Fractional Kinetic Equations Research Articles

Stable equilibrium based on Lévy statistics: Stochastic collision models approach.

We investigate equilibrium properties of two very different stochastic collision models: (i) the Rayleigh particle and (ii) the driven Maxwell gas. For both models the equilibrium velocity distribution is a Lévy distribution, the Maxwell distribution being a special case. We show how these models are related to fractional kinetic equations. Our work demonstrates that a stable power-law equilibrium, which is independent of details of the underlying models, is a natural generalization of Maxwell's velocity distribution.

Fractional dynamics and nonlinear harmonic responses in dielectric relaxation of disordered liquids.

The problem of the nonlinear dielectric response due to the application of a strong electric field is reconsidered in the context of fractional kinetic equations. To accomplish that, we start from a fractional noninertial Fokker-Planck equation and restrict ourselves to the case of anomalous subdiffusive processes characterized by the critical exponent alpha ranging from 0 to 1, the limit of normal diffusion. In particular, we evaluate the first- and third-order nonlinear harmonic components of the electric polarization in the case of either a pure ac field or a strong dc bias field superimposed on a weak ac field. The stationary regime is therefore calculated from an infinite set of differential recurrence relations by using a perturbation method. The results so obtained are illustrated by three-dimensional dispersion and absorption plots in order to show the influence of alpha. Cole-Cole diagrams are also presented, allowing one to see that the arcs become more and more flattened as alpha-->0, and corresponding to a broadening of the absorption peaks as effectively observed in complex liquids. The theoretical model is supported by comparison with experimental data of the third-order nonlinear dielectric permittivity of a ferroelectric liquid crystal.

Nonequilibrium stationary states in the earth's magnetotail: Stochastic acceleration processes and nonthermal distribution functions

A self-consistent kinetic description of turbulent plasma and fields at the nonequilibrium stationary states (NESS's) of the Earth's magnetotail is proposed. We argue that the inherent dynamics of the NESS's is manifested in the low-frequency power law fluctuation spectrum ∼ ƒ −1 (i.e., the flicker noise). At the higher frequebcies, the fluctuation spectrum has a steeper shape ∼ ƒ −7 3 and is related to the structural characteristics of the magnetotail turbulence. The basic dynamical processes operating at the NESS are described by a nonlinear fractional kinetic equation which includes the following principal effects: (1) stochastic particle acceleration in the turbulent magnetic field varying with time, and (2) self-interaction of the medium associated with the reproduction of the turbulent field by the energetic particles accelerated therein. We find that the particle energy distribution in presence of the self-interaction events reveals a high-energy nonthermal tail ƒ(ε) ∝ ε −η . The slope of the distribution, η, obeys the condition 6 ≤ η < 7; this reflects the fundamental dynamical properties underlying magnetic field and plasma coupling in self-organized electromagnetic systems. The results obtained are in close agreement with spacecraft observational data on the Earth's magnetotail.

Space- and time-fractional diffusion and wave equations, fractional Fokker–Planck equations, and physical motivation

Abstract We investigate the physical background and implications of a space- and time-fractional diffusion equation which corresponds to a random walker which combines competing long waiting times and Levy flight properties. Explicit solutions are examined, and the corresponding fractional Fokker–Planck–Smoluchowski equation is presented. The framework of fractional kinetic equations which control the systems relaxation to either Boltzmann–Gibbs equilibrium, or a far from equilibrium Levy form is explored, putting the fractional approach in some perspective from the standard non-equilibrium dynamics point of view, and its generalisation.

