Diffusion with two resetting points

Abstract

We study the problem of a target search by a Brownian particle subject to stochastic resetting to a pair of sites. The mean search time is minimized by an optimal resetting rate which does not vary smoothly, in contrast with the well-known single site case, but exhibits a discontinuous transition as the position of one resetting site is varied while keeping the initial position of the particle fixed, or vice-versa. The discontinuity vanishes at a "liquid-gas" critical point in position space. This critical point exists provided that the relative weight $m$ of the further site is comprised in the interval $[\mathrm{2.9028...},\mathrm{8.5603...}]$. When the initial position follows the resetting point distribution, a discontinuous transition also exists for the optimal rate as the distance between the resetting points is varied, provided that $m$ exceeds the critical value $m_c=\mathrm{6.6008...}$ This setup can be mapped onto an intermittent search problem with switching diffusion coefficients and represents a minimal model for the study of distributed resetting.

7 pages, 6 Figures, typos corrected

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Cited by in corpus (6)

• Instability in the quantum restart problem
• Random walks with stochastic resetting in complex networks: a discrete time approach
• Large Deviations in Switching Diffusion: from Free Cumulants to Dynamical Transitions
• Mean first-encounter times of simultaneous random walkers with resetting on networks
• Generalized diffusion process with nonlocal interactions: Continuous time random walk model and stochastic resetting
• Abrupt transitions in the optimization of diffusion with distributed resetting