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Journal ArticleDOI

The random walk's guide to anomalous diffusion: a fractional dynamics approach

01 Dec 2000-Physics Reports (North-Holland)-Vol. 339, Iss: 1, pp 1-77
TL;DR: Fractional kinetic equations of the diffusion, diffusion-advection, and Fokker-Planck type are presented as a useful approach for the description of transport dynamics in complex systems which are governed by anomalous diffusion and non-exponential relaxation patterns.
About: This article is published in Physics Reports.The article was published on 2000-12-01. It has received 7412 citations till now. The article focuses on the topics: Fractional dynamics & Continuous-time random walk.
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Journal ArticleDOI
TL;DR: In this article, the authors deal with the fractional Sobolev spaces W s;p and analyze the relations among some of their possible denitions and their role in the trace theory.
Abstract: This paper deals with the fractional Sobolev spaces W s;p . We analyze the relations among some of their possible denitions and their role in the trace theory. We prove continuous and compact embeddings, investigating the problem of the extension domains and other regularity results. Most of the results we present here are probably well known to the experts, but we believe that our proofs are original and we do not make use of any interpolation techniques nor pass through the theory of Besov spaces. We also present some counterexamples in non-Lipschitz domains.

3,555 citations

Journal ArticleDOI
26 Jan 2006-Nature
TL;DR: It is shown that human travelling behaviour can be described mathematically on many spatiotemporal scales by a two-parameter continuous-time random walk model to a surprising accuracy, and concluded that human travel on geographical scales is an ambivalent and effectively superdiffusive process.
Abstract: The website wheresgeorge.com invites its users to enter the serial numbers of their US dollar bills and track them across America and beyond. Why? “For fun and because it had not been done yet”, they say. But the dataset accumulated since December 1998 has provided the ideal raw material to test the mathematical laws underlying human travel, and that has important implications for the epidemiology of infectious diseases. Analysis of the trajectories of over half a million dollar bills shows that human dispersal is described by a ‘two-parameter continuous-time random walk’ model: our travel habits conform to a type of random proliferation known as ‘superdiffusion’. And with that much established, it should soon be possible to develop a new class of models to account for the spread of human disease. The dynamic spatial redistribution of individuals is a key driving force of various spatiotemporal phenomena on geographical scales. It can synchronize populations of interacting species, stabilize them, and diversify gene pools1,2,3. Human travel, for example, is responsible for the geographical spread of human infectious disease4,5,6,7,8,9. In the light of increasing international trade, intensified human mobility and the imminent threat of an influenza A epidemic10, the knowledge of dynamical and statistical properties of human travel is of fundamental importance. Despite its crucial role, a quantitative assessment of these properties on geographical scales remains elusive, and the assumption that humans disperse diffusively still prevails in models. Here we report on a solid and quantitative assessment of human travelling statistics by analysing the circulation of bank notes in the United States. Using a comprehensive data set of over a million individual displacements, we find that dispersal is anomalous in two ways. First, the distribution of travelling distances decays as a power law, indicating that trajectories of bank notes are reminiscent of scale-free random walks known as Levy flights. Second, the probability of remaining in a small, spatially confined region for a time T is dominated by algebraically long tails that attenuate the superdiffusive spread. We show that human travelling behaviour can be described mathematically on many spatiotemporal scales by a two-parameter continuous-time random walk model to a surprising accuracy, and conclude that human travel on geographical scales is an ambivalent and effectively superdiffusive process.

2,120 citations

Journal ArticleDOI
TL;DR: Fractional dynamics has experienced a firm upswing during the past few years, having been forged into a mature framework in the theory of stochastic processes as mentioned in this paper, and 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 Fokker-Planck equation.
Abstract: Fractional dynamics has experienced a firm upswing during the past few 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 Fokker–Planck equation (Metzler R and Klafter J 2000a, Phys. Rep. 339 1–77). It therefore appears timely to put these new works in a cohesive perspective. In this review we cover both the theoretical modelling of sub- and superdiffusive processes, placing emphasis on superdiffusion, and the discussion of applications such as the correct formulation of boundary value problems to obtain the first passage time density function. We also discuss extensively the occurrence of anomalous dynamics in various fields ranging from nanoscale over biological to geophysical and environmental systems.

