Optimal mean first-passage time of a Brownian searcher with resetting in one and two dimensions: experiments, theory and numerical tests

Abstract

We experimentally, numerically and theoretically study the optimal mean time needed by a Brownian particle, freely diffusing either in one or two dimensions, to reach, within a tolerance radius R tol, a target at a distance L from an initial position in the presence of resetting. The reset position is Gaussian distributed with width σ. We derived and tested two resetting protocols, one with a periodic and one with random (Poissonian) resetting times. We computed and measured the full first-passage probability distribution that displays spectacular spikes immediately after each resetting time for close targets. We study the optimal mean first-passage time as a function of the resetting period/rate for different target distances (values of the ratios b = L/σ) and target size (a = R tol/L). We find an interesting phase transition at a critical value of b, both in one and two dimensions. The details of the calculations as well as the experimental setup and limitations are discussed.