Active processes in one dimension

Demaerel, Thibaut; Maes, Christian

doi:10.1103/physreve.97.032604

Cited by 82 publications

(83 citation statements)

References 28 publications

(37 reference statements)

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“…We then look at the behaviour ofP (z) andQ(z) in Eqs. (44) and (45) near z = 0. In this limit both K(1 − z) and G 20 22 1 2 3 2 0 1 ; z diverge logarithmically whereas the Legendre and hypergeometric functions approach a constant value.…”

Section: Position Distribution For β =mentioning

confidence: 98%

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Exact stationary state of a run-and-tumble particle with three internal states in a harmonic trap

Basu¹,

Majumdar²,

Rosso³

et al. 2020

J. Phys. A: Math. Theor.

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We study the motion of a one-dimensional run-and-tumble particle with three discrete internal states in the presence of a harmonic trap of stiffness µ. The three internal states, corresponding to positive, negative and zero velocities respectively, evolve following a jump process with rate γ. We compute the stationary position distribution exactly for arbitrary values of µ and γ which turns out to have a finite support on the real line. We show that the distribution undergoes a shape-transition as β = γ/µ is changed. For β < 1, the distribution has a double-concave shape and shows algebraic divergences with an exponent (β − 1) both at the origin and at the boundaries. For β > 1, the position distribution becomes convex, vanishing at the boundaries and with a single, finite, peak at the origin. We also show that for the special case β = 1, the distribution shows a logarithmic divergence near the origin while saturating to a constant value at the boundaries.Exact stationary state of a three-state run-and-tumble particle

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Section: Position Distribution For β =mentioning

confidence: 98%

“…First, we note that the solutions in Eqs. (44) and (45) must satisfy the original first order equations (19) and (20) with β = 1 for all values of z. We then look at the behaviour ofP (z) andQ(z) in Eqs.…”

Section: Position Distribution For β =mentioning

confidence: 99%

Exact stationary state of a run-and-tumble particle with three internal states in a harmonic trap

Basu¹,

Majumdar²,

Rosso³

et al. 2020

J. Phys. A: Math. Theor.

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“…For example E. Coli bacteria runs for some time along a straight line and then tumbles to randomly choose a new direction of run [35]. In the last few years there has been a lot of interests in studying RTPs at the individual level as they show interesting phenomena like, accumulation near boundaries [36][37][38], clustering [39][40][41], passive to active transition [42,43], climbing against the hill [44], Kramer's escape problem [45] etc.…”

Section: Introductionmentioning

confidence: 99%

Generalised ‘Arcsine’ laws for run-and-tumble particle in one dimension

Singh¹,

Kundu²

2019

J. Stat. Mech.

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The 'Arcsine' laws of Brownian particles in one dimension describe distributions of three quantities: the time t m to reach maximum position, the time t r spent on the positive side and the time t of the last visit to the origin. Interestingly, the cumulative distribution of all the three quantities are same and given by Arcsine function. In this paper, we study distribution of these three times t m , t r and t in the context of single run-and-tumble particle in one dimension, which is a simple non-Markovian process. We compute exact distributions of these three quantities for arbitrary time and find that all three distributions have delta function part and a non-delta function part. Interestingly, we find that the distributions of t m and t r are identical (reminiscent of the Brownian particle case) when the initial velocities of the particle are chosen with equal probability. On the other hand, for t , only the non-delta function part is same with the other two. In addition, we find explicit expressions of the joint distributions of the maximum displacement and the time at which this maxima occurs. We verify all our analytical results through numerical simulations.

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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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Active processes in one dimension

Cited by 82 publications

References 28 publications

Exact stationary state of a run-and-tumble particle with three internal states in a harmonic trap

Exact stationary state of a run-and-tumble particle with three internal states in a harmonic trap

Generalised ‘Arcsine’ laws for run-and-tumble particle in one dimension

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

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