Fractional kinetics for relaxation and superdiffusion in a magnetic field

Fractional Fokker–Planck equation is proposed for the kinetic description of relaxation and superdiffusion processes in constant magnetic and random electric fields. It is assumed that the random electric field acting on a test charged particle is isotropic and possesses non-Gaussian Levy stable statistics. These assumptions provide one with a straightforward possibility to consider formation of anomalous stationary states and superdiffusion processes, both properties are inherent to strongly nonequilibrium plasmas of solar systems and thermonuclear devices. The fractional kinetic equation is solved, the properties of the solution are studied, and analytical results are compared with those of numerical simulation based on the solution of the Langevin equations with a noise source having Levy stable probability density. It is found, in particular, that the stationary states are essentially non-Maxwellian ones and, at the diffusion stage of relaxation, the characteristic displacement of a particle grows superdiffusively with time and is inversely proportional to the magnetic field.

Stationary solutions of the fractional kinetic equation with a symmetric power-law potential

The properties of stationary solutions of the one-dimensional fractional Einstein--Smoluchowski equation with a potential of the form x2m+2, m = 1, 2,..., and of the Riesz spatial fractional derivative of order α, 1 ≤ α ≤ 2, are studied analytically and numerically. We show that for 1 ≤ α < 2, the stationary distribution functions have power-law asymptotic approximations decreasing as x−(α+2m+1) for large values of the argument. We also show that these distributions are bimodal.

"Strange" Fermi processes and power-law nonthermal tails from a self-consistent fractional kinetic equation.

This study advocates the application of fractional dynamics to the description of anomalous acceleration processes in self-organized turbulent systems. Such processes (termed "strange" accelerations) involve both the non-Markovian fractal time acceleration events associated with a generalized stochastic Fermi mechanism, and the velocity-space Levy flights identified with nonlocal violent accelerations in turbulent media far from the (quasi)equilibrium. The "strange" acceleration processes are quantified by a fractional extension of the velocity-space transport equation with fractional time and phase space derivatives. A self-consistent nonlinear fractional kinetic equation is proposed for the stochastic fractal time accelerations near the turbulent nonequilibrium saturation state. The ensuing self-consistent energy distribution reveals a power-law superthermal tail psi(epsilon) proportional to epsilon(-eta) with slope 6 or =eta< or =7 depending on the type of acceleration process (persistent or antipersistent). The results obtained are in close agreement with observational data on the Earth's magnetotail.

Spectral Analysis of Fractional Kinetic Equations with Random Data

We present a spectral representation of the mean-square solution of the fractional kinetic equation (also known as fractional diffusion equation) with random initial condition. Gaussian and non-Gaussian limiting distributions of the renormalized solution of the fractional-in-time and in-space kinetic equation are described in terms of multiple stochastic integral representations.

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.

Large-scale behavior of the tokamak density fluctuations

An analysis of tokamak density fluctuations data permits the determination of two characteristic exponents. The exponents correspond to the powers of a power-law dependence of the distributions of the long-lasting monotonic change (“flight”) of the density and the time length of these changes. Speculation based on these results leads to construction of the fractional kinetic equation for the distribution function of the flights. The asymptotic transport properties of the particle density distribution function are directly connected with the exponents obtained from the density fluctuations data.

Generalized Diffusion−Advection Schemes and Dispersive Sedimentation: A Fractional Approach

We consider advection processes following anomalous statistics on an effective transport level, within the framework of fractional kinetic equations. We discuss different realistic situations such as Galilei variant and invariant advection, as well as dispersive sedimentation processes. Exact analytical solutions for the plume and the moments characterizing these transport processes are calculated. We introduce an estimation for the breakthrough of the tracer in the dispersive processes.

THE FRACTIONAL KINETIC EQUATION AND THERMONUCLEAR FUNCTIONS

The paper discusses the solution of a simple kinetic equation of thetype used for the computation of the change of the chemical compositionin stars like the Sun. Starting from the standard form of the kineticequation it is generalized to a fractional kinetic equation and itssolutions in terms of H-functions are obtained. The role of thermonuclearfunctions, which are also represented in terms of G- and H-functions,in such a fractional kinetic equation is emphasized. Results containedin this paper are related to recent investigations of possibleastrophysical solutions of the solar neutrino problem.

