2004
DOI: 10.1088/0305-4470/37/31/r01
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The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics

Abstract: Abstract. Fractional dynamics has experienced a firm upswing during the last years, having been forged into a mature framework in the theory of stochastic processes. A large number of research papers developing fractional dynamics further, or applying it to various systems have appeared since our first review article on the fractional FokkerPlanck equation [R. Metzler and J. Klafter, Phys. Rep. 339 (2000) 1-77]. It therefore appears timely to put these new works in a cohesive perspective. In this review we cov… Show more

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Cited by 2,175 publications
(1,924 citation statements)
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References 352 publications
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“…25) with Gaussian white noise ζ(u) and white Lévy noise τ (u) > 0 with 0 < α < walk theory by generating non-trivial jump dynamics. The latter approach has in turn been used, e.g., to understand measurements of anomalous photo currents in copy machines [39], microsphere diffusion in the cell membrane [45], translocations of biomolecules through membrane pores [77] and even dynamics of prices in financial markets [78]. It was demonstrated that this Langevin description leads to the time-fractional Fokker-Planck equation [75,76] …”
Section: Time-fractional Kineticsmentioning
confidence: 99%
See 1 more Smart Citation
“…25) with Gaussian white noise ζ(u) and white Lévy noise τ (u) > 0 with 0 < α < walk theory by generating non-trivial jump dynamics. The latter approach has in turn been used, e.g., to understand measurements of anomalous photo currents in copy machines [39], microsphere diffusion in the cell membrane [45], translocations of biomolecules through membrane pores [77] and even dynamics of prices in financial markets [78]. It was demonstrated that this Langevin description leads to the time-fractional Fokker-Planck equation [75,76] …”
Section: Time-fractional Kineticsmentioning
confidence: 99%
“…Paradigmatic examples are diffusion processes where the long-time mean square displacement does not grow linearly in time: That is, x 2 ∼ t α , where the angular brackets denote an ensemble average, does not increase with α = 1 as expected for Brownian motion but either subdiffusively with α < 1 or superdiffusively with α > 1 [36,37,38]. After pioneering work on amorphous semiconductors [39], anomalous transport phenomena have more recently been observed in a wide variety of complex systems, such as plasmas [40], nanopores [41], epidemic spreading [42], biological cell migration [43] and glassy materials [44], to mention a few [45,46]. This raises the question to which extent conventional FRs are valid for anomalous dynamics.…”
Section: Introductionmentioning
confidence: 95%
“…We pay a particular attention to jump-type processes which are omnipresent in Nature (see [3] and references therein). Their characterization is primarily provided by jump transition rates between different states of the system under consideration.…”
Section: Introductionmentioning
confidence: 99%
“…In recent years, the use of fractional derivatives in dynamical models of physical processes exhibiting anomalously slow or fast diffusion has diffused (cf. the surveys [MK04,SKB02]). Fractional calculus includes various extensions of the usual derivative from integer to real order.…”
Section: Application To the Fractional Diffusionmentioning
confidence: 99%