Anomalous Diffusion and Relaxation Close to Thermal Equilibrium: A Fractional Fokker-Planck Equation Approach

Metzler, Ralf; Barkai, Eli; Klafter, J.

doi:10.1103/physrevlett.82.3563

Cited by 749 publications

(692 citation statements)

References 34 publications

(44 reference statements)

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“…After findingQ(s) andP (s) explicitly we can return to method A and use Eqs. (15)(16)(17)(18)(19)(20)(21)(22)(23)(24)(25)(26) in order to write down the final solution.…”

Section: B Methods Bmentioning

confidence: 99%

“…The use of fractional-differential equations like Eqs. (8,11) became quite common in recent years [8,22], especially in the context of anomalous diffusion [8,23,24]. Several other fractional oscillator equations were considered in the literature [25,26,27] and general solutions for fractional-differential equations of the type Eq.…”

Section: Stat-mech] 26 Feb 2008mentioning

confidence: 99%

“…In particular fractional approach to sub-diffusion naturally leads to anomalous response functions commonly found in many systems e.g. the ColeCole relaxation [24,38,39]. In the next three subsections we will explore the response of FLE with and without the Harmonic potential and also investigate the behavior of the imaginary part of the complex susceptibility, i.e.…”

Section: Response To An External Fieldmentioning

confidence: 99%

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Fractional Langevin equation: Overdamped, underdamped, and critical behaviors

2008

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The dynamical phase diagram of the fractional Langevin equation is investigated for harmonically bound particle. It is shown that critical exponents mark dynamical transitions in the behavior of the system. Four different critical exponents are found. (i) αc = 0.402 ± 0.002 marks a transition to a non-monotonic under-damped phase, (ii) αR = 0.441... marks a transition to a resonance phase when an external oscillating field drives the system, (iii) αχ 1 = 0.527... and (iv) αχ 2 = 0.707... marks transition to a double peak phase of the "loss" when such an oscillating field present. As a physical explanation we present a cage effect, where the medium induces an elastic type of friction. Phase diagrams describing over-damped, under-damped regimes, motion and resonances, show behaviors different from normal.

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“…After findingQ(s) andP (s) explicitly we can return to method A and use Eqs. (15)(16)(17)(18)(19)(20)(21)(22)(23)(24)(25)(26) in order to write down the final solution.…”

Section: B Methods Bmentioning

confidence: 99%

Section: Stat-mech] 26 Feb 2008mentioning

confidence: 99%

Section: Response To An External Fieldmentioning

confidence: 99%

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Fractional Langevin equation: Overdamped, underdamped, and critical behaviors

2008

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“…3.2.3, p. 52 for which x 2 ∝ t α with α = 1. Various authors have attempted to describe such motion with fractional Fokker-Planck operators (Metzler et al, 1999;Carreras et al, 1999b).…”

Section: Long-time Tails and Socmentioning

confidence: 99%

Fundamental statistical descriptions of plasma turbulence in magnetic fields

Krommes¹

2002

Physics Reports

257

397

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A pedagogical review of the historical development and current status (as of early 2000) of systematic statistical theories of plasma turbulence is undertaken. Emphasis is on conceptual foundations and methodology, not practical applications. Particular attention is paid to equations and formalism appropriate to strongly magnetized, fully ionized plasmas. Extensive reference to the literature on neutral-fluid turbulence is made, but the unique properties and problems of plasmas are emphasized throughout. Discussions are given of quasilinear theory, weak-turbulence theory, resonance-broadening theory, and the clump algorithm. Those are developed independently, then shown to be special cases of the direct-interaction approximation (DIA), which provides a central focus for the article. Various methods of renormalized perturbation theory are described, then unified with the aid of the generating-functional formalism of Martin, Siggia, and Rose. A general expression for the renormalized dielectric function is deduced and discussed in detail. Modern approaches such as decimation and PDF methods are described. Derivations of DIA-based Markovian closures are discussed. The eddy-damped quasinormal Markovian closure is shown to be nonrealizable in the presence of waves, and a new realizable Markovian closure is presented. The test-field model and a realizable modification thereof are also summarized. Numerical solutions of various closures for some plasmaphysics paradigms are reviewed. The variational approach to bounds on transport is developed. Miscellaneous topics include Onsager symmetries for turbulence, the interpretation of entropy balances for both kinetic and fluid descriptions, self-organized criticality, statistical interactions between disparate scales, and the roles of both mean and random shear. Appendices are provided on Fourier transform conventions, dimensional and scaling analysis, the derivations of nonlinear gyrokinetic and gyrofluid equations, stochasticity criteria for quasilinear theory, formal aspects of resonance-broadening theory, Novikov's theorem, the treatment of weak inhomogeneity, the derivation of the Vlasov weak-turbulence wave kinetic equation from a fully renormalized description, some features of a code for solving the direct-interaction approximation and related Markovian closures, the details of the solution of the EDQNM closure for a solvable three-wave model, and the notation used in the article.

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“…17,28,29 Subdiffusive dynamics of an overdamped coordinate can be described using a fractional Fokker-Planck equation for Ornstein-Uhlenbeck process, developed by Metzler, Barkai, and Klafter. 30,31,32 The one-dimensional FFPE for the distribution W R (X 1 ,t 1 ;X 0 ,0) subject to the initial condition…”

Section: The Modelmentioning

confidence: 99%

Multipoint Fluorescence Quenching-Time Statistics for Single Molecules with Anomalous Diffusion

Barsegov

Mukamel

2003

J. Phys. Chem. A

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Three-point fluorescence lifetime correlation functions are computed for the fluorescence resonance energy transfer (FRET) and electron transfer (ET) quenching mechanisms in a single donor-acceptor system of which the distance X undergoes anomalous diffusion in a harmonic potential with short-time variance scaling as ∼t R . The three-point joint probability distribution of X and its moments are calculated by solving the fractional Fokker-Planck equation (FFPE). For R ) 1, the process is stationary and the two-and three-point joint probability distribution is centered around the time-dependent average donor-acceptor separation, 〈X(t)〉. For R < 1, the distribution slows down, becomes nonstationary, and remains centered at initial separation X(0) even for times exceeding the bath correlation time scale.

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Anomalous Diffusion and Relaxation Close to Thermal Equilibrium: A Fractional Fokker-Planck Equation Approach

Cited by 749 publications

References 34 publications

Fractional Langevin equation: Overdamped, underdamped, and critical behaviors

Fractional Langevin equation: Overdamped, underdamped, and critical behaviors

Fundamental statistical descriptions of plasma turbulence in magnetic fields

Multipoint Fluorescence Quenching-Time Statistics for Single Molecules with Anomalous Diffusion

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