标题： 群体重置的一般理论及其在回避中的应用 显示英文标题

标题： General Theory for Group Resetting with Application to Avoidance

摘要： 我们提出了一种用于多粒子系统在漂移势中的群体重置动力学的一般理论框架。 虽然传统的重置模型通常关注单个粒子，但我们的设置考虑了一组粒子，其集体动力学决定了重置过程。 最近，重置也被用作一种调节机制，以避免不利结果，例如防止大坝中水位过高或减少金融投资组合的杠杆率。 在这里，我们将当前的重置理论扩展到群体动力学，应用范围从抗生素压力下的细菌进化到多个搜索者的优化算法。 结合更新理论和极值统计学，我们推导出一个福克-普朗克方程，用于描述群体质心的空间分布，将其视为一个有效粒子。 这种形式化方法使我们能够解析计算关键可观测量，包括稳态平均位置、方差以及无量纲的风险度量——方差系数。 我们的结果展示了群体大小和重置率如何影响避开危险区域的概率。 这种理论方法为通过重置设计最优的群体级搜索和规避策略开辟了新的视角。 显示英文摘要

摘要： We present a general theoretical framework for group resetting dynamics in multi-particle systems in a drift potential. While traditional resetting models typically focus on a single particle, our setup considers a group of particles whose collective dynamics determine the resetting process. More recently, resetting has also been used as a regulatory mechanism to avoid adverse outcomes, such as preventing critically high water levels in dams or deleveraging financial portfolios. Here, we extend current resetting theories to group dynamics, with applications ranging from bacterial evolution under antibiotic pressure to multiple-searcher optimization algorithms. Combining renewal theory and extreme value statistics, we derive a Fokker-Planck equation for the spatial distribution of the group's center of mass, treated as an effective particle. This formalism allows us to analytically calculate essential observables including the stationary mean position, variance, and a dimensionless measure of risk--the squared coefficient of variation. Our results demonstrate how the group size and resetting rate affect the probability of avoiding the danger region. This theoretical approach opens new perspectives on designing optimal group-level search and avoidance strategies through resetting.