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

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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“…Substituting Eqs. (28) and (29) in Eq. (20) and using well known identities involving the hypergeometric function, we get,…”

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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“…Over the recent few years, this model has been substantially studied and a variety of results are known. Examples include -position distribution in free space as well as in confining potential [21][22][23][24][25][26][27], condensation transition [28][29][30], persistent properties [31][32][33], extremal properties [34,35], path functionals [34,36], current fluctuations [37], interacting multiple RTPs [27,[38][39][40][41], etc.…”

Section: Introductionmentioning

confidence: 99%

Mean area of the convex hull of a run and tumble particle in two dimensions

Singh,

Kundu,

Majumdar

et al. 2021

Preprint

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We investigate the statistics of the convex hull for a single run-and-tumble particle in two dimensions. Run-and-tumble particle (RTP), also known as persistent random walker, has gained significant interest in the recent years due to its biological application in modelling the motion of bacteria. We consider two different statistical ensembles depending on whether (i) the total number of tumbles n or (ii) the total observation time t is kept fixed. Benchmarking the results on perimeter, we study the statistical properties of the area of the convex hull for RTP. Exploiting the connections to extreme value statistics, we obtain exact analytical expressions for the mean area for both ensembles. For fixed-t ensemble, we show that the mean possesses a scaling form in γt (with γ being the tumbling rate) and the corresponding scaling function is exactly computed. Interestingly, we find that it exhibits crossover from ∼ t 3 scaling at small times t γ −1 to ∼ t scaling at large times t γ −1 . On the other hand, for fixed-n ensemble, the mean expectedly grows linearly with n for n 1. All our analytical findings are supported with numerical simulations.

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“…where D 0 is the diffusion coefficient. The study of the extremal statistics has been performed for a variety of stochastic processes like Brownian motion, random walk and their generalisations [19][20][21][22][23][24][25][26][27], random acceleration [28][29][30], active particles [31,32], fractional Brownian motion [33][34][35], continuous time random walk [36], random matrices [37][38][39], fluctuating interfaces [40][41][42], transport models [43][44][45], finance [46] and other physical systems [47][48][49][50][51] (see [52][53][54][55][56][57][58][59][60][61] for review). The subject of EVS has found applications in ecology [62], computer science [63][64][65] and convex ...…”

Section: Introductionmentioning

confidence: 99%

“…and hence the name arcsine laws. Over the years, these quantities have been studied in different contexts like Brownian motion, random walks and their generalisations [69][70][71][72][73][74][75][76], random acceleration [28,77], continuous time random walk [36,78], fractional Brownian motion [34], run and tumble particle [31,79], finance [46,80,81], renewal processes and other processes [82][83][84][85][86]. Quite recently, arcsine laws have also been studied both experimentally and theoretically in stochastic thermodynamics [87,88].…”

Section: Introductionmentioning

confidence: 99%

Extreme value statistics and arcsine laws for heterogeneous diffusion processes

Singh

2021

Preprint

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Heterogeneous diffusion with spatially changing diffusion coefficient arises in many experimental systems like protein dynamics in the cell cytoplasm, mobility of cajal bodies and confined hardsphere fluids. Here, we showcase a simple model of heterogeneous diffusion where the diffusion coefficient D(x) varies in power-law way, i.e. D(x) ∼ |x| −α with the exponent α > −1. This model exhibits anomalous scaling of the mean squared displacement (MSD) of the form ∼ t 2 2+α and weak ergodicity breaking in the sense that ensemble averaged and time averaged MSDs do not converge. In this paper, we look at the extreme value statistics of this model and derive, for all α, the exact probability distributions of the maximum spatial displacement M (t) and arg-maximum tm(t) (i.e. the time at which this maximum is reached) till duration t. In the second part of our paper, we analyze the statistical properties of the residence time tr(t) and the last-passage time t (t) and compute their distributions exactly for all values of α. Our study unravels that the heterogeneous version (α = 0) displays many rich and contrasting features compared to that of the standard Brownian motion (SBm). For example, while for SBm (α = 0), the distributions of tm(t), tr(t) and t (t) are all identical (á la "arcsine laws" due to Lévy), they turn out to be significantly different for non-zero α. Another interesting property of tr(t) is the existence of a critical α (which we denote by αc = −0.3182) such that the distribution exhibits a local maximum at tr = t/2 for α < αc whereas it has minima at tr = t/2 for α ≥ αc. The underlying reasoning for this difference hints at the very contrasting natures of the process for α ≥ αc and α < αc which we thoroughly examine in our paper. All our analytical results are backed by extensive numerical simulations.

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

Cited by 76 publications

References 51 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

Mean area of the convex hull of a run and tumble particle in two dimensions

Extreme value statistics and arcsine laws for heterogeneous diffusion processes

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