Brownian motion under intermittent harmonic potentials

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.

J. Phys. A: Math. Theor. (2021)