Diffusion under time-dependent resetting

Pal, A.; Kundu, Anupam; Evans, Martin R.

doi:10.1088/1751-8113/49/22/225001

Cited by 275 publications

(335 citation statements)

References 40 publications

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“…We also thank B. Meerson for helpful discussions and S. Reuveni for useful comments on the manuscript. Added Note: As final revisions were being made, we became aware of related work [46], in which the authors mathematically showed that deterministic reset leads to the smallest search time for the case of a single searcher, as we also observed.…”

Section: Discussionmentioning

confidence: 61%

Stochastic search with Poisson and deterministic resetting

Bhat¹,

Bacco²,

Redner³

2016

J. Stat. Mech.

140

136

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Abstract.We investigate a stochastic search process in one, two, and three dimensions in which N diffusing searchers that all start at x 0 seek a target at the origin. Each of the searchers is also reset to its starting point, either with rate r, or deterministically, with a reset time T . In one dimension and for a small number of searchers, the search time and the search cost are minimized at a non-zero optimal reset rate (or time), while for sufficiently large N , resetting always hinders the search. In general, a single searcher leads to the minimum search cost in one, two, and three dimensions. When the resetting is deterministic, several unexpected feature arise for N searchers, including the search time being independent of T for 1/T → 0 and the search cost being independent of N over a suitable range of N . Moreover, deterministic resetting typically leads to a lower search cost than in stochastic resetting.

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Section: Discussionmentioning

confidence: 61%

Stochastic search with Poisson and deterministic resetting

Bhat¹,

Bacco²,

Redner³

2016

J. Stat. Mech.

140

136

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“…There, a diffusive particle is studied when it may occasionally reset its position with a constant probability and the authors find that a non-equilibrium steady state (NESS) is reached and the mean first passage time of the overall process is finite and attains a minimum in terms of the resetting rate. 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].…”

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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Diffusion under time-dependent resetting

Cited by 275 publications

References 40 publications

Stochastic search with Poisson and deterministic resetting

Stochastic search with Poisson and deterministic resetting

Transport properties of random walks under stochastic noninstantaneous resetting

Continuous-time random walks with reset events

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