18 research outputs found
Fractional Quantum Mechanics
A path integral approach to quantum physics has been developed. Fractional
path integrals over the paths of the L\'evy flights are defined. It is shown
that if the fractality of the Brownian trajectories leads to standard quantum
and statistical mechanics, then the fractality of the L\'evy paths leads to
fractional quantum mechanics and fractional statistical mechanics. The
fractional quantum and statistical mechanics have been developed via our
fractional path integral approach. A fractional generalization of the
Schr\"odinger equation has been found. A relationship between the energy and
the momentum of the nonrelativistic quantum-mechanical particle has been
established. The equation for the fractional plane wave function has been
obtained. We have derived a free particle quantum-mechanical kernel using Fox's
H function. A fractional generalization of the Heisenberg uncertainty relation
has been established. Fractional statistical mechanics has been developed via
the path integral approach. A fractional generalization of the motion equation
for the density matrix has been found. The density matrix of a free particle
has been expressed in terms of the Fox's H function. We also discuss the
relationships between fractional and the well-known Feynman path integral
approaches to quantum and statistical mechanics.Comment: 27 page
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 with polynomial nonlinearity of order have the
noninteger power-like behavior of order near the singularity.Comment: LaTeX, 19 pages, 2 figure
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
Solutions of a particle with fractional -potential in a fractional dimensional space
A Fourier transformation in a fractional dimensional space of order \la
(0<\la\leq 1) is defined to solve the Schr\"odinger equation with Riesz
fractional derivatives of order \a. This new method is applied for a particle
in a fractional -potential well defined by V(x) =-
\gamma\delta^{\la}(x), where and \delta^{\la}(x) is the
fractional Dirac delta function. A complete solutions for the energy values and
the wave functions are obtained in terms of the Fox H-functions. It is
demonstrated that the eigen solutions are exist if 0< \la<\a. The results for
\la= 1 and \a=2 are in exact agreement with those presented in the standard
quantum mechanics
Towards synthesis of solar wind and geomagnetic scaling exponents: a fractional Levy motion model
Mandelbrot introduced the concept of fractals to describe the non-Euclidean shape of many aspects of the natural world. In the time series context, he proposed the use of fractional Brownian motion (fBm) to model non-negligible temporal persistence, the ‘Joseph Effect’; and Lévy flights to quantify large discontinuities, the ‘Noah Effect’. In space physics, both effects are manifested in the intermittency and long-range correlation which are by now well-established features of geomagnetic indices and their solar wind drivers. In order to capture and quantify the Noah and Joseph effects in one compact model, we propose the application of the ‘bridging’ fractional Lévy motion (fLm) to space physics. We perform an initial evaluation of some previous scaling results in this paradigm, and show how fLm can model the previously observed exponents. We suggest some new directions for the future