General Theory for Group Resetting with Application to Avoidance

TL;DR

This work develops a general theory for group resetting in stochastic search under a drift potential by mapping the multi-particle dynamics to an effective center-of-mass coordinate $\zeta(t)$ that undergoes diffusion with coefficient $D/n$ and resetting at rate $r$. Resetting statistics are governed by a renewal equation for the reset-position distribution $R(\zeta,t)$, which couples to a Fokker-Planck equation for $P(\zeta,t)$. In the key extreme-value resetting scenario, the kernel $K_n(\zeta|\zeta';\tau)$ converges to a Gumbel distribution $\mathrm{Gumbel}(\zeta;\mu,\beta)$ for large $n$, enabling analytic expressions for stationary moments and scaling laws; for a harmonic potential, $\mu$ and $\beta$ are determined from OU propagators. The results show that the stationary mean $\langle\zeta\rangle_s$ grows as $\propto\sqrt{\ln n}$ with group size and increases with reset rate, while the SCV $\sigma_\zeta^2/\langle\zeta\rangle_s^2$ decreases with $n$ and $r$, informing the effectiveness of group avoidance. The framework is validated against simulations and extended to group-avoidance problems, with potential applications in swarm-optimization and artificial-selection protocols that exploit group-level resetting strategies.

Abstract

We present a general theoretical framework for group resetting dynamics in a potential landscape. While traditional resetting models typically focus on a single particle, we consider a group of particles whose collective dynamics govern the resetting. We extend existing resetting theories to cover extreme-value group resetting. This has applications from bacterial evolution under antibiotic pressure to swarm-search optimization. Using renewal theory, we derive a Fokker-Planck equation for the spatial distribution of the group's center of mass, treated as an effective particle. This formalism yields analytical expressions for key observables such as the stationary mean position and variance. We also study a group avoidance problem, where the particles must avoid an undesirable region. Such problems have recently been studied in contexts such as preventing critically high water levels in dams and controlling excessive financial leverage. Our framework offers new insight into how resetting can optimize group-level search and avoidance strategies.

Figures (4)

• Figure 1: Schematic figure of group resetting. (a) Reset dynamics. All particles relocate to the position of the selected particle in the group. So does the center of mass (CM). (b) Avoidance problem. The drift potential drags the CM towards the minimum, marking the start of the undesired or dangerous area (red). (c) The spatial distributions before and after the group reset (black lines). All particles instantaneously relocate to the rightmost particle before resuming diffusion.
• Figure 2: Particle trajectories and average positions. (a) Simulated particle trajectories with group resetting (grey) and their center of mass (CM, blue). The red line marks the boundary of the dangerous region $(x \leq 0)$. Parameters: $(n, r, k, D) = (10, 1, 1, 2)$. (b) Average positions over time for $n=100$ particles. The blue symbols represent the average CM trajectory, and the orange ones show the average from effective particle simulations. The green line corresponds to the first moment of $P(\zeta,t)$ from Eq. \ref{['eq:FP']}, and the dashed line indicates the stationary position $\langle\zeta\rangle_s$ [Eq. \ref{['eq:st_sol']}]. (inset) Position distributions at $t=10$. Histograms show simulation results for $\zeta$ and $x_\textrm{CM}$, while the green line is the stationary distribution $P_s(\zeta)$ calculated numerically from Eq. \ref{['eq:FP']}. The dashed line shows the first moment of $P_s(\zeta)$.
• Figure 3: [(a) and (b)] The stationary mean position $\langle \zeta \rangle_s$ and [(c) and (d)] the squared coefficient of variation (SCV) $\sigma^2_\zeta/\langle\zeta\rangle_s^2$ with respect to [(a) and (c)] $n$ for $(r,k,D) = (1,1,2)$, and [(b) and (d)] $r$ for $(n,k,D)=(100,1,2)$. $\langle \zeta \rangle_s$ increases with $n$ and $r$, while the SCV decreases for both parameters. However, the SCV saturates at large $n$. The blue and orange symbols indicate simulation results for groups of particles and the effective center of mass, respectively. The green lines represent theoretical results, and the dashed lines depict scaling trends.
• Figure 4: (a) Stationary distributions $P_s(\zeta)$ for different parameter values of $n$ and $r$. These distributions share the same squared coefficient of variation (SCV) value, $\sigma_\zeta^2 / \langle \zeta \rangle^2_s = 3$, but differ in their detailed shapes. Red area represents the dangerous region. (b) Avoidance probability $P_a$ as a function of SCV for different group size $n$. Symbols represent simulation results, and lines represent the numerical solution of Eq. \ref{['eq:FP']}. The dashed line indicates $\sigma_\zeta^2 / \langle \zeta \rangle^2_s = 3$. The parameters are $(k,D)=(1,2)$.