Noncrossing run-and-tumble particles on a line

Doussal, Pierre Le; Majumdar, Satya N.; Schehr, Grégory

doi:10.1103/physreve.100.012113

Cited by 69 publications

(108 citation statements)

References 33 publications

(58 reference statements)

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“…These terms arise from the events in which the RTP has not changed its direction till time t and such events occur with probability e −γt . From these expressions, we can obtain the total probability density P (x, t|x, α, 0) = a[P + (x, t|0, +, 0)+P − (x, t|0, +, 0)]+ b[P + (x, t|0, −, 0) + P − (x, t|0, −, 0)] of finding the particle at x in time t and check that at x = M it matches with the result given in [46]. Using the large z asymptotic of I ν (z) e z / √ 2πz for both ν = 0 and 1, one can show that in the limit v → ∞ and γ → ∞ keeping v 2 /(2γ) = D fixed, the individual probability densities in Eqs.…”

Section: Propagator Of a Rtp In Presence Of An Absorbing Barrier At Xsupporting

confidence: 53%

“…Starting from the Langevin equation Eq. (2), it is easy to show [36,46] that these propagators satisfy the following forward master equations…”

Section: Propagator Of a Rtp In Presence Of An Absorbing Barrier At Xmentioning

confidence: 99%

“…For finite t, the particle, starting from the origin, can never reach x → −∞ in finite duration t which implies P ± (x → −∞, t|0, ±, 0) = 0. To see how the boundary condition at x = M appear, we follow [46] where the problem of finding the propagator of a single RTP in presence of an absorbing boundary (at the origin) has recently been considered. In this paper, the final result of the total probability P (x, t|0, 0) = a[P + (x, t|0, +, 0) + P − (x, t|0, +, 0)] + b[P + (x, t|0, −, 0) + P − (x, t|0, −, 0)] at the wall has been presented explicitly.…”

Section: Propagator Of a Rtp In Presence Of An Absorbing Barrier At Xmentioning

confidence: 99%

“…(7) have to be solved are P ± (x → −∞, t|0, ±, 0) = 0 at x → −∞ and P − (M, t|0, α, 0) = 0 at x = M . It turns out that these boundary conditions are enough to find the propagator P ± (x, t|0, α, 0) uniquely [46].…”

Section: Propagator Of a Rtp In Presence Of An Absorbing Barrier At Xmentioning

confidence: 99%

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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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Section: Propagator Of a Rtp In Presence Of An Absorbing Barrier At Xsupporting

confidence: 53%

“…Starting from the Langevin equation Eq. (2), it is easy to show [36,46] that these propagators satisfy the following forward master equations…”

Section: Propagator Of a Rtp In Presence Of An Absorbing Barrier At Xmentioning

confidence: 99%

Section: Propagator Of a Rtp In Presence Of An Absorbing Barrier At Xmentioning

confidence: 99%

Section: Propagator Of a Rtp In Presence Of An Absorbing Barrier At Xmentioning

confidence: 99%

See 2 more Smart Citations

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

Singh¹,

Kundu²

2019

J. Stat. Mech.

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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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On the Consistency of the Reaction-Telegraph Process Within Finite Domains

Tilles

Petrovskii

2019

J Stat Phys

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Reaction-telegraph equation (RTE) is a mathematical model that has often been used to describe natural phenomena, with specific applications ranging from physics to social sciences. In particular, in the context of ecology, it is believed to be a more realistic model to describe animal movement than the more traditional approach based on the reaction-diffusion equations. Indeed, the reaction-telegraph equation arises from more realistic microscopic assumptions about individual animal movement (the correlated random walk) and hence could be expected to be more relevant than the diffusion-type models that assume the simple, unbiased Brownian motion. However, the RTE has one significant drawback as its solutions are not positively defined. It is not clear at which stage of the RTE derivation the realism of the microscopic description is lost and/or whether the RTE can somehow be 'improved' to guarantee the solutions positivity. Here we show that the origin of the problem is twofold. Firstly, the RTE is not fully equivalent to the Cattaneo system from which it is obtained; the equivalence can only be achieved in a certain parameter range and only for the initial conditions containing a finite number of Fourier modes. Secondly, the Dirichlet type boundary conditions routinely used for reaction-diffusion equations appear to be meaningless if used for the RTE resulting in solutions with unrealistic properties. We conclude that, for the positivity to be regained, one has to use the Cattaneo system with boundary conditions of Robin type or Neumann type, and we show how relevant classes of solutions can be obtained. Response to Reviewers:Reply to Reviewer 1 All changes made in the manuscript are marked in blue colour.1) The referee is right: there are restrictions on what current fields can be realised within a given density field, and the observed 'population appears from nowhere' feature indeed comes from the imposition of a positive rate of change of density on the

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Noncrossing run-and-tumble particles on a line

Cited by 69 publications

References 33 publications

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

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

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

On the Consistency of the Reaction-Telegraph Process Within Finite Domains

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