Run and tumble particle under resetting: a renewal approach

Evans, Martin R.; Majumdar, Satya N.

doi:10.1088/1751-8121/aae74e

Cited by 245 publications

(277 citation statements)

References 63 publications

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“…For a fixed configuration, let us denote by ( )  t r the probability of this configuration in the system undergoing resetting at rate r (and by ( )  t 0 the probability of this configuration in the system without resetting), conditional on the fixed initial configuration to which the system is instantaneously brought back at each resetting event. Following the renewal argument of [29,32], the probability distribution ( )  t r can be expressed in terms of the probability distribution  0 . The expression consists of two terms corresponding to the following alternative.…”

Section: Resetting Eventsmentioning

confidence: 99%

“…Characterisation of such a non-equilibrium stationary sate has recently become a major focus of activity in statistical physics [23][24][25]. The field enjoys a broad range of applications including RNA polymerisation processes [26,27], active matter [28][29][30] and randomised searching problems [31] (see the review [32] and references therein).…”

Section: Introductionmentioning

confidence: 99%

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Non-conserving zero-range processes with extensive rates under resetting

Grange

2020

J. Phys. Commun.

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We consider a non-conserving zero-range process with hopping rate proportional to the number of particles at each site. Particles are added to the system with a site-dependent creation rate, and vanish with a uniform annihilation rate. On a fully-connected lattice with a large number of sites, the meanfield geometry leads to a negative binomial law for the number of particles at each site, with parameters depending on the hopping, creation and annihilation rates. This model can be mapped to population dynamics (if the creation rates are reproductive fitnesses in a haploid population, and the hopping rate is the mutation rate). It can also be mapped to a Bianconi-Barabási model of a growing network with random rewiring of links (if creation rates are the rates of acquisition of links by nodes, and the hopping rate is the rewiring rate). The steady state has recently been worked out and gives rise to occupation numbers that reproduce Kingman's house-of-cards model of selection and mutation. In this paper we solve the master equation using a functional method, which yields integral equations satisfied by the occupation numbers. The occupation numbers are shown to forget initial conditions at an exponential rate that decreases linearly with the fitness level. Moreover, they can be computed exactly in the Laplace domain, which allows to obtain the steady state of the system under resetting. The result modifies the house-of-cards result by simply adding a skewed version of the initial conditions, and by adding the resetting rate to the hopping rate.

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Section: Resetting Eventsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Non-conserving zero-range processes with extensive rates under resetting

Grange

2020

J. Phys. Commun.

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“…The model was further extended to study other, i.e. non-diffusive stochastic processes under resetting [25,[52][53][54][55][56][57][58][59][60][61][62].…”

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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“…Then the particle tumbles again and so on.non-Boltzmann distribution in the steady state in the presence of a confining potential [22,[26][27][28][29], motilityinduced phase separation [23], jamming [30] etc. Variants of the RTP model where the speed v ≥ 0 of the particle is renewed after each tumbling by drawing it from a probability density function (PDF) W (v) [31,32] or where the RTP undergoes random resetting to its initial position at a constant rate [34,35] have also been studied.In the d = 1 case, the first-passage properties of the RTP model and of its variants have been widely studied [24,[36][37][38][39]. Several recent studies investigated the survival probability of an RTP in d = 1, both in the absence and in the presence of a confining potential/wall [27,[37][38][39][40].…”

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confidence: 99%

Universal Survival Probability for a d -Dimensional Run-and-Tumble Particle

et al. 2020

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We consider an active run-and-tumble particle (RTP) in d dimensions and compute exactly the probability S(t) that the x-component of the position of the RTP does not change sign up to time t. When the tumblings occur at a constant rate, we show that S(t) is independent of d for any finite time t (and not just for large t), as a consequence of the celebrated Sparre Andersen theorem for discrete-time random walks in one dimension. Moreover, we show that this universal result holds for a much wider class of RTP models in which the speed v of the particle after each tumbling is random, drawn from an arbitrary probability distribution. We further demonstrate, as a consequence, the universality of the record statistics in the RTP problem.The first time t f at which a stochastic process reaches a fixed target level is a fundamental observable with many applications. Statistics of t f plays a crucial role in various situations, including e.g., the encounter of two molecules in a chemical reaction [1], the capture of a prey in a hunting scenario [2], or the escape of a comet from the solar system [3, 4]. In the context of finance, agents often use limit orders to buy/sell a stock only when its price is below/above a target value. Thus, it is important to estimate if and when that target value will be reached and this question has been intensively studied during decades (for recent reviews see [2,[5][6][7][8][9]). Due to the ubiquity of these problems, novel applications are constantly being identified, raising in turn new challenging questions.In recent years, tremendous efforts have been devoted to the study of statistical fluctuations in active matter systems [10][11][12][13]. In contrast to a passive matter such as a Brownian motion (BM), whose dynamics is driven by thermal fluctuations of the environment, this class of active non-equilibrium systems is characterized by selfpropelled motility based on continuous consumption of energy from the environment. For example, models of active matter have been used to describe vibrating granular matter [14], active gels [15,16], bacteria [17,18] or collective motion of "animals" [15,[19][20][21]. In this context, one of the most studied model is the run-andtumble particle (RTP) [22,23], also known as "persistent random walk" [24,25]. In the simplest version of the model, an RTP performs a ballistic motion along a certain direction at a constant speed v 0 ≥ 0 ("run") during a certain "time of flight" τ . Following this run, it "tumbles", i.e., chooses a new direction uniformly at random and then performs a new run along this direction again with speed v 0 during a random time τ and so on (see Fig. 1). Typically these tumblings occur with constant rate γ, i.e. the τ 's of different runs are independently distributed via exponential distribution p(τ ) = γe −γτ , though other distributions will also be considered later. Despite its simplicity, this RTP model exhibits complex interesting features such as clustering at boundaries [11], 0 0 1 1 O l l l 3 1 4 5 l l 6 2 l l n FIG. 1: Ty...

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Run and tumble particle under resetting: a renewal approach

Cited by 245 publications

References 63 publications

Non-conserving zero-range processes with extensive rates under resetting

Non-conserving zero-range processes with extensive rates under resetting

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

Universal Survival Probability for a d -Dimensional Run-and-Tumble Particle

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