Péclet number governs transition to acceleratory restart in drift-diffusion

Abstract

First-passage processes can be divided in two classes: those that are accelerated by the introduction of restart and those that display an opposite response. In physical systems, a transition between the two classes may occur as governing parameters are varied to cross a universal tipping point. However, a fully tractable model system to teach us how this transition unfolds is still lacking. To bridge this gap, we quantify the effect of stochastic restart on the first-passage time of a drift-diffusion process to an absorbing boundary. There, we find that the transition is governed by the Péclet number (Pe) -the ratio between the rates of advective and diffusive transport. When Pe > 1 the process is drift-controlled and restart can only hinder its completion. In contrast, when 0 ≤ Pe < 1 the process is diffusion-controlled and restart can speed-up its completion by a factor of ∼ 1/Pe. Such speedup occurs when the process is restarted at an optimal rate r r 0 (1 − Pe), where r 0 stands for the optimal restart rate in the pure-diffusion limit. The transition considered herein stands at the core of restart phenomena and is relevant to a large variety of processes that are driven to completion in the presence of noise. Each of these processes has unique characteristics, but our analysis reveals that the restart transition resembles other phase transitions -some of its central features are completely generic. arXiv:1811.08239v4 [cond-mat.stat-mech]

First passage under stochastic resetting in an interval

“…Appendix C: Proof of Eq. (38) In this Appendix, we will provide the proofs of the following relations…”

“…Second, restart has emerged as a conceptual framework to study search processes [29][30][31][32][33][34][35][36][37][38][39]. Consider a simple diffusive searcher looking for a target.…”

Time-dependent density of diffusion with stochastic resetting is invariant to return speed

“…The simple model of diffusion with stochastic resetting has been extended and generalized to cover: diffusion in the presence of a potential [6][7][8], in a domain [9][10][11][12], and arbitrary dimensions [13]; diffusion in the presence of non-exponential resetting time distributions e.g., deterministic [14], intermittent [4], non-Markovian [15], non-stationary [16], with general time dependent resetting rates [17], as well as other protocols [18]; and diffusion in the presence of interactions [19][20][21]. The effect of resetting on random walks [22,23], continuous time random walks [24][25][26], Lévy flights [27,28], and other forms of stochastic motion [29][30][31][32][33], has also been studied.…”

Landau-like expansion for phase transitions in stochastic resetting

“…No surprise that this observation has taken the center stage and lead to a myriad of studies. In particular, restart renders mean first passage time finite for many diffusive [1,2,[25][26][27]29,30] and nondiffusive search processes [18,21,28]. Further studies revealed existence of the dominant restart strategy which will globally optimize the MFPT [19,20,23], and a genre of universality relations associated with the optimally restarted processes [22,24].…”

“…First-order transition in ORR was observed in the case of a nondiffusive search with the aid of Lévy flight [17] or a search comprising of a drifted random walker with exponential flights subjected to restart [21]. The continuous transition in the restart rate, on the other hand, was observed in a system of Brownian walker in a domain or force field subject to resetting [25][26][27]. Although each variant above carried with it some unique and alluring features, there exists no unified framework which can describe the existence of both the transitions in a single system and other higher-order features such as a shift in transition from being continuous to first order.…”