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

Abstract

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...

Fractional Parker equation for the transport of cosmic rays: steady-state solutions

“…Thus, the transport properties strongly depend on the turbulence features and the adopted numerical representation. The implications of superdiffusion for particle acceleration and transport at shocks have recently been considered by Perri & Zimbardo (2012b); Zimbardo & Perri (2013); Lazarian & Yan (2014), and these works have attracted considerable theoretical interest (Uchaikin et al 2015;Rocca et al 2015Rocca et al , 2016Saenko 2016). Nondiffusive processes can be described by several tools, including Lévy walks, Lévy flights, and fractional derivatives (e.g., Perrone et al 2013).…”

Numerical Solution to Anomalous Diffusion Equations for Levy Walks

“…Let us consider the case 0 < α < 1. In this case, the Laplace transform of the density (23) has the form (see [33]):…”

“…In the case 1 < α < 2, the distribution (23) has a mathematical expectation. In view of the fact that the Laplace transform of density (23) takes the form (see [33]):…”

“…To estimate the density at the points x = ±v 0 T * , the function ψ(x * , x i , T * , t i ) has the form (33), to estimate the density at the points x = ±v 0 T * the function ψ(x * , x i , T * , t i ) takes the form (34). We finally obtain that the density at the point x * at the moment of time T * , is estimated by the formula:…”

“…In the work [28], kinetic equations of anomalous diffusion with a finite velocity are obtained, the root-mean-square deviation is investigated, and an exact solution to the kinetic equation in the one-dimensional case is obtained. The work [29] is devoted to the study of statistical moments for the case of Levy random walks with a random finite velocity without traps, and the case of multidimensional walks with traps of arbitrary type with constant velocity is considered in the works and the case of multidimensional walks with traps of arbitrary type with constant velocity is considered in the works [30][31][32][33]. The work [34] examines the influence of the final velocity on the spatial distribution of particles in Levy walks with exponential traps.…”