Non-equilibrium steady states of stochastic processes with intermittent resetting

Eule, Stephan; Metzger, Jakob

doi:10.1088/1367-2630/18/3/033006

Cited by 180 publications

(201 citation statements)

References 17 publications

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“…Stochastic motion with stochastic resetting is of considerable interest due to its broad applicability in statistical [1][2][3][4][5][6][7], chemical [8][9][10][11][12], and biological physics [13,14]; and due to its importance in computer science [15,16], computational physics [17,18], population dynamics [19][20][21], queuing theory [22][23][24] and the theory of search and first-passage [25][26][27]. Particularly, in statistical physics, such motion has become a focal point of recent studies owing to the rich non-equilibrium [2][3][4][5][6][28][29][30] and first-passage [31][32][33][34][35][36] phenomena it displays.…”

Section: Introductionmentioning

confidence: 99%

Invariants of motion with stochastic resetting and space-time coupled returns

2019

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Motion under stochastic resetting serves to model a myriad of processes in physics and beyond, but in most cases studied to date resetting to the origin was assumed to take zero time or a time decoupled from the spatial position at the resetting moment. However, in our world, getting from one place to another always takes time and places that are further away take more time to be reached. We thus set off to extend the theory of stochastic resetting such that it would account for this inherent spatio-temporal coupling. We consider a particle that starts at the origin and follows a certain law of stochastic motion until it is interrupted at some random time. The particle then returns to the origin via a prescribed protocol. We study this model and surprisingly discover that the shape of the steady-state distribution which governs the stochastic motion phase does not depend on the return protocol. This shape invariance then gives rise to a simple, and generic, recipe for the computation of the full steady state distribution. Several case studies are analyzed and a class of processes whose steady state is completely invariant with respect to the speed of return is highlighted. For processes in this class we recover the same steady-state obtained for resetting with instantaneous returns-irrespective of whether the actual return speed is high or low. Our work significantly extends previous results on motion with stochastic resetting and is expected to find various applications in statistical, chemical, and biological physics.

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Section: Introductionmentioning

confidence: 99%

Invariants of motion with stochastic resetting and space-time coupled returns

2019

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“…The existence of a NESS has been further studied for different types of motion and resetting mechanisms [5][6][7][8][9][10][11][12][13][14][15][16][17][18], showing that they are not exclusive of diffusion with Markovian resets. Aside from these, other works have shown that the resetting does not always generate a NESS but transport is also possible when the resetting probability density function (PDF) is long-tailed [19][20][21][22] or when the resetting process is subordinated to the motion [10,12]. However, the above cited stochastic resetting models lack some realism as long as resetting is treated as an instantaneous process.…”

Section: Introductionmentioning

confidence: 99%

Transport properties of random walks under stochastic noninstantaneous resetting

2019

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Random walks with stochastic resetting provides a treatable framework to study interesting features about central-place motion. In this work, we introduce non-instantaneous resetting as a twostate model being a combination of an exploring state where the walker moves randomly according to a propagator and a returning state where the walker performs a ballistic motion with constant velocity towards the origin. We study the emerging transport properties for two types of reset time probability density functions (PDFs): exponential and Pareto. In the first case, we find the stationary distribution and a general expression for the stationary mean square displacement (MSD) in terms of the propagator. We find that the stationary MSD may increase, decrease or remain constant with the returning velocity. This depends on the moments of the propagator. Regarding the Pareto resetting PDF we also study the stationary distribution and the asymptotic scaling of the MSD for diffusive motion. In this case, we see that the resetting modifies the transport regime, making the overall transport sub-diffusive and even reaching a stationary MSD., i.e., a stochastic localization. This phenomena is also observed in diffusion under instantaneous Pareto resetting. We check the main results with stochastic simulations of the process.PACS numbers:

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“…More recently, in 2011, Evans and Majumdar [16,17] studied a diffusing model with a resetting term in a Fokker-Planck equation, derived from microscopical considerations. Afterwards, several analysis and generalizations of this formulation have been performed, including: The incorporation of an absorbing state [18]; the generalizations to d-spatial dimensions [19]; the presence of a general potential [20]; the inclusion of time dependency in the resetting rate [21] or a general distribution for the reset time [22]; a study of large deviations in Markovian processes [23]; a comparison with deterministic resetting [24]; the relocation to a previously position [25]; analyses on general properties of the first-passage time [26,27]; or the possibility that internal properties drive the reset mechanism of the system [28].…”

Section: Introductionmentioning

confidence: 99%

Continuous-time random walks with reset events

2017

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In this paper, we consider a stochastic process that may experience random reset events which relocate the system to its starting position. We focus our attention on a one-dimensional, monotonic continuous-time random walk with a constant drift: the process moves in a fixed direction between the reset events, either by the effect of the random jumps, or by the action of a deterministic bias. However, the orientation of its motion is randomly determined after each restart. As a result of these alternating dynamics, interesting properties do emerge. General formulas for the propagator as well as for two extreme statistics, the survival probability and the mean first-passage time, are also derived. The rigor of these analytical results is verified by numerical estimations, for particular but illuminating examples.

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Non-equilibrium steady states of stochastic processes with intermittent resetting

Cited by 180 publications

References 17 publications

Invariants of motion with stochastic resetting and space-time coupled returns

Invariants of motion with stochastic resetting and space-time coupled returns

Transport properties of random walks under stochastic noninstantaneous resetting

Continuous-time random walks with reset events

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