Theory of Fractional Lévy Kinetics for Cold Atoms Diffusing in Optical Lattices

Kessler, David A.; Barkai, Eli

doi:10.1103/physrevlett.108.230602

Cited by 106 publications

(123 citation statements)

References 38 publications

(58 reference statements)

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“…For α > 3, the usual Green-Kubo formula (4) holds. We can thus identify three different diffusion phases [49]: normal diffusion with hx 2 ðtÞi ∼ t when α > 3, anomalous superdiffusion with hx 2 ðtÞi ∼ t 4−α when 1 < α < 3, and Richardson-like diffusion with hx 2 ðtÞi ∼ t 3 when α < 1, which is known from turbulence [50] and has recently been observed as a transient behavior in ordinary Brownian motion [51].…”

Section: (28)mentioning

confidence: 99%

Scaling Green-Kubo Relation and Application to Three Aging Systems

et al. 2014

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The Green-Kubo formula relates the spatial diffusion coefficient to the stationary velocity autocorrelation function. We derive a generalization of the Green-Kubo formula that is valid for systems with longrange or nonstationary correlations for which the standard approach is no longer valid. For the systems under consideration, the velocity autocorrelation function hvðt þ τÞvðtÞi asymptotically exhibits a certain scaling behavior and the diffusion is anomalous, hx 2 ðtÞi ≃ 2D ν t ν . We show how both the anomalous diffusion coefficient D ν and the exponent ν can be extracted from this scaling form. Our scaling GreenKubo relation thus extends an important relation between transport properties and correlation functions to generic systems with scale-invariant dynamics. This includes stationary systems with slowly decaying power-law correlations, as well as aging systems, systems whose properties depend on the age of the system. Even for systems that are stationary in the long-time limit, we find that the long-time diffusive behavior can strongly depend on the initial preparation of the system. In these cases, the diffusivity D ν is not unique, and we determine its values, respectively, for a stationary or nonstationary initial state. We discuss three applications of the scaling Green-Kubo relation: free diffusion with nonlinear friction corresponding to cold atoms diffusing in optical lattices, the fractional Langevin equation with external noise recently suggested to model active transport in cells, and the Lévy walk with numerous applications, in particular, blinking quantum dots. These examples underline the wide applicability of our approach, which is able to treat very different mechanisms of anomalous diffusion.

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Section: (28)mentioning

confidence: 99%

Scaling Green-Kubo Relation and Application to Three Aging Systems

et al. 2014

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“…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.…”

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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 is a fundamental notion in physics and biology [1] with numerous applications including the transport of light governed by Lévy statistics [2], anomalous superdiffusion of cold atoms in optical lattices [3], T-cell motility in the brain [4], endosomal active transport within living cells [5]. In the last few years, the interest in the Lévy walk models increases rapidly in the context of Lévy flight foraging hypothesis according to which a Lévy transport is optimal to search for randomly located objects [6].…”

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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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Theory of Fractional Lévy Kinetics for Cold Atoms Diffusing in Optical Lattices

Cited by 106 publications

References 38 publications

Scaling Green-Kubo Relation and Application to Three Aging Systems

Scaling Green-Kubo Relation and Application to Three Aging Systems

Large Fluctuations for Spatial Diffusion of Cold Atoms

Single integrodifferential wave equation for a Lévy walk

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