118 research outputs found
Acceleration of bouncing balls in external fields
We introduce two models, the Fermi-Ulam model in an external field and a one
dimensional system of bouncing balls in an external field above a periodically
oscillating plate. For both models we investigate the possibility of unbounded
motion. In a special case the two models are equivalent
Infinitesimal Lyapunov functions for singular flows
We present an extension of the notion of infinitesimal Lyapunov function to
singular flows, and from this technique we deduce a characterization of
partial/sectional hyperbolic sets. In absence of singularities, we can also
characterize uniform hyperbolicity.
These conditions can be expressed using the space derivative DX of the vector
field X together with a field of infinitesimal Lyapunov functions only, and are
reduced to checking that a certain symmetric operator is positive definite at
the tangent space of every point of the trapping region.Comment: 37 pages, 1 figure; corrected the statement of Lemma 2.2 and item (2)
of Theorem 2.7; removed item (5) of Theorem 2.7 and its wrong proof since the
statement of this item was false; corrected items (1) and (2) of Theorem 2.23
and their proofs. Included Example 6 on smooth reduction of families of
quadratic forms. The published version in Math Z journal needs an errat
Optimum spectral window for imaging of art with optical coherence tomography
Optical Coherence Tomography (OCT) has been shown to have potential for important applications in the field of art conservation and archaeology due to its ability to image subsurface microstructures non-invasively. However, its depth of penetration in painted objects is limited due to the strong scattering properties of artists’ paints. VIS-NIR (400 nm – 2400 nm) reflectance spectra of a wide variety of paints made with historic artists’ pigments have been measured. The best spectral window with which to use optical coherence tomography (OCT) for the imaging of subsurface structure of paintings was found to be around 2.2 μm. The same spectral window would also be most suitable for direct infrared imaging of preparatory sketches under the paint layers. The reflectance spectra from a large sample of chemically verified pigments provide information on the spectral transparency of historic artists’ pigments/paints as well as a reference set of spectra for pigment identification. The results of the paper suggest that broadband sources at ~2 microns are highly desirable for OCT applications in art and potentially material science in general
Autocorrelation function of eigenstates in chaotic and mixed systems
We study the autocorrelation function of different types of eigenfunctions in
quantum mechanical systems with either chaotic or mixed classical limits. We
obtain an expansion of the autocorrelation function in terms of the correlation
length. For localized states, like bouncing ball modes or states living on
tori, a simple model using only classical input gives good agreement with the
exact result. In particular, a prediction for irregular eigenfunctions in mixed
systems is derived and tested. For chaotic systems, the expansion of the
autocorrelation function can be used to test quantum ergodicity on different
length scales.Comment: 30 pages, 12 figures. Some of the pictures are included in low
resolution only. For a version with pictures in high resolution see
http://www.physik.uni-ulm.de/theo/qc/ or http://www.maths.bris.ac.uk/~maab
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
Regular and chaotic interactions of two BPS dyons at low energy
We identify and analyze quasiperiodic and chaotic motion patterns in the time
evolution of a classical, non-Abelian Bogomol'nyi-Prasad-Sommerfield (BPS) dyon
pair at low energies. This system is amenable to the geodesic approximation
which restricts the underlying SU(2) Yang-Mills-Higgs dynamics to an
eight-dimensional phase space. We numerically calculate a representative set of
long-time solutions to the corresponding Hamilton equations and analyze
quasiperiodic and chaotic phase space regions by means of Poincare surfaces of
section, high-resolution power spectra and Lyapunov exponents. Our results
provide clear evidence for both quasiperiodic and chaotic behavior and
characterize it quantitatively. Indications for intermittency are also
discussed.Comment: 22 pages, 6 figures (v2 contains a few additional references, a new
paragraph on intermittency and minor stylistic corrections to agree with the
published version
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
Master equation approach to the conjugate pairing rule of Lyapunov spectra for many-particle thermostatted systems
The master equation approach to Lyapunov spectra for many-particle systems is
applied to non-equilibrium thermostatted systems to discuss the conjugate
pairing rule. We consider iso-kinetic thermostatted systems with a shear flow
sustained by an external restriction, in which particle interactions are
expressed as a Gaussian white randomness. Positive Lyapunov exponents are
calculated by using the Fokker-Planck equation to describe the tangent vector
dynamics. We introduce another Fokker-Planck equation to describe the
time-reversed tangent vector dynamics, which allows us to calculate the
negative Lyapunov exponents. Using the Lyapunov exponents provided by these two
Fokker-Planck equations we show the conjugate pairing rule is satisfied for
thermostatted systems with a shear flow in the thermodynamic limit. We also
give an explicit form to connect the Lyapunov exponents with the
time-correlation of the interaction matrix in a thermostatted system with a
color field.Comment: 10 page
About ergodicity in the family of limacon billiards
By continuation from the hyperbolic limit of the cardioid billiard we show
that there is an abundance of bifurcations in the family of limacon billiards.
The statistics of these bifurcation shows that the size of the stable intervals
decreases with approximately the same rate as their number increases with the
period. In particular, we give numerical evidence that arbitrarily close to the
cardioid there are elliptic islands due to orbits created in saddle node
bifurcations. This shows explicitly that if in this one parameter family of
maps ergodicity occurs for more than one parameter the set of these parameter
values has a complicated structure.Comment: 17 pages, 9 figure
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]
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