Two-Dimensional Force-Free Relaxation and Superdiffusion Governed by Fractional Kinetic Equations

Starting from the integral Chapman-Kolmogorov equation and the Langevin equations with a multivariate Levy noise, we consider possible generalization of a one-dimensional fractional kinetic equations. It is assumed that the Levy noise has independent components. As the result, we get fractional Fokker-Planck equation for the probability density function in a phase space and fractional Einstein-Smoluchowski equation for the probability density function in a real space. We consider force-free superdiffusion and relaxation described by fractional equations and carry out numerical simulation of these processes by solving the corresponding Langevin equations. Analytical and numerical results agree quantitatively.

From continuous time random walks to the fractional fokker-planck equation

We generalize the continuous time random walk (CTRW) to include the effect of space dependent jump probabilities. When the mean waiting time diverges we derive a fractional Fokker-Planck equation (FFPE). This equation describes anomalous diffusion in an external force field and close to thermal equilibrium. We discuss the domain of validity of the fractional kinetic equation. For the force free case we compare between the CTRW solution and that of the FFPE.

Quantum Lévy Processes and Fractional Kinetics

Exotic stochastic processes are shown to emerge in the quantum evolution of complex systems. Using influence function techniques, we consider the dynamics of a system coupled to a chaotic subsystem described through random matrix theory. We find that the reduced density matrix can display dynamics given by L\'evy stable laws. The classical limit of these dynamics can be related to fractional kinetic equations. In particular, we derive a fractional extension of Kramers equation.

Fractional kinetic equations: solutions and applications.

Fractional generalization of the diffusion equation includes fractional derivatives with respect to time and coordinate. It had been introduced to describe anomalous kinetics of simple dynamical systems with chaotic motion. We consider a symmetrized fractional diffusion equation with a source and find different asymptotic solutions applying a method which is similar to the method of separation of variables. The method has a clear physical interpretation presenting the solution in a form of decomposition of the process of fractal Brownian motion and Levy-type process. Fractional generalization of the Kolmogorov-Feller equation is introduced and its solutions are analyzed. (c) 1997 American Institute of Physics.

Fractional kinetics and accelerator modes

Kinetic properties of chaotic dynamics are studied for the web-map. It is shown that depending on the values of the perturbation parameter K, there are alternative possibilities: kinetics, of the quasilinear (normal, i.e. Gaussian) type or kinetics of the anomalous (Lévyan) type. If K is close to the values K = ℓ π when the accelerator modes occur, then there is superdiffusion with anomalous values of the transport exponent. We consider self-similar properties of trajectories near the islands in the phase space. Fractional kinetic equation is proposed and fractal exponents are obtained. Local properties of trajectories (exit time) and nonlocal ones (Poincaré cycles) are studied and compared for the normal and anomalous kinetics. It is shown that the power-like law of the trapping distribution function imposes the same kind of asymptotics for the Poincaré cycles distribution.

Self-similarity, renormalization, and phase space nonuniformity of Hamiltonian chaotic dynamics.

A detailed description of fractional kinetics is given in connection to islands' topology in the phase space of a system. The method of renormalization group is applied to the fractional kinetic equation in order to obtain characteristic exponents of the fractional space and time derivatives, and an analytic expression for the transport exponents. Numerous simulations for the web-map and standard map demonstrate different results of the theory. Special attention is applied to study the singular zone, a domain near the island boundary with a self-similar hierarchy of subislands. The birth and collapse of islands of different types are considered. (c) 1997 American Institute of Physics.

Fractional kinetic equation for Hamiltonian chaos

Hamiltonian chaotic dynamics of particles (or passive particles in fluids) can be described by a fractional generalization of the Fokker-Planck-Kolmogorov equation (FFPK) which is defined by two fractional critical exponents (α, β) responsible for the space and time derivatives of the distribution function correspondingly. A renormalization method has been proposed to determine (α, β) from the first principles (ie. from the Hamiltonian). The anomalous transport exponent μ is derived as μ = β/α or μ = β/2α for the first order mean displacement in self-similar transport.