Infinite densities for Lévy walks

Rebenshtok, A.; Denisov, S. I.; Hänggi, Peter; Barkai, Eli

doi:10.1103/physreve.90.062135

Cited by 73 publications

(195 citation statements)

References 80 publications

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“…For the arbitrary μ from the interval 0 < μ < 1, one can find from (35) that the solution to (34) is the Lamperti distribution [8]. Note that at the small times t τ 0 , it follows from (30) δ(x + vt) representing two waves traveling in opposite directions.…”

Section: Gamma Pdf G(τ2λ)mentioning

confidence: 99%

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Single integrodifferential wave equation for a Lévy walk

Fedotov

2016

Phys. Rev. E

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We derive the single integrodifferential wave equation for the probability density function of the position of a classical one-dimensional Lévy walk with continuous sample paths. This equation involves a classical wave operator together with memory integrals describing the spatiotemporal coupling of the Lévy walk. It is valid at all times, not only in the long time limit, and it does not involve any large-scale approximations. It generalizes the well-known telegraph or Cattaneo equation for the persistent random walk with the exponential switching time distribution. Several non-Markovian cases are considered when the particle's velocity alternates at the gamma and power-law distributed random times. In the strong anomalous case we obtain the asymptotic solution to the integrodifferential wave equation. We implement the nonlinear reaction term of Kolmogorov-PetrovskyPiskounov type into our equation and develop the theory of wave propagation in reaction-transport systems involving Lévy diffusion.

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Section: Gamma Pdf G(τ2λ)mentioning

confidence: 99%

“…The shape of the PDF at several successive times can be found in [8]. It would be interesting to use a new wave equation for the nonnormalizable density problem for superdiffusive anomalous transport [35].…”

Section: Gamma Pdf G(τ2λ)mentioning

confidence: 99%

Single integrodifferential wave equation for a Lévy walk

Fedotov

2016

Phys. Rev. E

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“…In the case that around the origin the PDF is represented by a Lévy distribution of the form t −1/γ L γ (x/t 1/γ ) [36], one will find (by "stitching" this limit function and the ICD at a central region of x, as in [15]) the following relation between the scaling exponents: α − β + αγ = 1. Indeed, in our case, (α, β) = (3/2, 1 + 1/(2D)) gives the correct γ = (1 + D)/3D = ν.…”

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confidence: 99%

“…Largedeviations theory mainly deals with thin-tailed processes where extreme events are rare, but in Lévy processes these large fluctuations are dominant. To study the fluctuations in this system, we will show that the relevant tool is the asymptotic moment-generating function, which yields an infinite-covariant density (ICD) [12,15]. We will discuss the generality of this approach and its results below.…”

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confidence: 99%

Large Fluctuations for Spatial Diffusion of Cold Atoms

2017

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Large-deviations theory deals with tails of probability distributions and the rare events of random processes, for example spreading packets of particles. Mathematically, it concerns the exponential fall-of of the density of thin-tailed systems. Here we investigate the spatial density Pt(x) of laser cooled atoms, where at intermediate length scales the shape is fat-tailed. We focus on the rare events beyond this range, which dominate important statistical properties of the system. Through a novel friction mechanism induced by the laser fields, the density is explored with the recently proposed nonnormalized infinite-covariant density approach. The small and large fluctuations give rise to a bi-fractal nature of the spreading packet.PACS numbers: 05.40. Jc,46.65.+g In diffusion processes such as Brownian motion, the concentration of particles starting at the origin spreads out like a Gaussian, which is fully characterized by the mean squared-displacement. This is the result of the widely applicable Gaussian central limit theorem (CLT) [1]. Of no less importance is large-deviations theory [2], which deals with the rare fluctuations of processes such as simple coin tossing random walks (see e.g., [3]), extreme variations of the surface height in the KardarParisi-Zhang model [4] and the tails of the position distribution in single-file diffusion [5,6]. Mathematically, a prerequisite of the theory is that the cumulant generating function be "well behaved", i.e. smooth and differentiable. Large-deviations theory works when the decay of the probability of the observable of interest is exponential (see details in [2]). However many systems do not meet this requirement [2], for example Lévy fat-tailed processes [7][8][9], where the decay rate is a power-law. This is the case for a cloud of atoms undergoing Sisyphus laser-cooling [10], where both theoretically [11,12] and experimentally [13], it was shown that the central part of the spreading particle packet is described by the Lévy CLT [14]. The latter deals with the sum of independent identically-distributed random variables, but unlike the classical Gaussian CLT, here the summands' own distribution is heavy-tailed. As a result, Lévy's CLT yields an infinite mean squared-displacement for the sum [14], and consequently also the second cumulant. Largedeviations theory mainly deals with thin-tailed processes where extreme events are rare, but in Lévy processes these large fluctuations are dominant. To study the fluctuations in this system, we will show that the relevant tool is the asymptotic moment-generating function, which yields an infinite-covariant density (ICD) [12,15]. We will discuss the generality of this approach and its results below.In an experimental situation, diverging moments are unphysical. For example, although the experiment in [13] shows a nice fit of the particles' density to a symmetric Lévy distribution, clearly at finite times no particles traveling at finite velocities can ever be found infinitely far from their origin. The finiteness of all t...

