Brownian motion under intermittent harmonic potentials

Abstract

We study the effects of an intermittent harmonic potential of strength μ = μ 0 ν—that switches on and off stochastically at a constant rate γ, on an overdamped Brownian particle with damping coefficient ν. 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) γ ≪ μ 0 which shows a non-trivial distribution, in addition as μ 0 → ∞, we get back the result for resetting with refractory period; (ii) γ ≫ μ 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 γ = 2nμ 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.