arXiv subjects

A. V. Chechkin

Publications and source records attributed to A. V. Chechkin.

43 records · Page 3Linked to original sources

arXiv2000

Levy Anomalous Diffusion and Fractional Fokker--Planck Equation

We demonstrate that the Fokker-Planck equation can be generalized into a 'Fractional Fokker-Planck' equation, i.e. an equation which includes fractional space differentiations, in order to encompass the wide class of anomalous diffusions due to a Levy stable stochastic forcing. A precise determination of this equation is obtained by substituting a Levy stable source to the classical gaussian one in the Langevin equation. This yields not only the anomalous diffusion coefficient, but a non trivial fractional operator which corresponds to the possible asymmetry of the Levy stable source. Both of them cannot be obtained by scaling arguments. The (mono-) scaling behaviors of the Fractional Fokker-Planck equation and of its solutions are analysed and a generalization of the Einstein relation for the anomalous diffusion coefficient is obtained. This generalization yields a straightforward physical interpretation of the parameters of Levy stable distributions. Furthermore, with the help of important examples, we show the applicability of the Fractional Fokker-Planck equation in physics.

nlin.CD↗

arXiv1999

Linear Relaxation Processes Governed by Fractional Symmetric Kinetic Equations

We get fractional symmetric Fokker - Planck and Einstein - Smoluchowski kinetic equations, which describe evolution of the systems influenced by stochastic forces distributed with stable probability laws. These equations generalize known kinetic equations of the Brownian motion theory and contain symmetric fractional derivatives over velocity and space, respectively. With the help of these equations we study analytically the processes of linear relaxation in a force - free case and for linear oscillator. For a weakly damped oscillator we also get kinetic equation for the distribution in slow variables. Linear relaxation processes are also studied numerically by solving corresponding Langevin equations with the source which is a discrete - time approximation to a white Levy noise. Numerical and analytical results agree quantitatively.

cond-mat.stat-mech↗

arXiv1999

Self - affinity of ordinary Levy motion, spurious multi - affinity and pseudo - Gaussian relations

The ordinary Levy motion is a random process whose stationary independent increments are statistically self-affine and distributed with a stable probability law characterized by the Levy index alpha, 0 < alpha < 2. The divergence of statistical moments of the order q > alpha leads to an important role of the finite sample effects. The objective of this paper is to study the influence of these effects on the self-affine properties of the ordinary Levy motion, namely, on the '1/alpha laws', that is, time dependence of the q-th order structure function and of the range. Analytical estimates and simulations of the finite sample effects clearly demonstrates three phenomena: spurious multi-affinity of the Levy motion, strong dependence of the structure function on the sample size at q > alpha, and pseudo-Gaussian behavior of the second-order structure function and of the normalized range. We discuss these phenomena in detail and propose the modified Hurst method for empirical rescaled range analysis.

cond-mat.stat-mech↗

arXiv1999

A Model for Persistent Levy Motion

We propose the model, which allows us to approximate fractional Levy noise and fractional Levy motion. Our model is based (i) on the Gnedenko limit theorem for an attraction basin of stable probability law, and (ii) on regarding fractional noise as the result of fractional integration/differentiation of a white Levy noise. We investigate self - affine properties of the approximation and conclude that it is suitable for modeling persistent Levy motion with the Levy index between 1 and 2.

cond-mat.stat-mech↗

arXiv1999

Fractional Brownian Motion Approximation Based on Fractional Integration of a White Noise

We study simple approximations to fractional Gaussian noise and fractional Brownian motion. The approximations are based on spectral properties of the noise. They allow one to consider the noise as the result of fractional integration/differentiation of a white Gaussian noise. We study correlation properties of the approximation to fractional Gaussian noise and point to the peculiarities of persistent and anti-persistent behaviors. We also investigate self-similar properties of the approximation to fractional Brownian motion, namely, ``tH laws`` for the structure function and the range. We conclude that the models proposed serve as a convenient tool for the natural processes modelling and testing and improvement of the methods aimed at analysis and interpretation of experimental data.

cond-mat.stat-mech↗

arXiv1999

A Model for Ordinary Levy Motion

We propose a simple model based on the Gnedenko limit theorem for simulation and studies of the ordinary Levy motion, that is, a random process, whose increments are independent and distributed with a stable probability law. We use the generalized structure function for characterizing anomalous diffusion rate and propose to explore the modified Hurst method for empirical rescaled range analysis. We also find that the structure function being estimated from the ordinary Levy motion sample paths as well as the (ordinary) Hurst method lead to spurious ''pseudo-Gaussian'' relations.

cond-mat.stat-mech↗

arXiv1997

Anomalies of Transport in Reflectionally Noninvariant Turbulence

We consider the transport of passive admixture in locally homogeneous isotropic reflectionally noninvariant turbulence of incompressible fluid. It is shown that anomalous convective flow appears which direction does not coincide with that of a mean flow.

chao-dyn↗