3,297 research outputs found
Towards a combined fractional mechanics and quantization
A fractional Hamiltonian formalism is introduced for the recent combined
fractional calculus of variations. The Hamilton-Jacobi partial differential
equation is generalized to be applicable for systems containing combined Caputo
fractional derivatives. The obtained results provide tools to carry out the
quantization of nonconservative problems through combined fractional canonical
equations of Hamilton type.Comment: This is a preprint of a paper whose final and definite form will be
published in: Fract. Calc. Appl. Anal., Vol. 15, No 3 (2012). Submitted
21-Feb-2012; revised 29-May-2012; accepted 03-June-201
Separating the regular and irregular energy levels and their statistics in Hamiltonian system with mixed classical dynamics
We look at the high-lying eigenstates (from the 10,001st to the 13,000th) in
the Robnik billiard (defined as a quadratic conformal map of the unit disk)
with the shape parameter . All the 3,000 eigenstates have been
numerically calculated and examined in the configuration space and in the phase
space which - in comparison with the classical phase space - enabled a clear
cut classification of energy levels into regular and irregular. This is the
first successful separation of energy levels based on purely dynamical rather
than special geometrical symmetry properties. We calculate the fractional
measure of regular levels as which is in remarkable
agreement with the classical estimate . This finding
confirms the Percival's (1973) classification scheme, the assumption in
Berry-Robnik (1984) theory and the rigorous result by Lazutkin (1981,1991). The
regular levels obey the Poissonian statistics quite well whereas the irregular
sequence exhibits the fractional power law level repulsion and globally
Brody-like statistics with . This is due to the strong
localization of irregular eigenstates in the classically chaotic regions.
Therefore in the entire spectrum we see that the Berry-Robnik regime is not yet
fully established so that the level spacing distribution is correctly captured
by the Berry-Robnik-Brody distribution (Prosen and Robnik 1994).Comment: 20 pages, file in plain LaTeX, 7 figures upon request submitted to J.
Phys. A. Math. Gen. in December 199
Dynamic and spectral mixing in nanosystems
In the framework of simple spin-boson Hamiltonian we study an interplay
between dynamic and spectral roots to stochastic-like behavior. The Hamiltonian
describes an initial vibrational state coupled to discrete dense spectrum
reservoir. The reservoir states are formed by three sequences with rationally
independent periodicities typical for vibrational states in many nanosize
systems. We show that quantum evolution of the system is determined by a
dimensionless parameter which is characteristic number of the reservoir states
relevant for the initial vibrational level dynamics. Our semi-quantitative
analytic results are confirmed by numerical solution of the equation of motion.
We anticipate that predicted in the paper both kinds of stochastic-like
behavior (namely, due to spectral mixing and recurrence cycle dynamic mixing)
can be observed by femtosecond spectroscopy methods in nanosystems.Comment: 6 pages, 4 figure
Holder exponents of irregular signals and local fractional derivatives
It has been recognized recently that fractional calculus is useful for
handling scaling structures and processes. We begin this survey by pointing out
the relevance of the subject to physical situations. Then the essential
definitions and formulae from fractional calculus are summarized and their
immediate use in the study of scaling in physical systems is given. This is
followed by a brief summary of classical results. The main theme of the review
rests on the notion of local fractional derivatives. There is a direct
connection between local fractional differentiability properties and the
dimensions/ local Holder exponents of nowhere differentiable functions. It is
argued that local fractional derivatives provide a powerful tool to analyse the
pointwise behaviour of irregular signals and functions.Comment: 20 pages, Late
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