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 $t\to\infty$) have been obtained for $N$ dimensional case where the free path of particles demonstrates the power-law distribution $p_{\xi}(x)=\alpha x_0^\alpha x^{-\alpha-1}$, $x\to\infty$, $0<\alpha<2$. 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 ($1<\alpha<2$) and propagation with distribution at the infinite mathematical expectation of the free path of the particle ($0<\alpha<1$). In the case $1<\alpha<2$, 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 $0<\alpha<1$, 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.