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

TL;DR

This paper investigates the optimal mean first-passage time of a Brownian particle with resetting in one and two dimensions through experiments, theory, and simulations, revealing phase transitions related to resetting protocols and target parameters.

Contribution

It introduces and tests two resetting protocols, derives the full first-passage distribution, and uncovers phase transitions in the optimal search time based on target distance and size.

Findings

Spectacular spikes in first-passage distribution after resets.

Existence of a phase transition at a critical target distance ratio.

Optimal resetting rate depends on target parameters and shows distinct regimes.

Abstract

We study experimentally, numerically and theoretically the optimal mean time needed by a Brownian particle, freely diffusing either in one or two dimensions, to reach, within a tolerance radius $R_{\text tol}$, 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 ($a=R_\text{tol}/L$). 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 experimental setup and limitations are discussed.