519 research outputs found
The Cosmic No-Hair Theorem and the Nonlinear Stability of Homogeneous Newtonian Cosmological Models
The validity of the cosmic no-hair theorem is investigated in the context of
Newtonian cosmology with a perfect fluid matter model and a positive
cosmological constant. It is shown that if the initial data for an expanding
cosmological model of this type is subjected to a small perturbation then the
corresponding solution exists globally in the future and the perturbation
decays in a way which can be described precisely. It is emphasized that no
linearization of the equations or special symmetry assumptions are needed. The
result can also be interpreted as a proof of the nonlinear stability of the
homogeneous models. In order to prove the theorem we write the general solution
as the sum of a homogeneous background and a perturbation. As a by-product of
the analysis it is found that there is an invariant sense in which an
inhomogeneous model can be regarded as a perturbation of a unique homogeneous
model. A method is given for associating uniquely to each Newtonian
cosmological model with compact spatial sections a spatially homogeneous model
which incorporates its large-scale dynamics. This procedure appears very
natural in the Newton-Cartan theory which we take as the starting point for
Newtonian cosmology.Comment: 16 pages, MPA-AR-94-
Self-attraction effect and correction on three absolute gravimeters
The perturbations of the gravitational field due to the mass distribution of
an absolute gravimeter have been studied. The so called Self Attraction Effect
(SAE) is crucial for the measurement accuracy, especially for the International
Comparisons, and for the uncertainty budget evaluation. Three instruments have
been analysed: MPG-2, FG5-238 and IMPG-02. The SAE has been calculated using a
numerical method based on FEM simulation. The observed effect has been treated
as an additional vertical gravity gradient. The correction (SAC) to be applied
to the computed g value has been associated with the specific height level,
where the measurement result is typically reported. The magnitude of the
obtained corrections is of order 1E-8 m/s2.Comment: 14 pages, 8 figures, submitted to Metrologi
Chaos and the continuum limit in nonneutral plasmas and charged particle beams
This paper examines discreteness effects in nearly collisionless N-body
systems of charged particles interacting via an unscreened r^-2 force, allowing
for bulk potentials admitting both regular and chaotic orbits. Both for
ensembles and individual orbits, as N increases there is a smooth convergence
towards a continuum limit. Discreteness effects are well modeled by Gaussian
white noise with relaxation time t_R = const * (N/log L)t_D, with L the Coulomb
logarithm and t_D the dynamical time scale. Discreteness effects accelerate
emittance growth for initially localised clumps. However, even allowing for
discreteness effects one can distinguish between orbits which, in the continuum
limit, feel a regular potential, so that emittance grows as a power law in
time, and chaotic orbits, where emittance grows exponentially. For sufficiently
large N, one can distinguish two different `kinds' of chaos. Short range
microchaos, associated with close encounters between charges, is a generic
feature, yielding large positive Lyapunov exponents X_N which do not decrease
with increasing N even if the bulk potential is integrable. Alternatively,
there is the possibility of larger scale macrochaos, characterised by smaller
Lyapunov exponents X_S, which is present only if the bulk potential is chaotic.
Conventional computations of Lyapunov exponents probe X_N, leading to the
oxymoronic conclusion that N-body orbits which look nearly regular and have
sharply peaked Fourier spectra are `very chaotic.' However, the `range' of the
microchaos, set by the typical interparticle spacing, decreases as N increases,
so that, for large N, this microchaos, albeit very strong, is largely
irrelevant macroscopically. A more careful numerical analysis allows one to
estimate both X_N and X_S.Comment: 13 pages plus 17 figure
Dynamical elastic bodies in Newtonian gravity
Well-posedness for the initial value problem for a self-gravitating elastic
body with free boundary in Newtonian gravity is proved. In the material frame,
the Euler-Lagrange equation becomes, assuming suitable constitutive properties
for the elastic material, a fully non-linear elliptic-hyperbolic system with
boundary conditions of Neumann type. For systems of this type, the initial data
must satisfy compatibility conditions in order to achieve regular solutions.
Given a relaxed reference configuration and a sufficiently small Newton's
constant, a neigborhood of initial data satisfying the compatibility conditions
is constructed
Chaotic Orbits in Thermal-Equilibrium Beams: Existence and Dynamical Implications
Phase mixing of chaotic orbits exponentially distributes these orbits through
their accessible phase space. This phenomenon, commonly called ``chaotic
mixing'', stands in marked contrast to phase mixing of regular orbits which
proceeds as a power law in time. It is operationally irreversible; hence, its
associated e-folding time scale sets a condition on any process envisioned for
emittance compensation. A key question is whether beams can support chaotic
orbits, and if so, under what conditions? We numerically investigate the
parameter space of three-dimensional thermal-equilibrium beams with space
charge, confined by linear external focusing forces, to determine whether the
associated potentials support chaotic orbits. We find that a large subset of
the parameter space does support chaos and, in turn, chaotic mixing. Details
and implications are enumerated.Comment: 39 pages, including 14 figure
Chaotic mixing in noisy Hamiltonian systems
This paper summarises an investigation of the effects of low amplitude noise
and periodic driving on phase space transport in 3-D Hamiltonian systems, a
problem directly applicable to systems like galaxies, where such perturbations
reflect internal irregularities and.or a surrounding environment. A new
diagnsotic tool is exploited to quantify how, over long times, different
segments of the same chaotic orbit can exhibit very different amounts of chaos.
