Controlling inertial active Brownian motion via stochastic resetting

Abstract

Inertia is intrinsic to many living and synthetic active systems, from animals and robotic agents to colloidal swimmers, and it strongly shapes transport. Many such systems employ intermittent restart protocols to regulate exploration. Stochastic resetting provides a theoretical framework for these strategies and a route to control nonequilibrium steady states, yet the role of inertia in reset-controlled active dynamics remains poorly understood. Here we study an inertial active Brownian particle in two dimensions under complete and partial stochastic resetting. We derive exact stationary second- and fourth-order scalar moments for the different protocols wherever their stationary limits exist, and obtain the full transient evolution of the mean-squared velocity and displacement under complete resetting. Under complete resetting, inertia strongly enhances localization: at large reset rates, the steady-state mean-squared displacement decays as $r^{-3}$, rather than following the overdamped scaling $r^{-1}$. Kurtosis-based phase diagrams reveal strongly non-Gaussian position and velocity statistics, including activity-induced re-entrant behavior as a function of reset rate and inertia. The transient solutions show that resetting shifts the velocity-relaxation rates, whereas the terminal relaxation of the displacement is governed solely by the mean resetting time $1/r$, with inertia controlling the faster transient modes. Comparing complete resetting with the partial protocols, we find that resetting the velocity together with the position substantially enhances localization. Resetting the orientation to a fixed or a random direction drives the mean-squared velocity toward opposite limits. These results establish inertia, reset frequency, and the choice of reset variables as coupled control parameters for localization, relaxation, and non-Gaussian fluctuations in active systems.

arXiv comment29 pages, 9 figures, 4 tables