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

Electroproduction of Charmonia off Protons and Nuclei

Elastic virtual photoproduction of charmonia on nucleons is calculated in a parameter free way with the light-cone dipole formalism and the same input: factorization in impact parameters, light-cone wave functions for the photons and the charmonia, and the universal phenomenological dipole cross section which is fitted to other data. The charmonium wave functions are calculated with four known realistic potentials, and two models for the dipole cross section are tested. Very good agreement with data for the cross section of charmonium electroproduction is found in a wide range of $s$ and $Q^2$. Using the ingredients from those calculations we calculate also exclusive electroproduction of charmonia off nuclei. Here new effects become important, (i) color filtering of the $c\bar c$ pair on its trajectory through nuclear matter, (ii) dependence on the finite lifetime of the $c\bar c$ fluctuation (coherence length) and (iii) gluon shadowing in a nucleus compared to the one in a nucleon. Total coherent and incoherent cross sections for C, Cu and Pb as functions of $s$ are presented. The results can be tested with future electron-nucleus colliders or in the peripheral collisions of relativistic heavy ions.Comment: Talk at 2-nd International Workshop on Hadron Physics, 25-29 September 2002, Coimbra, Portugal. To appear in the Workshop Proceedings (will be published by the American Institute of Physics

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

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 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 Systems and Fractional Bogoliubov Hierarchy Equations

We consider the fractional generalizations of the phase volume, volume element and Poisson brackets. These generalizations lead us to the fractional analog of the phase space. We consider systems on this fractional phase space and fractional analogs of the Hamilton equations. The fractional generalization of the average value is suggested. The fractional analogs of the Bogoliubov hierarchy equations are derived from the fractional Liouville equation. We define the fractional reduced distribution functions. The fractional analog of the Vlasov equation and the Debye radius are considered.Comment: 12 page

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 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

Corrosion Resistance of Fe-Cr-Al-Si Alloys with Low Chromium Content

Within the framework of this work, alloys with a chromium content of 5 to14 wt%, aluminum from 0 to 4 wt% and silicon from 0 to 4 wt%. The samples were tested for resistance to oxidation with calm dry air (800 ∘C, 0.1 MPa) for 60 hours; in high parameter water (350 ∘C, 16 MPa), for 300 hours; in steam (400 ∘C, 10 MPa), for 72hours and superheated steam (1100 ∘C, 0.1 MPa) for 1 hour.The compositions most resistant to corrosion under the specified conditions were determined, and the existence of a synergistic effect of silicon and aluminum asalloying elements of iron alloys was confirmed. Keywords: fuel cladding; PWR; tolerant fuel, ferrite steel; corrosive resistance steel

"Unusual" metals in two dimensions: one-particle model of the metal-insulator transition at T=0

The conductance of disordered nano-wires at T=0 is calculated in one-particle approximation by reducing the original multi-dimensional problem for an open bounded system to a set of exactly one-dimensional non-Hermitian problems for mode propagators. Regarding two-dimensional conductor as a limiting case of three-dimensional disordered quantum waveguide, the metallic ground state is shown to result from its multi-modeness. On thinning the waveguide (in practice, e. g., by means of the ``pressing'' external electric field) the electron system undergoes a continuous phase transition from metallic to insulating state. The result predicted conform qualitatively to the observed anomalies of the resistance of different planar electron and hole systems.Comment: 7 pages, LATEX-2