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

Abstract

We study experimentally, numerically and theoretically the optimal mean time\nneeded by a Brownian particle, freely diffusing either in one or two\ndimensions, to reach, within a tolerance radius $R_{\\text tol}$, a target at a\ndistance $L$ from an initial position in the presence of resetting. The reset\nposition is Gaussian distributed with width $\\sigma$. We derived and tested two\nresetting protocols, one with a periodic and one with random (Poissonian)\nresetting times. We computed and measured the full first-passage probability\ndistribution that displays spectacular spikes immediately after each resetting\ntime for close targets. We study the optimal mean first-passage time as a\nfunction of the resetting period/rate for different target distances (values of\nthe ratios $b=L/\\sigma$) and target size ($a=R_\\text{tol}/L$). We find an\ninteresting phase transition at a critical value of $b$, both in one and two\ndimensions. The details of the calculations as well as experimental setup and\nlimitations are discussed.\n