General theory for group resetting with application to avoidance

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 insights into how resetting can optimize group-level search and avoidance strategies.