2,119 citations


Cites background or methods from "The random walk's guide to anomalou..."

  • ...The MFPT diverges both in the absence of a bias and under a constant drift, pertaining to both finite as well as semi-infinite domains (Metzler and Klafter 2000e, Rangarajan and Ding 2000a, 2000b, 2003, Scher et al 2002a, Barkai 2001)22....

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  • ...Conversely, in the overdamped case, the FKKE (14) corresponds in position space to the fractional Fokker– Planck equation (FFPE) (Metzler et al 1999a, 1999b, Metzler and Klafter 2000b, 2000c) ∂P ∂t = 0D1−αt ( − ∂ ∂x F (x) mηα + Kα ∂2 ∂x2 ) P(x, t), (16) where ηα ≡ ητα/τ ∗ and Kα ≡ kBT /(mηα)....

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  • ...In this limit, one can obtain simple reductions of the H-function solution, in the following three cases (Metzler and Nonnenmacher 2002): (i) Cauchy propagator α = µ = 1, P(x, t) = 1 2πK11 t 1 1 + x2 /( K11 t ) (52) with the long-tailed asymptotics P(x, t) ∼ (2π)−1x−2....

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  • ...…the LF in the presence of the drift V defined by the equation ∂ ∂t P (x, t) = ( ∂ ∂x V + Kµ ∂µ ∂|x|µ ) P(x, t) (35) is the solution PV =0(x, t) of equation (34) taken at position x − V t , i.e., PV (x, t) = PV =0(x−V t, t) (Jespersen et al 1999, Metzler and Compte 2000, Metzler and Klafter 2000a)....

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  • ...We note that the solution of certain classes of fractional equations is intimately related to the Fox H-function and related special functions (Mathai and Saxena 1978, Srivastava et al 1982, Saxena and Saigo 2001)....

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Journal ArticleDOI
TL;DR: The Hermiticity of the fractional Hamilton operator and the parity conservation law for fractional quantum mechanics are established and the energy spectra of a hydrogenlike atom and of a fractional oscillator in the semiclassical approximation are found.
Abstract: Some properties of the fractional Schrodinger equation are studied. We prove the Hermiticity of the fractional Hamilton operator and establish the parity conservation law for fractional quantum mechanics. As physical applications of the fractional Schrodinger equation we find the energy spectra of a hydrogenlike atom (fractional "Bohr atom") and of a fractional oscillator in the semiclassical approximation. An equation for the fractional probability current density is developed and discussed. We also discuss the relationships between the fractional and standard Schrodinger equations.

1,391 citations

References
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01 Jan 1982
TL;DR: This book is a blend of erudition, popularization, and exposition, and the illustrations include many superb examples of computer graphics that are works of art in their own right.
Abstract: "...a blend of erudition (fascinating and sometimes obscure historical minutiae abound), popularization (mathematical rigor is relegated to appendices) and exposition (the reader need have little knowledge of the fields involved) ...and the illustrations include many superb examples of computer graphics that are works of art in their own right." Nature

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"The random walk's guide to anomalou..." refers background in this paper

  • ...In fact, the LeH vy #ight trajectory can be assigned a fractal dimension d f "k [38,42,89] and is commonly supposed to be an e$cient search strategy of living organisms [141,142,146]....

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Book
01 Jan 1962

24,003 citations

Book
31 Dec 1959
TL;DR: In this paper, a classic account describes the known exact solutions of problems of heat flow, with detailed discussion of all the most important boundary value problems, including boundary value maximization.
Abstract: This classic account describes the known exact solutions of problems of heat flow, with detailed discussion of all the most important boundary value problems.

21,807 citations

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01 Jan 1956
TL;DR: Though it incorporates much new material, this new edition preserves the general character of the book in providing a collection of solutions of the equations of diffusion and describing how these solutions may be obtained.
Abstract: Though it incorporates much new material, this new edition preserves the general character of the book in providing a collection of solutions of the equations of diffusion and describing how these solutions may be obtained

20,495 citations