Reducing mean first passage times with intermittent confining potentials: a realization of resetting processes

TL;DR

This paper investigates a physically realizable intermittent confining potential protocol for Brownian particles that optimizes mean first passage times, revealing phase transition behavior and non-equilibrium stationary states.

Contribution

It introduces a novel intermittent potential switching protocol that can be experimentally implemented, showing rich optimization and phase transition phenomena in search processes.

Findings

Intermittent switching reduces mean first passage time below Kramer's time.

A continuous phase transition occurs in optimal switching rates as potential stiffness varies.

Potential intermittency enhances target encounter efficiency for stiff potentials.

Abstract

During a random search, resetting the searcher's position from time to time to the starting point often reduces the mean completion time of the process. Although many different resetting models have been studied over the past ten years, only a few can be physically implemented. Here we study theoretically a protocol that can be realised experimentally and which exhibits unusual optimization properties. A Brownian particle is subject to an arbitrary confining potential $v(x)$ which is switched on and off intermittently at fixed rates. Motion is constrained between an absorbing wall located at the origin and a reflective wall. When the walls are sufficiently far apart, the interplay between free diffusion during the "off" phases and attraction toward the potential minimum during the "on" phases gives rise to rich behaviours, not observed in ideal resetting models. For potentials of the form $v(x)=k|x-x_0|^n/n$, with $n>0$, the switch-on and switch-off rates that minimise the mean first passage time (MFPT) to the origin undergo a continuous phase transition as the potential stiffness $k$ is varied. When $k$ is above a critical value $k_c$, potential intermittency enhances target encounter: the minimal MFPT is lower than the Kramer's time and is attained for a non-vanishing pair of switching rates. We focus on the harmonic case $n=2$, extending previous results for the piecewise linear potential ($n=1$) in unbounded domains. We also study the non-equilibrium stationary states emerging in this process.