Fractional Generalization of Gradient Systems

We consider a fractional generalization of gradient systems. We use differential forms and exterior derivatives of fractional orders. Examples of fractional gradient systems are considered. We describe the stationary states of these systems.Comment: 11 pages, LaTe

Fractional Variations for Dynamical Systems: Hamilton and Lagrange Approaches

Fractional generalization of an exterior derivative for calculus of variations is defined. The Hamilton and Lagrange approaches are considered. Fractional Hamilton and Euler-Lagrange equations are derived. Fractional equations of motion are obtained by fractional variation of Lagrangian and Hamiltonian that have only integer derivatives.Comment: 21 pages, LaTe

Dynamics of Fractal Solids

We describe the fractal solid by a special continuous medium model. We propose to describe the fractal solid by a fractional continuous model, where all characteristics and fields are defined everywhere in the volume but they follow some generalized equations which are derived by using integrals of fractional order. The order of fractional integral can be equal to the fractal mass dimension of the solid. Fractional integrals are considered as an approximation of integrals on fractals. We suggest the approach to compute the moments of inertia for fractal solids. The dynamics of fractal solids are described by the usual Euler's equations. The possible experimental test of the continuous medium model for fractal solids is considered.Comment: 12 pages, LaTe

Phase-Space Metric for Non-Hamiltonian Systems

We consider an invariant skew-symmetric phase-space metric for non-Hamiltonian systems. We say that the metric is an invariant if the metric tensor field is an integral of motion. We derive the time-dependent skew-symmetric phase-space metric that satisfies the Jacobi identity. The example of non-Hamiltonian systems with linear friction term is considered.Comment: 12 page

Path Integral for Quantum Operations

In this paper we consider a phase space path integral for general time-dependent quantum operations, not necessarily unitary. We obtain the path integral for a completely positive quantum operation satisfied Lindblad equation (quantum Markovian master equation). We consider the path integral for quantum operation with a simple infinitesimal generator.Comment: 24 pages, LaTe

Fractional Derivative as Fractional Power of Derivative

Definitions of fractional derivatives as fractional powers of derivative operators are suggested. The Taylor series and Fourier series are used to define fractional power of self-adjoint derivative operator. The Fourier integrals and Weyl quantization procedure are applied to derive the definition of fractional derivative operator. Fractional generalization of concept of stability is considered.Comment: 20 pages, LaTe

Psi-Series Solution of Fractional Ginzburg-Landau Equation

One-dimensional Ginzburg-Landau equations with derivatives of noninteger order are considered. Using psi-series with fractional powers, the solution of the fractional Ginzburg-Landau (FGL) equation is derived. The leading-order behaviours of solutions about an arbitrary singularity, as well as their resonance structures, have been obtained. It was proved that fractional equations of order $alpha$ with polynomial nonlinearity of order $s$ have the noninteger power-like behavior of order $\alpha/(1-s)$ near the singularity.Comment: LaTeX, 19 pages, 2 figure

Transport Equations from Liouville Equations for Fractional Systems

We consider dynamical systems that are described by fractional power of coordinates and momenta. The fractional powers can be considered as a convenient way to describe systems in the fractional dimension space. For the usual space the fractional systems are non-Hamiltonian. Generalized transport equation is derived from Liouville and Bogoliubov equations for fractional systems. Fractional generalization of average values and reduced distribution functions are defined. Hydrodynamic equations for fractional systems are derived from the generalized transport equation.Comment: 11 pages, LaTe

Fractional Liouville and BBGKI Equations

We consider the fractional generalizations of Liouville equation. The normalization condition, phase volume, and average values are generalized for fractional case.The interpretation of fractional analog of phase space as a space with fractal dimension and as a space with fractional measure are discussed. The fractional analogs of the Hamiltonian systems are considered as a special class of non-Hamiltonian systems. The fractional generalization of the reduced distribution functions are suggested. The fractional analogs of the BBGKI equations are derived from the fractional Liouville equation.Comment: 20 page

Fractional Fokker-Planck Equation for Fractal Media

We consider the fractional generalizations of equation that defines the medium mass. We prove that the fractional integrals can be used to describe the media with noninteger mass dimensions. Using fractional integrals, we derive the fractional generalization of the Chapman-Kolmogorov equation (Smolukhovski equation). In this paper fractional Fokker-Planck equation for fractal media is derived from the fractional Chapman-Kolmogorov equation. Using the Fourier transform, we get the Fokker-Planck-Zaslavsky equations that have fractional coordinate derivatives. The Fokker-Planck equation for the fractal media is an equation with fractional derivatives in the dual space.Comment: 17 page