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“…Lévy walk [1][2][3] is an important concept which describes a wide spectrum of physical and biological processes involving stochastic transport [4][5][6][7][8][9][10]. Cold atoms moving in dissipative optical lattices [11], endosomal active transport in living cells [12], and T-cells migrating in the brain tissue [13] are just several examples where Lévy walks were reported.…”

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confidence: 99%

Emergence of Lévy walks in systems of interacting individuals

Fedotov¹,

Korabel²

2017

Phys. Rev. E

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We propose a model of superdiffusive Lévy walk as an emergent nonlinear phenomenon in systems of interacting individuals. The aim is to provide a qualitative explanation of recent experiments [G. Ariel et al., Nat. Commun. 6, 8396 (2015)] revealing an intriguing behavior: swarming bacteria fundamentally change their collective motion from simple diffusion into a superdiffusive Lévy walk dynamics. We introduce microscopic mean-field kinetic equations in which we combine two key ingredients: (1) alignment interactions between individuals and (2) non-Markovian effects. Our interacting run-and-tumble model leads to the superdiffusive growth of the mean-squared displacement and the power-law distribution of run length with infinite variance. The main result is that the superdiffusive behavior emerges as a cooperative effect without using the standard assumption of the power-law distribution of run distances from the inception. At the same time, we find that the collision and repulsion interactions lead to the density-dependent exponential tempering of power-law distributions. This qualitatively explains the experimentally observed transition from superdiffusion to the diffusion of mussels as their density increases [M. [15][16][17][18]. Recently an intriguing behavior of swarming bacteria was found: they fundamentally change their collective motion from simple diffusion into a superdiffusive Lévy walk dynamics [19]. The extraordinary feature of this movement is that the emergence of a superdiffusive motility is a result of the interactions between bacteria rather than the standard mechanism of controlling the individual frequency of tumbling. However, it is still an open question how Lévy walks emerge in systems of interacting self-propelled particles. The current theory of Lévy walk [8] assumes noninteracting particles and power-law distribution of traveled distances from the inception.The collective behavior of large groups of interacting individuals such as bird flocks, fish schools, and the collective migration of cells or bacteria is another rapidly growing area of active matter research [20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38]. There exist two main types of models used for a collective behavior: (1) Lagrangian models describing the movements of self-propelled particles in terms of nonlinear equations for the positions and velocities of all particles [20][21][22][23], and (2) kinetic models involving partial differential equations for the population densities [24][25][26][27][28][29][30]. In this Rapid Communication we are only concerned with the nonlinear kinetic and macroscopic equations. They have been used to describe interactions between individuals and investigate the formation of a large variety of spatiotemporal self-organized aggregations. Most of these models converge to the density-dependent diffusive transport and do not lead to non-Markovian Lévy walks.In this Rapid Communication we propose a kinetic nonlinear Lévy walk model for interacting individuals in which...

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Infinite densities for Lévy walks

Cited by 73 publications

References 80 publications

Single integrodifferential wave equation for a Lévy walk

Single integrodifferential wave equation for a Lévy walk

Large Fluctuations for Spatial Diffusion of Cold Atoms

Emergence of Lévy walks in systems of interacting individuals

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