97 research outputs found
Lyapunov exponents from geodesic spread in configuration space
The exact form of the Jacobi–Levi-Civita (JLC) equation for geodesic spread is here explicitly worked out at arbitrary dimension for the configuration space manifold [Formula Presented] of a standard Hamiltonian system, equipped with the Jacobi (or kinetic energy) metric [Formula Presented] As the Hamiltonian flow corresponds to a geodesic flow on [Formula Presented] the JLC equation can be used to study the degree of instability of the Hamiltonian flow. It is found that the solutions of the JLC equation are closely resembling the solutions of the standard tangent dynamics equation which is used to compute Lyapunov exponents. Therefore the instability exponents obtained through the JLC equation are in perfect quantitative agreement with usual Lyapunov exponents. This work completes a previous investigation that was limited only to two degrees of freedom systems. © 1997 The American Physical Society
Lyapunov exponents from geodesic spread in configuration space
The exact form of the Jacobi -- Levi-Civita (JLC) equation for geodesic
spread is here explicitly worked out at arbitrary dimension for the
configuration space manifold M_E = {q in R^N | V(q) < E} of a standard
Hamiltonian system, equipped with the Jacobi (or kinetic energy) metric g_J. As
the Hamiltonian flow corresponds to a geodesic flow on (M_E,g_J), the JLC
equation can be used to study the degree of instability of the Hamiltonian
flow. It is found that the solutions of the JLC equation are closely resembling
the solutions of the standard tangent dynamics equation which is used to
compute Lyapunov exponents. Therefore the instability exponents obtained
through the JLC equation are in perfect quantitative agreement with usual
Lyapunov exponents. This work completes a previous investigation that was
limited only to two-degrees of freedom systems.Comment: REVTEX file, 10 pages, 2 figure
On the clustering phase transition in self-gravitating N-body systems
The thermodynamic behaviour of self-gravitating -body systems has been
worked out by borrowing a standard method from Molecular Dynamics: the time
averages of suitable quantities are numerically computed along the dynamical
trajectories to yield thermodynamic observables. The link between dynamics and
thermodynamics is made in the microcanonical ensemble of statistical mechanics.
The dynamics of self-gravitating -body systems has been computed using two
different kinds of regularization of the newtonian interaction: the usual
softening and a truncation of the Fourier expansion series of the two-body
potential. particles of equal masses are constrained in a finite three
dimensional volume. Through the computation of basic thermodynamic observables
and of the equation of state in the plane, new evidence is given of the
existence of a second order phase transition from a homogeneous phase to a
clustered phase. This corresponds to a crossover from a polytrope of index
, i.e. , to a perfect gas law , as is shown by
the isoenergetic curves on the plane. The dynamical-microcanonical
averages are compared to their corresponding canonical ensemble averages,
obtained through standard Monte Carlo computations. A major disagreement is
found, because the canonical ensemble seems to have completely lost any
information about the phase transition. The microcanonical ensemble appears as
the only reliable statistical framework to tackle self-gravitating systems.
Finally, our results -- obtained in a ``microscopic'' framework -- are compared
with some existing theoretical predictions -- obtained in a ``macroscopic''
(thermodynamic) framework: qualitative and quantitative agreement is found,
with an interesting exception.Comment: 19 pages, 20 figure
Weak and strong chaos in Fermi-Pasta-Ulam models and beyond
We briefly review some of the most relevant results that our group obtained in the past, while investigating the dynamics of the Fermi-Pasta-Ulam (FPU) models. The first result is the numerical evidence of the existence of two different kinds of transitions in the dynamics of the FPU models: (i) A stochasticity threshold (ST), characterized by a value of the energy per degree of freedom below which the overwhelming majority of the phase space trajectories are regular (vanishing Lyapunov exponents). It tends to vanish as the number N of degrees of freedom is increased. (ii) A strong stochasticity threshold (SST), characterized by a value of the energy per degree of freedom at which a crossover appears between two different power laws of the energy dependence of the largest Lyapunov exponent, which phenomenologically corresponds to the transition between weak and strong chaotic regimes. It is stable with N. The second result is the development of a Riemannian geometric theory to explain the origin of Hamiltonian chaos. Starting this theory has been motivated by the inadequacy of the approach based on homoclinic intersections to explain the origin of chaos in systems of arbitrarily large N, or arbitrarily far from quasi-integrability, or displaying a transition between weak and strong chaos. Finally, the third result stems from the search for the transition between weak and strong chaos in systems other than FPU. Actually, we found that a very sharp SST appears as the dynamical counterpart of a thermodynamic phase transition, which in turn has led, in the light of the Riemannian theory of chaos, to the development of a topological theory of phase transitions. (C) 2005 American Institute of Physics
The gas turbulence in planetary nebulae: quantification and multi-D maps from long-slit, wide-spectral range echellogram
This methodological paper is part of a short series dedicated to the
long-standing astronomical problem of de-projecting the bi-dimensional,
apparent morphology of a three-dimensional distribution of gas. We focus on the
quantification and spatial recovery of turbulent motions in planetary nebulae
(and other classes of expanding nebulae) by means of long-slit echellograms
over a wide spectral range. We introduce some basic theoretical notions,
discuss the observational methodology, and develop an accurate procedure
disentangling all broadening components of the velocity profile in all spatial
positions of each spectral image. This allows us to extract random, non-thermal
motions at unprecedented accuracy, and to map them in 1-, 2- and 3-dimensions.
