572 research outputs found
Stabilization of the Yang-Mills chaos in non-Abelian Born-Infeld theory
We investigate dynamics of the homogeneous time-dependent SU(2) Yang-Mills
fields governed by the non-Abelian Born-Infeld lagrangian which arises in
superstring theory as a result of summation of all orders in the string slope
parameter . It is shown that generically the Born-Infeld dynamics is
less chaotic than that in the ordinary Yang-Mills theory, and at high enough
field strength the Yang-Mills chaos is stabilized. More generally, a smothering
effect of the string non-locality on behavior of classical fields is
conjectured.Comment: 7 pages, 5 figure
Accelerating cycle expansions by dynamical conjugacy
Periodic orbit theory provides two important functions---the dynamical zeta
function and the spectral determinant for the calculation of dynamical averages
in a nonlinear system. Their cycle expansions converge rapidly when the system
is uniformly hyperbolic but greatly slowed down in the presence of
non-hyperbolicity. We find that the slow convergence can be associated with
singularities in the natural measure. A properly designed coordinate
transformation may remove these singularities and results in a dynamically
conjugate system where fast convergence is restored. The technique is
successfully demonstrated on several examples of one-dimensional maps and some
remaining challenges are discussed
Arnol'd Tongues and Quantum Accelerator Modes
The stable periodic orbits of an area-preserving map on the 2-torus, which is
formally a variant of the Standard Map, have been shown to explain the quantum
accelerator modes that were discovered in experiments with laser-cooled atoms.
We show that their parametric dependence exhibits Arnol'd-like tongues and
perform a perturbative analysis of such structures. We thus explain the
arithmetical organisation of the accelerator modes and discuss experimental
implications thereof.Comment: 20 pages, 6 encapsulated postscript figure
Signatures of Classical Periodic Orbits on a Smooth Quantum System
Gutzwiller's trace formula and Bogomolny's formula are applied to a
non--specific, non--scalable Hamiltonian system, a two--dimensional anharmonic
oscillator. These semiclassical theories reproduce well the exact quantal
results over a large spatial and energy range.Comment: 12 pages, uuencoded postscript file (1526 kb
Periodic orbit spectrum in terms of Ruelle--Pollicott resonances
Fully chaotic Hamiltonian systems possess an infinite number of classical
solutions which are periodic, e.g. a trajectory ``p'' returns to its initial
conditions after some fixed time tau_p. Our aim is to investigate the spectrum
tau_1, tau_2, ... of periods of the periodic orbits. An explicit formula for
the density rho(tau) = sum_p delta (tau - tau_p) is derived in terms of the
eigenvalues of the classical evolution operator. The density is naturally
decomposed into a smooth part plus an interferent sum over oscillatory terms.
The frequencies of the oscillatory terms are given by the imaginary part of the
complex eigenvalues (Ruelle--Pollicott resonances). For large periods,
corrections to the well--known exponential growth of the smooth part of the
density are obtained. An alternative formula for rho(tau) in terms of the zeros
and poles of the Ruelle zeta function is also discussed. The results are
illustrated with the geodesic motion in billiards of constant negative
curvature. Connections with the statistical properties of the corresponding
quantum eigenvalues, random matrix theory and discrete maps are also
considered. In particular, a random matrix conjecture is proposed for the
eigenvalues of the classical evolution operator of chaotic billiards
Chaotic Properties of Dilute Two and Three Dimensional Random Lorentz Gases I: Equilibrium Systems
We compute the Lyapunov spectrum and the Kolmogorov-Sinai entropy for a
moving particle placed in a dilute, random array of hard disk or hard sphere
scatterers - i.e. the dilute Lorentz gas model. This is carried out in two
ways: First we use simple kinetic theory arguments to compute the Lyapunov
spectrum for both two and three dimensional systems. In order to provide a
method that can easily be generalized to non-uniform systems we then use a
method based upon extensions of the Lorentz-Boltzmann (LB) equation to include
variables that characterize the chaotic behavior of the system. The extended LB
