更新重置机制下的间歇性连续时间随机游走

中文摘要 AI 辅助

随机重置作为复杂和无序环境中一种实用且高效的搜索策略，长期以来一直是研究者关注的话题。基于跳跃与重置之间的竞争，本文提出并研究了随机重置下的间歇性连续时间随机游走（CTRWs），采用跳跃和重置中较小的等待时间作为更新（renewal）时间，其中跳跃和重置的等待时间可以具有任意分布。在每个更新事件之后，系统将不考虑先前历史，重新开始新的跳跃和重置等待时间。我们研究了具有更新重置的间歇性CTRWs的主控方程和Montroll-Weiss方程，以及马尔可夫重置情形。我们证明了当跳跃和重置等待时间遵循任意指数分布和幂律分布时，在更新重置机制下非平衡稳态的存在性。对于指数分布和高斯分布的跳跃长度，我们检查了粒子的均方位移（MSDs）以确定其单调性和渐近稳定性。此外，我们计算了首次到达时间以量化搜索效率，并验证了在指数跳跃和重置等待时间分布（WTDs）、幂律跳跃和指数重置WTDs，以及指数跳跃和幂律重置WTDs下，更新重置的间歇性CTRWs导致到达任何固定位置的有限平均首次到达时间（MFAT）。然而，对于幂律跳跃和重置WTDs，MFAT发散。基于竞争机制的间歇性CTRW模型可应用于许多物理场景，例如动物在一次不成功的觅食尝试后返回巢穴的觅食策略，或智能机器人在长时间运行后返回能量补给点的工作规划。

英文摘要

Stochastic resetting as a practical and efficient search strategy in complex and disordered environments has long been a topic of interest to researchers. Based on the competition between jumping and resetting, this article proposes and investigates intermittent continuous-time random walks (CTRWs) under stochastic resetting, using the smaller waiting time for jump and reset as the renewal time, where the waiting times for both jump and reset can have arbitrary distributions. After each renewal event, the system will proceed with new waiting times for jump and reset regardless of their previous histories. We study the governing equation and Montroll-Weiss equation with renewal resetting, as well as the Markovian resetting for intermittent CTRWs. We prove the existence of non-equilibrium stationary states within the renewal reset mechanism when the jump and reset waiting times follow any exponential and power law distributions. For exponential and Gaussian distributed jump lengths, we examine the mean square displacements (MSDs) of particles to determine their monotonicity and asymptotic stability. Moreover, we calculate the first-arrival time to quantify search efficiency, and validate the intermittent CTRWs under renewal resetting lead to a finite mean first-arrival time (MFAT) to any fixed position for exponential jump and reset waiting time distributions (WTDs), power-law jump and exponential reset WTDs, as well as exponential jump and power-law reset WTDs. However, the MFAT diverges for power-law jump and reset WTDs. The intermittent CTRW model, which is based on the competition mechanism, can be applied to many physical scenarios, such as the foraging strategy of animals that return to their nests after an unsuccessful foraging attempt, or the work planning of intelligent robots that return to energy replenishment points after prolonged operation.

发表机构

• School of Mathematical Sciences, Chengdu University of Technology（成都理工大学数学科学学院）

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