Anomalous Diffusion of Inertial, Weakly Damped Particles

Friedrich, Rudolf; Jenko, F.; Baule, Adrian; Eule, Stephan

doi:10.1103/physrevlett.96.230601

Cited by 117 publications

(167 citation statements)

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“…To describe a Lévy walk spatiotemporal coupling, two dynamical equations have been suggested in [12]. Another approach to describe the superdiffusive behavior is based on the analysis of joint PDF p(x,v,t) of the particle's position x and velocity v. Various fractional generalizations of Kramers-Fokker-Planck equation for p(x,v,t) have been derived [13][14][15].…”

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confidence: 99%

Single integrodifferential wave equation for a Lévy walk

Fedotov

2016

Phys. Rev. E

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We derive the single integrodifferential wave equation for the probability density function of the position of a classical one-dimensional Lévy walk with continuous sample paths. This equation involves a classical wave operator together with memory integrals describing the spatiotemporal coupling of the Lévy walk. It is valid at all times, not only in the long time limit, and it does not involve any large-scale approximations. It generalizes the well-known telegraph or Cattaneo equation for the persistent random walk with the exponential switching time distribution. Several non-Markovian cases are considered when the particle's velocity alternates at the gamma and power-law distributed random times. In the strong anomalous case we obtain the asymptotic solution to the integrodifferential wave equation. We implement the nonlinear reaction term of Kolmogorov-PetrovskyPiskounov type into our equation and develop the theory of wave propagation in reaction-transport systems involving Lévy diffusion.

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confidence: 99%

Single integrodifferential wave equation for a Lévy walk

Fedotov

2016

Phys. Rev. E

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“…The properties of time-integrated observables of the ξ -driven processes, which are expressed as functionals of their fluctuating trajectories, are also an open problem. For functionals of CTRWs, closedform evolution equations can be derived that generalize the Feynman-Kac framework to anomalous processes [59,60]. A further generalization to anomalous processes with arbitrary waiting time distributions has recently been obtained [34], which highlights the connection between the waiting time distribution and the memory kernel appearing in the fractional evolution equations.…”

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confidence: 99%

Langevin formulation of a subdiffusive continuous-time random walk in physical time

Cairoli

Baule

2015

Phys. Rev. E

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Systems living in complex nonequilibrated environments often exhibit subdiffusion characterized by a sublinear power-law scaling of the mean square displacement. One of the most common models to describe such subdiffusive dynamics is the continuous-time random walk (CTRW). Stochastic trajectories of a CTRW can be described in terms of the subordination of a normal diffusive process by an inverse Lévy-stable process. Here, we propose an equivalent Langevin formulation of a force-free CTRW without subordination. By introducing a different type of non-Gaussian noise, we are able to express the CTRW dynamics in terms of a single Langevin equation in physical time with additive noise. We derive the full multipoint statistics of this noise and compare it with the scaled Brownian motion (SBM), an alternative stochastic model describing subdiffusive dynamics. Interestingly, these two noises are identical up to the second order correlation functions, but different in the higher order statistics. We extend our formalism to general waiting time distributions and force fields and compare our results with those of the SBM. In the presence of external forces, our proposed noise generates a different class of stochastic processes, resembling a CTRW but with forces acting at all times.

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“…Friedrich and his co-workers discuss the CTRW model with position-velocity coupling PDF [5]. Carmi and Barkai use the CTRW model with functional of path and position coupling PDF [1].…”

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“…Carmi and Barkai use the CTRW model with functional of path and position coupling PDF [1]. Based on the CTRW models with coupling PDFs, they all derive the deterministic equations; and mathematically an important operator, fractional substantial derivative, is introduced [1,2,5,13,14].…”

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Discretized fractional substantial calculus

Chen¹,

Deng²

2014

ESAIM: M2AN

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Abstract. This paper discusses the properties and the numerical discretizations of the fractional substantial integraland the fractional substantial derivativewhere Ds = ∂ ∂x + σ = D + σ, σ can be a constant or a function without related to x, say σ(y); and m is the smallest integer that exceeds µ. The Fourier transform method and fractional linear multistep method are used to analyze the properties or derive the discretized schemes. And the convergences of the presented discretized schemes with the global truncation error O(h p ) (p = 1, 2, 3, 4, 5) are theoretically proved and numerically verified.

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Anomalous Diffusion of Inertial, Weakly Damped Particles

Cited by 117 publications

References 17 publications

Single integrodifferential wave equation for a Lévy walk

Single integrodifferential wave equation for a Lévy walk

Langevin formulation of a subdiffusive continuous-time random walk in physical time

Discretized fractional substantial calculus

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