We present the solution to practical problems in the multi-dimensional
turbulence-analysis of a testing-planetary nebula (NGC 7009), using the
three-step procedure (spatio-kinematics, tomography, and 3-D rendering)
developed at the Astronomical Observatory of Padua. In addition, we introduce
an observational paradigm valid for all spectroscopic parameters in all classes
of expanding nebulae. Unsteady, chaotic motions at a local scale constitute a
fundamental (although elusive) kinematical parameter of each planetary nebula,
providing deep insights on its different shaping agents and mechanisms, and on
their mutual interaction. The detailed study of turbulence, its stratification
within a target and (possible) systematic variation among different sub-classes
of planetary nebulae deserve long-slit, multi-position angle, wide-spectral
range echellograms containing emissions at low-, medium-, and high-ionization,
to be analyzed pixel-to-pixel with a straightforward and versatile methodology,
extracting all the physical information stored in each frame at best.Comment: 11 page, 10 figures, A&A in pres
UV (IUE) spectra of the central stars of high latitude planetary nebulae Hb7 and Sp3
We present an analysis of the UV (IUE) spectra of the central stars of Hb7
and Sp3. Comparison with the IUE spectrum of the standard star HD 93205 leads
to a spectral classification of O3V for these stars, with an effective
temperature of 50,000 K. From the P-Cygni profiles of CIV (1550 A), we derive
stellar wind velocities and mass loss rates of -1317 km/s +/- 300 km/s and
2.9X10^{-8} solar mass yr^{-1} and -1603 km/s +/- 400 km/s and 7X10^{-9} solar
mass yr^{-1} for Hb7 and Sp3 respectively. From all the available data, we
reconstruct the spectral energy distribution of Hb7 and Sp3.Comment: 4 pages, 3 figures, latex, accepted for publication in Astronomy &
Astrophysic
Very Late Thermal Pulses Influenced by Accretion in Planetary Nebulae
We consider the possibility that a mass of ~10^{-5}-10^{-3} Msun flows back
from the dense shell of planetary nebulae and is accreted by the central star
during the planetary nebula phase. This backflowing mass is expected to have a
significant specific angular momentum even in (rare) spherical planetary
nebulae, such that a transient accretion disk might be formed. This mass might
influence the occurrence and properties of a very late thermal pulse (VLTP),
and might even trigger it. For example, the rapidly rotating outer layer, and
the disk if still exist, might lead to axisymmetrical mass ejection by the
VLTP. Unstable burning of accreted hydrogen might result in a mild flash of the
hydrogen shell, also accompanied by axisymmetrical ejection.Comment: Submitted to New Astronom
Dynamical temperature study for classical planar spin systems
In this micro-canonical simulation the temperature and also the specific heat
are determined as averages of expressions easy to implement. The XY-chain is
studied for a test. The second order transition on a cubic lattice and the
first order transition on an fcc lattice are analyzed in greater detail to have
a more severe test about the feasibility of this micro-canonical method.Comment: 9 pages in Latex(revtex), 7 PS-figure
- …