79 research outputs found
Spectral Statistics in the Quantized Cardioid Billiard
The spectral statistics in the strongly chaotic cardioid billiard are
studied. The analysis is based on the first 11000 quantal energy levels for odd
and even symmetry respectively. It is found that the level-spacing distribution
is in good agreement with the GOE distribution of random-matrix theory. In case
of the number variance and rigidity we observe agreement with the random-matrix
model for short-range correlations only, whereas for long-range correlations
both statistics saturate in agreement with semiclassical expectations.
Furthermore the conjecture that for classically chaotic systems the normalized
mode fluctuations have a universal Gaussian distribution with unit variance is
tested and found to be in very good agreement for both symmetry classes. By
means of the Gutzwiller trace formula the trace of the cosine-modulated heat
kernel is studied. Since the billiard boundary is focusing there are conjugate
points giving rise to zeros at the locations of the periodic orbits instead of
exclusively Gaussian peaks.Comment: 20 pages, uu-encoded ps.Z-fil
Tunable Lyapunov exponent in inverse magnetic billiards
The stability properties of the classical trajectories of charged particles
are investigated in a two dimensional stadium-shaped inverse magnetic domain,
where the magnetic field is zero inside the stadium domain and constant
outside. In the case of infinite magnetic field the dynamics of the system is
the same as in the Bunimovich billiard, i.e., ergodic and mixing. However, for
weaker magnetic fields the phase space becomes mixed and the chaotic part
gradually shrinks. The numerical measurements of the Lyapunov exponent
(performed with a novel method) and the integrable/chaotic phase space volume
ratio show that both quantities can be smoothly tuned by varying the external
magnetic field. A possible experimental realization of the arrangement is also
discussed.Comment: 4 pages, 6 figure
Coin Tossing as a Billiard Problem
We demonstrate that the free motion of any two-dimensional rigid body
colliding elastically with two parallel, flat walls is equivalent to a billiard
system. Using this equivalence, we analyze the integrable and chaotic
properties of this new class of billiards. This provides a demonstration that
coin tossing, the prototypical example of an independent random process, is a
completely chaotic (Bernoulli) problem. The related question of which billiard
geometries can be represented as rigid body systems is examined.Comment: 16 pages, LaTe
Decay of Classical Chaotic Systems - the Case of the Bunimovich Stadium
The escape of an ensemble of particles from the Bunimovich stadium via a
small hole has been studied numerically. The decay probability starts out
exponentially but has an algebraic tail. The weight of the algebraic decay
tends to zero for vanishing hole size. This behaviour is explained by the slow
transport of the particles close to the marginally stable bouncing ball orbits.
It is contrasted with the decay function of the corresponding quantum system.Comment: 16 pages, RevTex, 3 figures are available upon request from
[email protected], to be published in Phys.Rev.
Classical and quantum chaos in a circular billiard with a straight cut
We study classical and quantum dynamics of a particle in a circular billiard
with a straight cut. This system can be integrable, nonintegrable with soft
chaos, or nonintegrable with hard chaos, as we vary the size of the cut. We use
a quantum web to show differences in the quantum manifestations of classical
chaos for these three different regimes.Comment: LaTeX2e, 8 pages including 3 Postscript figures and 4 GIF figures,
submitted to Phys. Rev.
Chaos in Andreev Billiards
A new type of classical billiard - the Andreev billiard - is investigated
using the tangent map technique. Andreev billiards consist of a normal region
surrounded by a superconducting region. In contrast with previously studied
billiards, Andreev billiards are integrable in zero magnetic field, {\it
regardless of their shape}. A magnetic field renders chaotic motion in a
generically shaped billiard, which is demonstrated for the Bunimovich stadium
by examination of both Poincar\'e sections and Lyapunov exponents. The issue of
the feasibility of certain experimental realizations is addressed.Comment: ReVTeX3.0, 4 pages, 3 figures appended as postscript file (uuencoded
with uufiles
Chaotic eigenfunctions in momentum space
We study eigenstates of chaotic billiards in the momentum representation and
propose the radially integrated momentum distribution as useful measure to
detect localization effects. For the momentum distribution, the radially
integrated momentum distribution, and the angular integrated momentum
distribution explicit formulae in terms of the normal derivative along the
billiard boundary are derived. We present a detailed numerical study for the
stadium and the cardioid billiard, which shows in several cases that the
radially integrated momentum distribution is a good indicator of localized
eigenstates, such as scars, or bouncing ball modes. We also find examples,
where the localization is more strongly pronounced in position space than in
momentum space, which we discuss in detail. Finally applications and
generalizations are discussed.Comment: 30 pages. The figures are included in low resolution only. For a
version with figures in high resolution see
http://www.physik.uni-ulm.de/theo/qc/ulm-tp/tp99-2.htm
Analyzing symmetry breaking within a chaotic quantum system via Bayesian inference
Bayesian inference is applied to the level fluctuations of two coupled
microwave billiards in order to extract the coupling strength. The coupled
resonators provide a model of a chaotic quantum system containing two coupled
symmetry classes of levels. The number variance is used to quantify the level
fluctuations as a function of the coupling and to construct the conditional
probability distribution of the data. The prior distribution of the coupling
parameter is obtained from an invariance argument on the entropy of the
posterior distribution.Comment: Example from chaotic dynamics. 8 pages, 7 figures. Submitted to PR
Quantum Chaos, Irreversible Classical Dynamics and Random Matrix Theory
The Bohigas--Giannoni--Schmit conjecture stating that the statistical
spectral properties of systems which are chaotic in their classical limit
coincide with random matrix theory is proved. For this purpose a new
semiclassical field theory for individual chaotic systems is constructed in the
framework of the non--linear -model. The low lying modes are shown to
be associated with the Perron--Frobenius spectrum of the underlying
irreversible classical dynamics. It is shown that the existence of a gap in the
Perron-Frobenius spectrum results in a RMT behavior. Moreover, our formalism
offers a way of calculating system specific corrections beyond RMT.Comment: 4 pages, revtex, no figure
The origin of power-law distributions in deterministic walks: the influence of landscape geometry
We investigate the properties of a deterministic walk, whose locomotion rule
is always to travel to the nearest site. Initially the sites are randomly
distributed in a closed rectangular ( landscape and, once
reached, they become unavailable for future visits. As expected, the walker
step lengths present characteristic scales in one () and two () dimensions. However, we find scale invariance for an intermediate
geometry, when the landscape is a thin strip-like region. This result is
induced geometrically by a dynamical trapping mechanism, leading to a power law
distribution for the step lengths. The relevance of our findings in broader
contexts -- of both deterministic and random walks -- is also briefly
discussed.Comment: 7 pages, 11 figures. To appear in Phys. Rev.
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