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 subject to complete stochastic resetting of position, velocity, and orientation in two dimensions. Using a moment-generating framework together with the Final-Value Theorem, we derive closed-form steady-state moments up to fourth order as functions of inertia, activity, and reset rate. We show that inertia fundamentally modifies reset-controlled transport: at large reset rates the steady-state mean-squared displacement is suppressed much more strongly than in the overdamped limit, yielding enhanced localization near the reset point. At the same time, position excess-kurtosis phase diagrams reveal strongly non-Gaussian steady states characterized by a sharp central peak coexisting with heavy tails in the position distribution, indicating rare long excursions enabled by inertial persistence. The tail weight varies non-monotonically with reset rate, reflecting a competition between inertial momentum relaxation and resetting that selects an optimal regime maximizing rare excursions. Our results provide experimentally testable signatures of inertial effects in reset-controlled active systems.

Figures (6)

• Figure 1: Inertial effects in active Brownian particles under stochastic resetting. (a) Schematic diagram of the inertial active Brownian particle position $\mathbf{r}$, velocity $\mathbf{v}$, and orientation $\hat{\mathbf{u}}$ evolve(see Eqs. \ref{['eom:disp']}--\ref{['eom:resetting']}), with intermittent complete resetting to the initial state $(\mathbf{r}(0),\mathbf{v}(0),\hat{\mathbf{u}}(0))\equiv (\mathbf{r}_0,\mathbf{v}_0,\hat{\mathbf{u}}_0)$. (b) In the overdamped regime (low inertia), the particle exhibits an instantaneous orientational response: velocity direction $\hat{ {\bf v}}$ is immediately slaved to the propulsion direction ${\hat{\mathbf{u}}}$. In the underdamped regime (high inertia), the particle shows a delayed response, where velocity direction $\hat{ {\bf v}}$ relaxes over a finite time due to inertia. (c) Comparison of steady state single particle trajectory for inertia $M=0.01$ and $1$ at reset rate $r = 1$ and activity $\mathrm{Pe} = 10$. (d) Steady state positions snapshot of $2 \times 10^4$ particles for inertia $M=0.01$, $1$, and $10$ at reset rate $r = 10$ and activity $\mathrm{Pe} = 10$. We consider the initial(reset) state ${\bf r}_0= 0$, ${\bf v}_0 = 0$, and ${\hat{\mathbf{u}}}_0 = (1,0)$.
• Figure 2: Steady-state second moments under stochastic resetting. (a,b) Mean-squared velocity (MSV) and (c,d) mean-squared displacement (MSD) of inertial active Brownian particles under complete stochastic resetting. Active particles ($\mathrm{Pe}=10$, solid lines, labelled A) are compared with passive Brownian particles ($\mathrm{Pe}=0$, dashed lines, labelled B). (a) MSV as a function of inertia $M$ for reset rates $r=0,\,0.1,\,2.0$. (b) MSV as a function of reset rate $r$ for $M=0.1,\,2.0$. (c) MSD as a function of inertia $M$ for $r=0.1,\,2.0$. (d) MSD as a function of reset rate $r$ for $M=0$ (overdamped), $0.1$, and $2.0$. Lines denote analytical predictions; symbols represent simulation data. Inertia and resetting suppress steady-state fluctuations, while activity enhances both MSV and MSD relative to the passive baseline.
• Figure 3: Phase diagrams using velocity excess kurtosis as primary metric. Steady state excess kurtosis of velocity ${\cal L}_r^{\rm st}$. Heat map in $r-M$ plane of (a) passive inertial Brownian particle under resetting $\mathrm{Pe} = 0$, (b) inertial ABP under resetting. The line plot of excess kurtosis in velocity as a function of $r$ in (c) and $M$ in (d) at representative parameter values (lines: theory; symbols: simulations). The dashed line plot $rM = 4$ in (a) represents the line where ${\cal L}_{r}^{\rm st} = 0.5$ and the dashed line in (b) corresponds to ${\cal L}_{r}^{\rm st} = 0$. Probability distribution for magnitude of velocity $|{\bf{v}}|= \sqrt{v_x^2 + v_y^2}$. (e)--(j) show the steady--state probability distributions of the velocity magnitude $| {\bf v}|$ at selected points (i)--(vi) in the phase diagram in (a) and (I)--(VI) in the phase diagram in (b), illustrating how inertia and resetting jointly enhance heavy-tailed velocity fluctuations.
• Figure 4: Phase diagrams using position excess kurtosis as primary metric. Steady-state excess kurtosis of position, ${\cal K}_r^{\rm st}$. Heat maps in the $r$--$M$ plane for (a) a passive inertial Brownian particle under resetting ($\mathrm{Pe}=0$) and (b) an inertial ABP under resetting. (c,d) Line plots of ${\cal K}_r^{\rm st}$ as a function of (c) reset rate $r$ and (d) inertia $M$ at representative parameter values (lines: theory; symbols: simulations). (e)--(j) Steady-state probability distributions of the displacement magnitude $| {\bf r}|=\sqrt{x^2+y^2}$ at selected points (i)--(vi) in panel (a) and (I)--(VI) in panel (b), illustrating the crossover from weakly non-Gaussian statistics to heavy-tailed, strongly non-Gaussian regimes and the emergence of re-entrant behavior at large inertia and finite activity.
• Figure 5: Steady--state fourth moment of velocity $\langle {\bf v}^4 \rangle_{r}^{\rm st}$((a)--(b)) and position $\langle {\bf r}^4 \rangle_{r}^{\rm st}$ ((c)--(d)) of inertial active Brownian particles under complete stochastic resetting, comparing active particles ($\mathrm{Pe} = 10$, solid lines, labeled "A") with passive Brownian particles ($\mathrm{Pe} = 0$, dashed lines, labeled "B"). (a) Fourth moment of velocity as a function of inertia $M$ for reset rates $r=0,\,0.1,\,2.0$. (b) Fourth moment of velocity as a function of reset rate $r$ for $M=0.1,\,2.0$. (c) Fourth moment of position as a function of inertia $M$ for reset rates $r=0.1,\,2.0$. (d) Fourth moment of position as a function of reset rate $r$ for $M=0$ (overdamped), $0.1$, and $2.0$. Dashed black lines indicate analytic scaling regimes ($M^{-2}$, $M^{-4}$, $r^{-2}$, $r^{-6}$) in the large--$M$ or large--$r$ limits. These results highlight how inertia and resetting jointly suppress fluctuations, with activity strongly enhancing both fourth moment of position and velocity compared to the Brownian baseline.