22 research outputs found
Wavefunction statistics in open chaotic billiards
We study the statistical properties of wavefunctions in a chaotic billiard
that is opened up to the outside world. Upon increasing the openings, the
billiard wavefunctions cross over from real to complex. Each wavefunction is
characterized by a phase rigidity, which is itself a fluctuating quantity. We
calculate the probability distribution of the phase rigidity and discuss how
phase rigidity fluctuations cause long-range correlations of intensity and
current density. We also find that phase rigidities for wavefunctions with
different incoming wave boundary conditions are statistically correlated.Comment: 4 pages, RevTeX; 1 figur
Anomalous diffusion associated with nonlinear fractional derivative Fokker-Planck-like equation: Exact time-dependent solutions
We consider the nonlinear Fokker-Planck-like equation with fractional
derivatives . Exact
time-dependent solutions are found for
(). By considering the long-distance {\it asymptotic}
behavior of these solutions, a connection is established, namely
(), with the solutions optimizing
the nonextensive entropy characterized by index . Interestingly enough,
this relation coincides with the one already known for L\'evy-like
superdiffusion (i.e., and ). Finally, for
we obtain which differs from the value
corresponding to the solutions available in the literature (
porous medium equation), thus exhibiting nonuniform convergence.Comment: 3 figure
Fractional Langevin equation
We investigate fractional Brownian motion with a microscopic random-matrix
model and introduce a fractional Langevin equation. We use the latter to study
both sub- and superdiffusion of a free particle coupled to a fractal heat bath.
We further compare fractional Brownian motion with the fractal time process.
The respective mean-square displacements of these two forms of anomalous
diffusion exhibit the same power-law behavior. Here we show that their lowest
moments are actually all identical, except the second moment of the velocity.
This provides a simple criterion which enables to distinguish these two
non-Markovian processes.Comment: 4 page
Chaos and flights in the atom-photon interaction in cavity QED
We study dynamics of the atom-photon interaction in cavity quantum
electrodynamics (QED), considering a cold two-level atom in a single-mode
high-finesse standing-wave cavity as a nonlinear Hamiltonian system with three
coupled degrees of freedom: translational, internal atomic, and the field. The
system proves to have different types of motion including L\'{e}vy flights and
chaotic walkings of an atom in a cavity. It is shown that the translational
motion, related to the atom recoils, is governed by an equation of a parametric
nonlinear pendulum with a frequency modulated by the Rabi oscillations. This
type of dynamics is chaotic with some width of the stochastic layer that is
estimated analytically. The width is fairly small for realistic values of the
control parameters, the normalized detuning and atomic recoil
frequency . It is demonstrated how the atom-photon dynamics with a
given value of depends on the values of and initial
conditions. Two types of L\'{e}vy flights, one corresponding to the ballistic
motion of the atom and another one corresponding to small oscillations in a
potential well, are found. These flights influence statistical properties of
the atom-photon interaction such as distribution of Poincar\'{e} recurrences
and moments of the atom position . The simulation shows different regimes of
motion, from slightly abnormal diffusion with at to a superdiffusion with at that
corresponds to a superballistic motion of the atom with an acceleration. The
obtained results can be used to find new ways to manipulate atoms, to cool and
trap them by adjusting the detuning .Comment: 16 pages, 7 figures. To be published in Phys. Rev.
Uncoupled continuous-time random walks: Solution and limiting behavior of the master equation
A detailed study is presented for a large class of uncoupled continuous-time
random walks (CTRWs). The master equation is solved for the Mittag-Leffler
survival probability. The properly scaled diffusive limit of the master
equation is taken and its relation with the fractional diffusion equation is
discussed. Finally, some common objections found in the literature are
thoroughly reviewed.Comment: Preprint version of an already published paper. 8 page
Continuous-time statistics and generalized relaxation equations
Using two simple examples, the continuous-time random walk as well as a two state Markov chain, the relation between generalized anomalous relaxation equations and semi-Markov processes is illustrated. This relation is then used to discuss continuous-time random statistics in a general setting, for statistics of convolution-type. Two examples are presented in some detail: the sum statistic and the maximum statistic
Global stabilization of feedforward systems under perturbations in sampling schedule
For nonlinear systems that are known to be globally asymptotically
stabilizable, control over networks introduces a major challenge because of the
asynchrony in the transmission schedule. Maintaining global asymptotic
stabilization in sampled-data implementations with zero-order hold and with
perturbations in the sampling schedule is not achievable in general but we show
in this paper that it is achievable for the class of feedforward systems. We
develop sampled-data feedback stabilizers which are not approximations of
continuous-time designs but are discontinuous feedback laws that are
specifically developed for maintaining global asymptotic stabilizability under
any sequence of sampling periods that is uniformly bounded by a certain
"maximum allowable sampling period".Comment: 27 pages, 5 figures, submitted for possible publication to SIAM
Journal Control and Optimization. Second version with added remark