Interplay between timescales governs the residual activity of a harmonically bound active Brownian particle

Abstract: Active microparticles in confining potentials manifest complex and intriguing dynamical phenomena, as their activity competes with confinement. The steady-state position distributions of harmonically bound active Brownian particles (HBABPs) exhibit a crossover from Boltzmann-like to bimodal, commonly recognized as passive to active transition, upon variation of the activity and the confinement strength. By studying optically trapped phoretically active Janus colloids, along with simulations and analytical calculations of HBABPs, we provide a comprehensive dynamical description emphasizing the resultant velocity to examine this understanding. Our results establish that the crossover is instead from active to passive-like, and is governed solely by the interplay between the persistence time $\tau_{\rm R}$, and the equilibration time in harmonic potential $\tau_k$. When $\tau_{\rm R} < \tau_k$, despite a Boltzmann-like position distribution, the HBABP retains a substantial resultant or residual active velocity, denoting an activity-dominated regime. In contrast, at $\tau_{\rm R} > \tau_k$, the restoring force fully counterbalances propulsion at a radial distance, where the HBABP exhibits harmonically bound Brownian particle (HBBP)-like dynamics, and the position distribution becomes bimodal. We further provide a quantitative measure of the residual activity, which decreases monotonically with $\tau_{\rm R} / \tau_k$, eventually converging to a nominal value corresponding to HBBP as $\tau_{\rm R} / \tau_k$ exceeds 1, corroborating our conclusions.