, and the Brownian bees model. For model A, we obtain an exact asymptotic expression for the MFPT which scales as 1/N. For model B and the Brownian bees model, we propose a sharp upper bound for the MFPT. The bound assumes an evaporation scenario, where the first passage requires multiple attempts of a single particle, which breaks away from the rest of the particles, to reach the target. The resulting MFPT for model B and the Brownian bees model scales exponentially with sqrt[N]. We verify this bound by performing highly efficient weighted-ensemble simulations of the first passage in model B.