The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics

Metzler, Ralf; Klafter, J.

doi:10.1088/0305-4470/37/31/r01

Cited by 2,102 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%

“…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%

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Anomalous Fluctuation Relations

Klages

Chechkin

Dieterich

2013

Nonequilibrium Statistical Physics of Small Systems

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Abstract. We study Fluctuation Relations (FRs) for dynamics that are anomalous, in the sense that the diffusive properties strongly deviate from the ones of standard Brownian motion. We first briefly review the concept of transient work FRs for stochastic dynamics modeled by the ordinary Langevin equation. We then introduce three generic types of dynamics generating anomalous diffusion: Lévy flights, long-time correlated Gaussian stochastic processes and time-fractional kinetics. By combining Langevin and kinetic approaches we calculate the work probability distributions in the simple nonequilibrium situation of a particle subject to a constant force. This allows us to check the transient FR for anomalous dynamics. We find a new form of FRs, which is intimately related to the validity of fluctuation-dissipation relations. Analogous results are obtained for a particle in a harmonic potential dragged by a constant force. We argue that these findings are important for understanding fluctuations in experimentally accessible systems. As an example, we discuss the anomalous dynamics of biological cell migration both in equilibrium and in nonequilibrium under chemical gradients.

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Section: Time-fractional Kineticsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 95%

Anomalous Fluctuation Relations

Klages

Chechkin

Dieterich

2013

Nonequilibrium Statistical Physics of Small Systems

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“…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%

Lévy flights in confining environments: Random paths and their statistics

Żaba

Garbaczewski

Stephanovich

2013

Physica A: Statistical Mechanics and its Applications

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We analyze a specific class of random systems that are driven by a symmetric Lévy stable noise. In view of the Lévy noise sensitivity to the confining "potential landscape" where jumps take place (in other words, to environmental inhomogeneities), the pertinent random motion asymptotically sets down at the Boltzmann-type equilibrium, represented by a probability density function (pdf) ρ * (x) ∼ exp[−Φ(x)]. Since there is no Langevin representation of the dynamics in question, our main goal here is to establish the appropriate path-wise description of the underlying jump-type process and next infer the ρ(x, t) dynamics directly from the random paths statistics. A priori given data are jump transition rates entering the master equation for ρ(x, t) and its target pdf ρ * (x). We use numerical methods and construct a suitable modification of the Gillespie algorithm, originally invented in the chemical kinetics context. The generated sample trajectories show up a qualitative typicality, e.g. they display structural features of jumping paths (predominance of small vs large jumps) specific to particular stability indices µ ∈ (0, 2).

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“…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%

On the Controllability of Anomalous Diffusions Generated by the Fractional Laplacian

Miller

2006

Math. Control Signals Syst.

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Abstract. This paper introduces a "spectral observability condition" for a negative self-adjoint operator which is the key to proving the null-controllability of the semigroup that it generates and to estimating the controllability cost over short times. It applies to the interior controllability of diffusions generated by powers greater than 1/2 of the Dirichlet Laplacian on manifolds, generalizing the heat flow. The critical fractional order 1/2 is optimal for a similar boundary controllability problem in dimension one. This is deduced from a subsidiary result of this paper, which draws consequences on the lack of controllability of some one dimensional output systems from Müntz-Szász theorem on the closed span of sets of power functions.In this paper, an observability condition on the spectral subspaces of a negative self-adjoint operator is introduced which ensures fast controllability, i.e. the semigroup generated by this operator is null-controllable in arbitrarily small time. In this asymptotic, it also ensures an upper bound for the controllability cost, i.e. the supremum, over every initial state with norm one, of the norm of the optimal input function which steers it to zero (cf. section 1). This spectral observability condition is the abstract version of a property proved for the Dirichlet Laplacian ∆ on a compact manifold observed on any non empty region in [LZ98, JL99] (cf.[Mil05] for non-compact manifolds).It applies to the semigroup generated by the fractional Laplacian on manifolds −(−∆) α as long as α > 1/2. This semigroup is widely used to describe physical systems exhibiting anomalous diffusions (cf. references in sect.2.1). Thus new interior null-controllability results for such fractional diffusions with non-constant coefficients in any dimension are deduced (the one dimension problem with constant coefficients and one dimensional input was recently considered in [MZ04]). In particular, as the control time T tends to 0, the controllability cost grows at most like C β exp(c β /T β ) where C β and c β are positive constants and β > 1/(2α − 1) (n.b. a lower bound of the same form with equality β = 1/(2α − 1) holds in the case α = 1 which corresponds to the heat equation). It is proved in the appendix that a similar one dimensional problem is not controllable from the boundary forThis last result is deduced from a more general remark of independent interest on the lack of controllability of any finite linear combination of eigenfunctions of systems with one dimensional input, based on the generalized Müntz theorem on the completeness of sets of exponentials.1. The main result in the abstract setting.After recalling the duality between controllability and observability for parabolic semigroups, this section states the main definition and theorem.1.1. The abstract setting. Let the generator A be a positive self-adjoint operator with domain D(A) on the Hilbert space H of states which we identify with its dual.2000 Mathematics Subject Classification. 93B05, 34K35, 58J35.

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The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics

Cited by 2,102 publications

References 352 publications

Anomalous Fluctuation Relations

Anomalous Fluctuation Relations

Lévy flights in confining environments: Random paths and their statistics

On the Controllability of Anomalous Diffusions Generated by the Fractional Laplacian

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