George M. Zaslavsky

Chaos, fractional kinetics, and anomalous transport

Abstract Chaotic dynamics can be considered as a physical phenomenon that bridges the regular evolution of systems with the random one. These two alternative states of physical processes are, typically, described by the corresponding alternative methods: quasiperiodic or other regular functions in the first case, and kinetic or other probabilistic equations in the second case. What kind of kinetics should be for chaotic dynamics that is intermediate between completely regular (integrable) and completely random (noisy) cases? What features of the dynamics and in what way should they be represented in the kinetics of chaos? These are the subjects of this paper, where the new concept of fractional kinetics is reviewed for systems with Hamiltonian chaos. Particularly, we show how the notions of dynamical quasi-traps, Poincare recurrences, Levy flights, exit time distributions, phase space topology prove to be important in the construction of kinetics. The concept of fractional kinetics enters a different area of applications, such as particle dynamics in different potentials, particle advection in fluids, plasma physics and fusion devices, quantum optics, and many others. New characteristics of the kinetics are involved to fractional kinetics and the most important are anomalous transport, superdiffusion, weak mixing, and others. The fractional kinetics does not look as the usual one since some moments of the distribution function are infinite and fluctuations from the equilibrium state do not have any finite time of relaxation. Different important physical phenomena: cooling of particles and signals, particle and wave traps, Maxwells Demon, etc. represent some domains where fractional kinetics proves to be valuable.

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 kinetic equation for Hamiltonian chaos

Abstract 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.

Anomalous diffusion and exit time distribution of particle tracers in plasma turbulence model

To explore the character of transport in a plasma turbulence model with avalanche transport, the motion of tracer particles has been followed. Both the time evolution of the moments of the distribution function of the tracer particle radial positions, 〈|r(t)−r(0)|n〉, and their finite scale Lyapunov number are used to determine the anomalous diffusion exponent, ν. The numerical results show that the transport mechanism is superdiffusive with an exponent ν close to 0.88±0.07. The distribution of the exit times of particles trapped into stochastic jets is also determined. These particles have the lowest separation rate at the low resonant surfaces.

Fractional Ginzburg-Landau equation for fractal media

We derive the fractional generalization of the Ginzburg–Landau equation from the variational Euler–Lagrange equation for fractal media. To describe fractal media we use the fractional integrals considered as approximations of integrals on fractals. Some simple solutions of the Ginzburg–Landau equation for fractal media are considered and different forms of the fractional Ginzburg–Landau equation or nonlinear Schrodinger equation with fractional derivatives are presented. The Agrawal variational principle and its generalization have been applied.

Fractional dynamics of coupled oscillators with long-range interaction

We consider a one-dimensional chain of coupled linear and nonlinear oscillators with long-range powerwise interaction. The corresponding term in dynamical equations is proportional to 1//n-m/alpha+1. It is shown that the equation of motion in the infrared limit can be transformed into the medium equation with the Riesz fractional derivative of order alpha, when 0<alpha<2. We consider a few models of coupled oscillators and show how their synchronization can appear as a result of bifurcation, and how the corresponding solutions depend on alpha. The presence of a fractional derivative also leads to the occurrence of localized structures. Particular solutions for fractional time-dependent complex Ginzburg-Landau (or nonlinear Schrodinger) equation are derived. These solutions are interpreted as synchronized states and localized structures of the oscillatory medium.

Some applications of fractional equations

Abstract We present two observations related to the application of linear (LFE) and nonlinear fractional equations (NFE). First, we give the comparison and estimates of the role of the fractional derivative term to the normal diffusion term in a LFE. The transition of the solution from normal to anomalous transport is demonstrated and the dominant role of the power tails in the long time asymptotics is shown. Second, wave propagation or kinetics in a nonlinear media with fractal properties is considered. A corresponding fractional generalization of the Ginzburg–Landau and nonlinear Schrodinger equations is proposed.

Ray dynamics in a long-range acoustic propagation experiment.

A ray-based wave-field description is employed in the interpretation of broadband basin-scale acoustic propagation measurements obtained during the Acoustic Thermometry of Ocean Climate programs 1994 Acoustic Engineering Test. Acoustic observables of interest are wavefront time spread, probability density function (PDF) of intensity, vertical extension of acoustic energy in the reception finale, and the transition region between temporally resolved and unresolved wavefronts. Ray-based numerical simulation results that include both mesoscale and internal-wave-induced sound-speed perturbations are shown to be consistent with measurements of all the aforementioned observables, even though the underlying ray trajectories are predominantly chaotic, that is, exponentially sensitive to initial and environmental conditions. Much of the analysis exploits results that relate to the subject of ray chaos; these results follow from the Hamiltonian structure of the ray equations. Further, it is shown that the collection of the many eigenrays that form one of the resolved arrivals is nonlocal, both spatially and as a function of launch angle, which places severe restrictions on theories that are based on a perturbation expansion about a background ray.

Dynamics with low-level fractionality

The notion of fractional dynamics is related to equations of motion with one or a few terms with derivatives of a fractional order. This type of equation appears in the description of chaotic dynamics, wave propagation in fractal media, and field theory. For the fractional linear oscillator the physical meaning of the derivative of order α<2 is dissipation. In systems with many spacially coupled elements (oscillators) the fractional derivative, along the space coordinate, corresponds to a long range interaction. We discuss a method of constructing a solution using an expansion in ɛ=n-α with small ɛ and positive integer n. The method is applied to the fractional linear and nonlinear oscillators and to fractional Ginzburg–Landau or parabolic equations.

Nonholonomic constraints with fractional derivatives

We consider the fractional generalization of nonholonomic constraints defined by equations with fractional derivatives and provide some examples. The corresponding equations of motion are derived using variational principle. We prove that fractional constraints can be used to describe the evolution of dynamical systems in which some coordinates and velocities are related to velocities through a power-law memory function.