Asymptotic densities of ballistic Lévy walks

Froemberg, D.; Schmiedeberg, Michael; Barkai, Eli; Zaburdaev, Vasily

doi:10.1103/physreve.91.022131

Cited by 53 publications

(87 citation statements)

References 42 publications

(81 reference statements)

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“…In the strong ballistic case (0 < μ < 1), the integral term is in the balance with classical wave equation, while for the subballistic superdiffusive case (1 < μ < 2), the memory term is in the balance with the Cattaneo (telegraph) equation. One can perform various asymptotic analysis of (30) and (33), obtain the pseudodifferential equations for the walker's PDF position [30][31][32][33] and determine the shape of PDF profiles [34].…”

Section: Gamma Pdf G(τ2λ)mentioning

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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Section: Gamma Pdf G(τ2λ)mentioning

confidence: 99%

Single integrodifferential wave equation for a Lévy walk

Fedotov

2016

Phys. Rev. E

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“…The densities of 1-dimensional ballistic Lévy walks have been found by Froemberg et al in [25], see also [26] for other approach to this problem. Recently, in [27] the PDFs of 2 and 3-dimensional ballistic Lévy walks were derived.…”

Section: Introductionmentioning

confidence: 99%

“…This technique works only for one dimensional ballistic processes. It was introduced by Godrche and Luck in [30] for inverting double Laplace transform and then generalized by Froemberg et al in [25] for the Fourier-Laplace transform. After its application we obtain (see Appendix A)…”

Section: Odd Number Of Dimensions -D = 2n +mentioning

confidence: 99%

“…In this appendix we present the method of inverting the Fourier-Laplace transform for 1D ballistic Lévy walks from [25] and [30]. The Fourier-Laplace transform of H 1 (x, t) is given by …”

Section: Undershooting -Even Number Of Dimensions D = 2n +mentioning

confidence: 99%

“…Now we are in position to apply the method of inverting the FourierLaplace transform used in [25,30], which will allow us to get an explicit formula for Φ 1 . This technique is based on a special representation of the function g 1 (ξ) and the Sokhotsky-Weierstrass theorem.…”

Section: Odd Number Of Dimensions -D = 2n +mentioning

confidence: 99%

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Method of calculating densities for isotropic ballistic Lévy walks

Magdziarz

Zorawik

2017

Communications in Nonlinear Science and Numerical Simulation

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We provide explicit formulas for asymptotic densities of d-dimensional isotropic Lévy walks, when d > 1. The densities of multidimensional undershooting and overshooting Lévy walks are presented as well. Interestingly, when the number of dimensions is odd the densities of all these Lévy walks are given by elementary functions. When d is even, we can express the densities as fractional derivatives of hypergeometric functions, which makes an efficient numerical evaluation possible.

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The influence of the finite velocity on spatial distribution of particles in the frame of Levy walk model

Saenko

2016

Physica A: Statistical Mechanics and its Applications

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Levy walk at the finite velocity is considered. To analyze the spatial and temporal characteristics of this process, the method of moments has been used. The asymptotic distributions of the moments (at t → ∞) have been obtained for N dimensional case where the free path of particles demonstrates the power-law distribution p ξ (x) = αx α 0 x −α−1 , x → ∞, 0 < α < 2. The three regimes of distribution have been distinguished: ballistic, diffusion and asymptotic. Introduction of the finite velocity requires considering of two problems: propagation with distribution at the finite mathematical expectation of the free path (1 < α < 2) and propagation with distribution at the infinite mathematical expectation of the free path of the particle (0 < α < 1). In the case 1 < α < 2, the asymptotic distribution is described by the Levy stable law and the effect of the finite velocity is reduced to a decrease of diffusivity. At 0 < α < 1, the situation is quite different. Here, the asymptotic distribution exhibits a U-or W-shape and is described as the ballistic regime of distribution. The obtained moments allow to reconstruct the distribution densities of particles in one-dimensional and three-dimensional cases.Here, we consider an effect of the finite velocity on spatial distribution of the particles at anomalous diffusion. It is well known that the anomalous diffusion is defined by the power-law dependence of the diffusion packet width on time ∆(t) ∝ D α t γ ,where D α is the diffusion coefficient [1,2,3,4,5]. Different regimes of this process are recognized depending on the exponent value γ: normal diffusion (γ = 1/2), subdiffusion (γ < 1/2), and superdiffusion (γ > 1/2). At γ = 1 and γ > 1, the quasi-ballistic and superballistic regimes, respectively, are established. For more details on each regimes see Refs. [6,7,8,9].Anomalous diffusion is essential for study due to a wide range of its application. Anomalous diffusion is known for relaxation processes in dielectrics [10], turbulent fluxes of the particles at the edge of the plasma cord [11,12,13,14,15,16], central region of plasma cord [17] in closed magnetic traps, study of mRNA molecule diffusion in cells [18] , geological and geophysical processes, and biological systems (see [19] and Refs. there). The anomalous diffusion models are employed to describe propagation of cosmic rays [20,21,22,23,24,25] and their acceleration [26,27,28,29], to study wandering of interstellar magnetic field lines [30,31,32,33], to describe heat transfer in the systems which do not obey the Fourier conductivity law [34], to develop dynamic models describing sequences of nucleotides in DNA molecule [35].The Continuous Time Random Walk (CTRW) model is used to describe anomalous diffusion [36,37,38,39,40,41,5]. In order to take into account the finite velocity, the variations of CTRW model have been introduced. Conventionally these modifications are classified into two groups: 1) coupled Levy walk and 2) velocity Levy walk. The first group models assume a walk to be sequence of instantaneous j...

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Asymptotic densities of ballistic Lévy walks

Cited by 53 publications

References 42 publications

Single integrodifferential wave equation for a Lévy walk

Single integrodifferential wave equation for a Lévy walk

Method of calculating densities for isotropic ballistic Lévy walks

The influence of the finite velocity on spatial distribution of particles in the frame of Levy walk model

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