Intermittent continuous-time random walks under renewal reset mechanism

Abstract

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.