Optimal area exploration by resetting active particles

TL;DR

The work addresses maximizing spatial exploration by active matter under stochastic resetting, linking resetting theory to practical search strategies. It combines vibrobot experiments in a circular arena with active Brownian particle simulations, using dynamics $\dot{\mathbf{x}}= v_0 \mathbf{n} + \sqrt{2D}\boldsymbol{\xi}$ and $\dot{\theta}=\sqrt{2D_r}\eta$. A non-monotonic dependence of the normalized covered area $\mathcal{A}$ on the resetting rate $r$ is demonstrated, with an activity-dependent optimal rate $r^*$ that scales roughly linearly with the self-propulsion speed $v_0$ (and a nearly constant $\mathcal{A}_*$ at optimality). The findings provide a simple, robust design principle for efficient search in active systems and motivate extensions to non-Poissonian resetting and broader confinements for potential robotic applications.

Abstract

Identifying optimal strategies for efficient spatial exploration is crucial, both for animals seeking food and for robotic search processes, where maximizing the covered area is a fundamental requirement. Here, we propose position resetting as an optimal protocol to enhance spatial exploration in active matter systems. Specifically, we show that the area covered by an active Brownian particle exhibits a non-monotonic dependence on the resetting rate, demonstrating that resetting can optimize spatial exploration. Our results are based on experiments with active granular particles undergoing Poissonian resetting and are supported by active Brownian dynamics simulations. The covered area is analytically predicted at both large and small resetting rates, resulting in a scaling relation between the optimal resetting rate and the self-propulsion speed.

Figures (6)

• Figure 1: Active granular particle under resetting. (a) Visualization of the experimental setup, with a vibrobot moving in a circular arena. The particle is reset (red arrow) back to its initial position with orientation randomization at random times extracted from an exponential distribution. This process lasts until the particle hits the boundary of the plate. (b)-(c) Time-trajectories of finite-sized active particles in a circular plate with radial size $R$ for three different values of the resetting rate, $r=0.5,1.0,2.5 \text{ Hz}$. The covered area is obtained by coloring the center-of-mass trajectory with a thickness corresponding to the particle diameter. (b) is obtained from experiments, while (c) by simulations. These typical trajectories qualitatively reveal a non-monotonic covered area with the resetting rate $r$, both in experiments and simulations.
• Figure 2: Resetting induces optimal covered area. Covered area $\mathcal{A}$, normalized by the plate area $\pi R^{2}$, as a function of the resetting rate $r$. The analysis is performed for three different shaker frequencies, as reported in the legend, corresponding to different particle parameters (see End Matter, Sec. \ref{['app:experimentaldetails']}). Dots represent data points, while the gray solid lines are guides to the eye. Optimal resetting rates are identified as the values of $r$ that maximize the covered area.
• Figure 3: Resetting-induced optimal area covered by an active particle. (a) Covered area $\mathcal{A}$ (color gradient) normalized by the area of the circular arena, as a function of the resetting rate $r$ and self-propulsion speed $v_0$. Colored dots represent simulation data; the background is obtained by interpolation. (b) Normalized $\mathcal{A}$ versus $r$ for selected $v_0$ values, indicated by arrows in (a). The two dashed lines denote the predicted $\sim r^{-2}$ scaling at large $r$. The cutoff time is $t_c = 5000\,\text{s}$. (c) Optimal resetting rate as a function of $v_0$, showing the predicted linear behavior (dotted black lines). Blue and red points correspond to cutoff times $t_c = 500\,\text{s}$ and $5000\,\text{s}$, respectively. The remaining simulation parameters are $D_r = 0.85\,\text{s}^{-1}$, $\gamma_T = 10.0\,\text{g}\,\text{s}^{-1}$, and $D_T = 1.59\,\text{mm}^2\,\text{s}^{-1}$.
• Figure 4: Vibrobot and setup illustration. (a) Schematic side-view of the vibrobot. (b) Illustration of the setup top-view of a particle moving on the vibrating plate.
• Figure 5: Discretized particle path and area, with $\delta = 0.133 r_p$ and $\mathcal{A}/(\pi R^2)\approx 0.1$. Unoccupied and occupied cells are shown in gray and yellow. Overlapping black path shows center of mass trajectories.