First passage time experiments are used to study how small perturbations of an
individual orbit can dramatically accelerate phase space transport, allowing
`sticky' chaotic orbits trapped near regular islands to become unstuck on
suprisingly short time scales. Small perturbations are also studied in the
context of orbit ensembles with the aim of understanding how such
irregularities can increase the efficacy of chaotic mixing. For both noise and
periodic driving, the effect of the perturbation scales roughly in amplitude.
For white noise, the details are unimportant: additive and multiplicative noise
tend to have similar effects and the presence or absence of a friction related
to the noise by a Fluctuation- Dissipation Theorem is largely irrelevant.
Allowing for coloured noise can significantly decrease the efficacy of the
perturbation, but only when the autocorrelation time, which vanishes for white
noise, becomes so large that t here is little power at frequencies comparable
to the natural frequencies of the unperturbed orbit. This suggests strongly
that noise-induced extrinsic diffusion, like modulational diffusion associated
with periodic driving, is a resonance phenomenon. Potential implications for
galaxies are discussed.Comment: 15 pages including 18 figures, uses MNRAS LaTeX macro
Interplay Between Chaotic and Regular Motion in a Time-Dependent Barred Galaxy Model
We study the distinction and quantification of chaotic and regular motion in
a time-dependent Hamiltonian barred galaxy model. Recently, a strong
correlation was found between the strength of the bar and the presence of
chaotic motion in this system, as models with relatively strong bars were shown
to exhibit stronger chaotic behavior compared to those having a weaker bar
component. Here, we attempt to further explore this connection by studying the
interplay between chaotic and regular behavior of star orbits when the
parameters of the model evolve in time. This happens for example when one
introduces linear time dependence in the mass parameters of the model to mimic,
in some general sense, the effect of self-consistent interactions of the actual
N-body problem. We thus observe, in this simple time-dependent model also, that
the increase of the bar's mass leads to an increase of the system's chaoticity.
We propose a new way of using the Generalized Alignment Index (GALI) method as
a reliable criterion to estimate the relative fraction of chaotic vs. regular
orbits in such time-dependent potentials, which proves to be much more
efficient than the computation of Lyapunov exponents. In particular, GALI is
able to capture subtle changes in the nature of an orbit (or ensemble of
orbits) even for relatively small time intervals, which makes it ideal for
detecting dynamical transitions in time-dependent systems.Comment: 21 pages, 9 figures (minor typos fixed) to appear in J. Phys. A:
Math. Theo
Concerning the Wave equation on Asymptotically Euclidean Manifolds
We obtain KSS, Strichartz and certain weighted Strichartz estimate for the
wave equation on , , when metric
is non-trapping and approaches the Euclidean metric like with
. Using the KSS estimate, we prove almost global existence for
quadratically semilinear wave equations with small initial data for
and . Also, we establish the Strauss conjecture when the metric is radial
with for .Comment: Final version. To appear in Journal d'Analyse Mathematiqu
Geometric optics and instability for semi-classical Schrodinger equations
We prove some instability phenomena for semi-classical (linear or) nonlinear
Schrodinger equations. For some perturbations of the data, we show that for
very small times, we can neglect the Laplacian, and the mechanism is the same
as for the corresponding ordinary differential equation. Our approach allows
smaller perturbations of the data, where the instability occurs for times such
that the problem cannot be reduced to the study of an o.d.e.Comment: 22 pages. Corollary 1.7 adde
Chaos and the continuum limit in the gravitational N-body problem II. Nonintegrable potentials
This paper continues a numerical investigation of orbits evolved in `frozen,'
time-independent N-body realisations of smooth time-independent density
distributions corresponding to both integrable and nonintegrable potentials,
allowing for N as large as 300,000. The principal focus is on distinguishing
between, and quantifying, the effects of graininess on initial conditions
corresponding, in the continuum limit, to regular and chaotic orbits. Ordinary
Lyapunov exponents X do not provide a useful diagnostic for distinguishing
between regular and chaotic behaviour. Frozen-N orbits corresponding in the
continuum limit to both regular and chaotic characteristics have large positive
X even though, for large N, the `regular' frozen-N orbits closely resemble
regular characteristics in the smooth potential. Viewed macroscopically both
`regular' and `chaotic' frozen-N orbits diverge as a power law in time from
smooth orbits with the same initial condition. There is, however, an important
difference between `regular' and `chaotic' frozen-N orbits: For regular orbits,
the time scale associated with this divergence t_G ~ N^{1/2}t_D, with t_D a
characteristic dynamical time; for chaotic orbits t_G ~ (ln N) t_D. At least
for N>1000 or so, clear distinctions exist between phase mixing of initially
localised orbit ensembles which, in the continuum limit, exhibit regular versus
chaotic behaviour. For both regular and chaotic ensembles, finite-N effects are
well mimicked, both qualitatively and quantitatively, by energy-conserving
white noise with amplitude ~ 1/N. This suggests strongly that earlier
investigations of the effects of low amplitude noise on phase space transport
in smooth potentials are directly relevant to real physical systems.Comment: 20 pages, including 21 FIGURES, uses RevTeX macro
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