Cosmic rays propagation along solar magnetic field lines: a fractional approach

Учайкин, В. В.; Sibatov, Renat T.; Byzykchi, Alexander Nikolaevich

doi:10.1685/journal.caim.480

Cited by 7 publications

(9 citation statements)

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“…In one-dimensional case the asymptotic distributions are of U-and W-shapes. A similar result was obtained in [7,61,62,64,67,68,69,70,50]. In addition, the fractional Laplacian in the fractional diffusion equation for finite velocity should be replaced by the material derivative of the fractional order [9,65,68,69].…”

Section: Resultssupporting

confidence: 70%

“…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]. However, all the reported results are for one-dimensional case and the reasons responsible for formation of such kinds of distributions are not discussed.…”

Section: Discussionmentioning

confidence: 85%

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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: Resultssupporting

confidence: 70%

Section: Discussionmentioning

confidence: 85%

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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“…In [38], we investigated asymmetrical one-dimensional random walks with finite constant speed and the asymptotically power-law distribution of path length ( ) for scattering to the right and to the left respectively. In case 0 1 α < < its asymptotics ( ) In case 1 2 α < < first moment exists, and asymptotic expression for the propagator is represented by means of one-dimentional α -stable density [39].…”

Section: Longitudinal Propagator With Ballistic Restrictionmentioning

confidence: 99%

Nonlocal Models of Cosmic Ray Transport in the Galaxy

Учайкин¹

2015

JAMP

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Studying the cosmic ray transport in the Galaxy, we deal with two interacting substances: charged particles and interstellar magnetic field. Two coupled local equations describe this complicated system, but elimination of one of them (say, the magnetic field equation) transforms remaining one (the cosmic rays equation) into the nonlocal form. The most popular nonlocal operators in the cosmic ray physics are integro-differential operators of fractional order. This report contains review of recent works in this direction, including original results of the author. In the last section, some specific problems are discussed: fractional operators with soft truncation of their kernels, nonlocal properties of fractional Laplacian, and a true form of the fractional material derivative.

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“…In this work, we present results of calculations of these processes performed with our model proposed in [12], which correctly describes both the tails and the preasymptotic part of the time profiles. Let us recall the main ideas of this model that make it different from the standard fractional-differential model.…”

Section: Interpreting Data On Solar Cosmic Ray Fluxes Via the Fractiomentioning

confidence: 99%

“…In [11][12][13][14], 1D anomalous diffu sion was examined as the process of the particles' random walk with asymptotic free path distribution (fractal walk).…”

mentioning

confidence: 99%

Interpreting data on solar cosmic ray fluxes via the fractional derivative method

Учайкин

Sibatov

Byzykchi

2015

Bull. Russ. Acad. Sci. Phys.

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Solar cosmic ray propagation through the interplanetary magnetic field is considered as a random process of particles traveling along magnetic lines at a finite velocity of free motion and with a free path dis tributed according to an inverse power law. The propagator is presented as a sum of direct (nonscattered) flux (singular part of solution) and multiple scattered flux (regular part). In the long time asymptotic, the regular part is described by an equation with a fractional order derivative. Using analytical expressions for the prop agator, we numerically calculate fluxes of energetic particles accelerated by shock waves generated by solar flares. The presented model is in better agreement with Ulysses and Voyager 2 data than the Perri-Zimbardo model and may therefore be recommended for use in interpreting the results of further experiments.

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Cosmic rays propagation along solar magnetic field lines: a fractional approach

Cited by 7 publications

References 0 publications

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

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

Nonlocal Models of Cosmic Ray Transport in the Galaxy

Interpreting data on solar cosmic ray fluxes via the fractional derivative method

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