Diffusion with resetting in bounded domains

Christou, Christos; Schadschneider, Andreas

doi:10.1088/1751-8113/48/28/285003

Cited by 137 publications

(138 citation statements)

References 32 publications

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“…In [49] the survival probability Q r (x 0 , t) for a single resetting site X r , on a finite domain 0 ≤ x ≤ L with reflecting boundaries and a partially absorbing site with absorption velocity b was considered. The solution was worked out from the forward master equation for the survival probability Q r (x 0 , t) and an eigenfunction expansion.…”

Section: Survival Probability With Resetting On a Finite Domainmentioning

confidence: 99%

Stochastic resetting and applications

Evans

Majumdar

Schehr

2020

J. Phys. A: Math. Theor.

531

600

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In this Topical Review we consider stochastic processes under resetting, which have attracted a lot of attention in recent years. We begin with the simple example of a diffusive particle whose position is reset randomly in time with a constant rate r, which corresponds to Poissonian resetting, to some fixed point (e.g. its initial position). This simple system already exhibits the main features of interest induced by resetting: (i) the system reaches a nontrivial nonequilibrium stationary state (ii) the mean time for the particle to reach a target is finite and has a minimum, optimal, value as a function of the resetting rate r. We then generalise to an arbitrary stochastic process (e.g. Lévy flights or fractional Brownian motion) and non-Poissonian resetting (e.g. power-law waiting time distribution for intervals between resetting events). We go on to discuss multiparticle systems as well as extended systems, such as fluctuating interfaces, under resetting. We also consider resetting with memory which implies resetting the process to some randomly selected previous time. Finally we give an overview of recent developments and applications in the field.

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Section: Survival Probability With Resetting On a Finite Domainmentioning

confidence: 99%

Stochastic resetting and applications

Evans

Majumdar

Schehr

2020

J. Phys. A: Math. Theor.

531

600

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“…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.…”

Section: Introductionmentioning

confidence: 99%

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

2019

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The canonical Evans-Majumdar model for diffusion with stochastic resetting to the origin assumes that resetting takes zero time: upon resetting the diffusing particle is teleported back to the origin to start its motion anew. However, in reality getting from one place to another takes a finite amount of time which must be accounted for as diffusion with resetting already serves as a model for a myriad of processes in physics and beyond. Here we consider a situation where upon resetting the diffusing particle returns to the origin at a finite (rather than infinite) speed. This creates a coupling between the particle's random position at the moment of resetting and its return time, and further gives rise to a non-trivial cross-talk between two separate phases of motion: the diffusive phase and the return phase. We show that each of these phases relaxes to the stead-state in a unique manner; and while this could have also rendered the total relaxation dynamics extremely non-trivial our analysis surprisingly reveals otherwise. Indeed, the time-dependent distribution describing the particle's position in our model is completely invariant to the speed of return. Thus, whether returns are slow or fast, we always recover the result originally obtained for diffusion with instantaneous returns to the origin.

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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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Diffusion with resetting in bounded domains

Cited by 137 publications

References 32 publications

Stochastic resetting and applications

Stochastic resetting and applications

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

Transport properties of random walks under stochastic noninstantaneous resetting

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