Enhanced superdiffusion and finite velocity of Levy flights

Zaburdaev, Vasily; Chukbar, K. V.

doi:10.1134/1.1458474

Cited by 47 publications

(39 citation statements)

References 6 publications

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“…As mentioned in the introduction accounting the finite speed with anomalous diffusion has been already studied. Summarizing the results, one can say that for the anomalous diffusion with an infinite expectation of the free-path accounting of the finite velocity leads to replacement of the fractional Laplacian in the fractional diffusion equation by the material derivative of fractional order [9,65,68,69]. The limit distributions in one-dimensional case are Ushaped and W-shaped [7,50,61,62,64,67,68,69,70] and described by the Lamperty distribution [68,69,71,72].…”

Section: Discussionmentioning

confidence: 84%

“…Further, Levy walks at the finite velocity have been studied in Ref. [9]. Authors show that at the infinite mathematical expectation of free paths taking into consideration of the finite velocity leads not only to decrease of the diffusion constant in fractional diffusion equation as reported in Ref.…”

mentioning

confidence: 85%

“…[64,50] study Levy walks at the finite velocity without traps and state that at the infinite mathematical expectation of free paths the asymptotic distributions of particles demonstrate U-shape and W-shape.Further, Levy walks at the finite velocity have been studied in Ref. [9]. Authors show that at the infinite mathematical expectation of free paths taking into consideration of the finite velocity leads not only to decrease of the diffusion constant in fractional diffusion equation as reported in Ref.[7] but also to replacement of the fractional Laplace operator with material derivative of fractional order in fractional diffusion equation.…”

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confidence: 91%

“…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.…”

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confidence: 99%

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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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Section: Discussionmentioning

confidence: 84%

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confidence: 85%

mentioning

confidence: 91%

mentioning

confidence: 99%

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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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“…The present progress in describing the radiative transfer for infinite velocity of carriers, Equation (1), may be extended to the case of a finite velocity [40,41] (which includes the case of the resonance radiative transfer in astrophysics), where the exact solution and the asymptotics were also obtained [42].…”

Section: Discussionmentioning

confidence: 99%