equations depend upon the number of dimensions and on whether one is computing
positive or negative Lyapunov exponents. In the latter case the extended LB
equation is closely related to an "anti-Lorentz-Boltzmann equation" where the
collision operator has the opposite sign from the ordinary LB equation. Finally
we compare our results with computer simulations of Dellago and Posch and find
very good agreement.Comment: 48 pages, 3 ps fig
On the rate of quantum ergodicity in Euclidean billiards
For a large class of quantized ergodic flows the quantum ergodicity theorem
due to Shnirelman, Zelditch, Colin de Verdi\`ere and others states that almost
all eigenfunctions become equidistributed in the semiclassical limit. In this
work we first give a short introduction to the formulation of the quantum
ergodicity theorem for general observables in terms of pseudodifferential
operators and show that it is equivalent to the semiclassical eigenfunction
hypothesis for the Wigner function in the case of ergodic systems. Of great
importance is the rate by which the quantum mechanical expectation values of an
observable tend to their mean value. This is studied numerically for three
Euclidean billiards (stadium, cosine and cardioid billiard) using up to 6000
eigenfunctions. We find that in configuration space the rate of quantum
ergodicity is strongly influenced by localized eigenfunctions like bouncing
ball modes or scarred eigenfunctions. We give a detailed discussion and
explanation of these effects using a simple but powerful model. For the rate of
quantum ergodicity in momentum space we observe a slower decay. We also study
the suitably normalized fluctuations of the expectation values around their
mean, and find good agreement with a Gaussian distribution.Comment: 40 pages, LaTeX2e. This version does not contain any figures. A
version with all figures can be obtained from
http://www.physik.uni-ulm.de/theo/qc/ (File:
http://www.physik.uni-ulm.de/theo/qc/ulm-tp/tp97-8.ps.gz) In case of any
problems contact Arnd B\"acker (e-mail: [email protected]) or Roman
Schubert (e-mail: [email protected]
How Chaotic is the Stadium Billiard? A Semiclassical Analysis
The impression gained from the literature published to date is that the
spectrum of the stadium billiard can be adequately described, semiclassically,
by the Gutzwiller periodic orbit trace formula together with a modified
treatment of the marginally stable family of bouncing ball orbits. I show that
this belief is erroneous. The Gutzwiller trace formula is not applicable for
the phase space dynamics near the bouncing ball orbits. Unstable periodic
orbits close to the marginally stable family in phase space cannot be treated
as isolated stationary phase points when approximating the trace of the Green
function. Semiclassical contributions to the trace show an - dependent
transition from hard chaos to integrable behavior for trajectories approaching
the bouncing ball orbits. A whole region in phase space surrounding the
marginal stable family acts, semiclassically, like a stable island with
boundaries being explicitly -dependent. The localized bouncing ball
states found in the billiard derive from this semiclassically stable island.
The bouncing ball orbits themselves, however, do not contribute to individual
eigenvalues in the spectrum. An EBK-like quantization of the regular bouncing
ball eigenstates in the stadium can be derived. The stadium billiard is thus an
ideal model for studying the influence of almost regular dynamics near
marginally stable boundaries on quantum mechanics.Comment: 27 pages, 6 figures, submitted to J. Phys.
On the canonically invariant calculation of Maslov indices
After a short review of various ways to calculate the Maslov index appearing
in semiclassical Gutzwiller type trace formulae, we discuss a
coordinate-independent and canonically invariant formulation recently proposed
by A Sugita (2000, 2001). We give explicit formulae for its ingredients and
test them numerically for periodic orbits in several Hamiltonian systems with
mixed dynamics. We demonstrate how the Maslov indices and their ingredients can
be useful in the classification of periodic orbits in complicated bifurcation
scenarios, for instance in a novel sequence of seven orbits born out of a
tangent bifurcation in the H\'enon-Heiles system.Comment: LaTeX, 13 figures, 3 tables, submitted to J. Phys.
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