General Theory for Group Resetting with Application to Avoidance

TL;DR

This paper develops a comprehensive theoretical framework for group resetting dynamics in potential landscapes, extending traditional models to include collective behavior and extreme-value resetting, with applications in biological and optimization contexts.

Contribution

It introduces a general theory for group resetting, deriving analytical expressions for key observables, and applies it to group avoidance problems in various real-world scenarios.

Findings

Derived a Fokker-Planck equation for group center of mass

Provided analytical expressions for stationary mean and variance

Applied framework to group avoidance in diverse contexts

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.