Lévy dynamics of enhanced diffusion: Application to turbulence

Shlesinger, Michael F.; West, Bruce J.; Klafter, J.

doi:10.1103/physrevlett.58.1100

Cited by 735 publications

(542 citation statements)

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“…The durations of these on periods and off periods are found to be distributed according to a power law whose exponent is such that the average on and off time is infinite (i.e., of the order of the measurement time), leading to aging in the intensity autocorrelation function [7,66]. This model is the familiar Lévy walk [12,31,36,67], which is a popular stochastic framework with many applications. As mentioned in the Introduction, Zumofen and Klafter [12] showed that the diffusivity in this model is sensitive to the initial conditions.…”

Section: Blinking Quantum Dots and Lévy Walkmentioning

confidence: 99%

“…In particular, when 0 < μ < 1, the mean waiting time is infinite and the velocity correlation function ages [34,35]. The Lévy walk has been successfully applied to various systems, including turbulence [36], search patterns of animals [37], and blinking quantum dots [38]. For the latter case, we show how our scaling Green-Kubo relation offers a straightforward way to connect the anomalous photon statistics to the intensity autocorrelations.…”

Section: Introductionmentioning

confidence: 99%

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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formula that is valid for systems with longrange or nonstationary correlations for which the standard approach is no longer valid.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: Blinking Quantum Dots and Lévy Walkmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Scaling Green-Kubo Relation and Application to Three Aging Systems

et al. 2014

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“…diffusion not accurately modeled by the usual advection dispersion equation. Anomalous diffusion has been used in modeling turbulent flow [4,12], and chaotic dynamics of classical conservative systems [14]. In viscoelasticity, fractional differential operators have been used to describe materials' constitutive equations [7].…”

Section: Introductionmentioning

confidence: 99%

Variational formulation for the stationary fractional advection dispersion equation

Ervin

Roop

2006

Numer. Methods Partial Differential Eq.

695

444

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In this paper a theoretical framework for the Galerkin finite element approximation to the steady state fractional advection dispersion equation is presented. Appropriate fractional derivative spaces are defined and shown to be equivalent to the usual fractional dimension Sobolev spaces H s . Existence and uniqueness results are proven, and error estimates for the Galerkin approximation derived. Numerical results are included which confirm the theoretical estimates.

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“…This is not surprising, because the parameter κ = 1/|q − 1| can almost trivially be related to the parameter q that appears in the non-extensive thermostatistics. This relation was implicitly used in a note investigating superdiffusion near the magnetopause (Treumann, 1997, see the appendix of that note) when referring to Lévy-flight statistics in the form proposed by Shlesinger et al (1987) 3 , though not referring to Tsallis' non-extensive statistics such that the coincidence was somehow accidental. It was independently elaborated by Milovanov and Zelenyi (2000), Leubner (2002) and others in various contexts.…”

Section: Introductionmentioning

confidence: 99%

“…The parameter q ∈ R had been introduced first by Renyi (1955Renyi ( , 1970 as power of a qgeneralised logarithmic Boltzmann entropy that found wide application in the theory of deterministic chaos and the related thermodynamics (cf., e.g., Beck and Schlögl, 1997). Tsallis (1988) referred to it in postulating his non-extensive, conveniently simple version of entropy which became the basis of the celebrated Tsallis-nonextensive thermostatistics (cf.…”

Section: Introductionmentioning

confidence: 99%

Generalised partition functions: inferences on phase space distributions

Treumann

Baumjohann

2016

Ann. Geophys.

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Abstract. It is demonstrated that the statistical mechanical partition function can be used to construct various different forms of phase space distributions. This indicates that its structure is not restricted to the Gibbs-Boltzmann factor prescription which is based on counting statistics. With the widely used replacement of the Boltzmann factor by a generalised Lorentzian (also known as the q-deformed exponential function, where κ = 1/|q − 1|, with κ, q ∈ R) both the kappa-Bose and kappa-Fermi partition functions are obtained in quite a straightforward way, from which the conventional Bose and Fermi distributions follow for κ → ∞. For κ = ∞ these are subject to the restrictions that they can be used only at temperatures far from zero. They thus, as shown earlier, have little value for quantum physics. This is reasonable, because physical κ systems imply strong correlations which are absent at zero temperature where apart from stochastics all dynamical interactions are frozen. In the classical large temperature limit one obtains physically reasonable κ distributions which depend on energy respectively momentum as well as on chemical potential. Looking for other functional dependencies, we examine Bessel functions whether they can be used for obtaining valid distributions. Again and for the same reason, no Fermi and Bose distributions exist in the low temperature limit. However, a classical Bessel-Boltzmann distribution can be constructed which is a Bessel-modified Lorentzian distribution. Whether it makes any physical sense remains an open question. This is not investigated here. The choice of Bessel functions is motivated solely by their convergence properties and not by reference to any physical demands. This result suggests that the Gibbs-Boltzmann partition function is fundamental not only to Gibbs-Boltzmann but also to a large class of generalised Lorentzian distributions as well as to the corresponding nonextensive statistical mechanics.

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Lévy dynamics of enhanced diffusion: Application to turbulence

Cited by 735 publications

References 15 publications

Scaling Green-Kubo Relation and Application to Three Aging Systems

Scaling Green-Kubo Relation and Application to Three Aging Systems

Variational formulation for the stationary fractional advection dispersion equation

Generalised partition functions: inferences on phase space distributions

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