Brownian motion under intermittent harmonic potentials

TL;DR

This paper analyzes how an intermittently switching harmonic potential affects Brownian motion, revealing stationary states, relaxation times, and an optimal switching rate for target search efficiency, with implications for experimental verification.

Contribution

It provides exact solutions for the non-equilibrium stationary states and first passage times of Brownian particles under stochastic potential switching, extending understanding of stochastic resetting models.

Findings

Stationary solutions exist in all parameter regimes.

Optimal switching rate minimizes mean first passage time.

MFPT reaches a finite value at high switching rates.

Abstract

We study the effects of an intermittent harmonic potential of strength $\mu = \mu_0 \nu$ -- that switches on and off stochastically at a constant rate $\gamma$, on an overdamped Brownian particle with damping coefficient $\nu$. This can be thought of as a realistic model for realisation of stochastic resetting. We show that this dynamics admits a stationary solution in all parameter regimes and compute the full time dependent variance for the position distribution and find the characteristic relaxation time. We find the exact non-equilibrium stationary state distributions in the limits -- (i) $\gamma\ll\mu_0 $ which shows a non-trivial distribution, in addition as $\mu_0\to\infty$, we get back the result for resetting with refractory period; (ii) $\gamma\gg\mu_0$ where the particle relaxes to a Boltzmann distribution of an Ornstein-Uhlenbeck process with half the strength of the original potential and (iii) intermediate $\gamma=2n\mu_0$ for $n=1, 2$. The mean first passage time (MFPT) to find a target exhibits an optimisation with the switching rate, however unlike instantaneous resetting the MFPT does not diverge but reaches a stationary value at large rates. MFPT also shows similar behavior with respect to the potential strength. Our results can be verified in experiments on colloids using optical tweezers.