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

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 sigma. 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/sigma) 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.

Automated summary

This study experimentally investigates the optimal mean time for a Brownian particle to reach a target with resetting in one or two dimensions. Different resetting protocols were derived and tested, revealing a phase transition at a critical value, with calculations, experimental setup, and limitations discussed.