531 research outputs found
Macroscopic detection of the strong stochasticity threshold in Fermi-Pasta-Ulam chains of oscillators
The largest Lyapunov exponent of a system composed by a heavy impurity
embedded in a chain of anharmonic nearest-neighbor Fermi-Pasta-Ulam oscillators
is numerically computed for various values of the impurity mass . A
crossover between weak and strong chaos is obtained at the same value
of the energy density (energy per degree of freedom)
for all the considered values of the impurity mass . The threshold \epsi
lon_{_T} coincides with the value of the energy density at which a
change of scaling of the relaxation time of the momentum autocorrelation
function of the impurity ocurrs and that was obtained in a previous work ~[M.
Romero-Bastida and E. Braun, Phys. Rev. E {\bf65}, 036228 (2002)]. The complete
Lyapunov spectrum does not depend significantly on the impurity mass . These
results suggest that the impurity does not contribute significantly to the
dynamical instability (chaos) of the chain and can be considered as a probe for
the dynamics of the system to which the impurity is coupled. Finally, it is
shown that the Kolmogorov-Sinai entropy of the chain has a crossover from weak
to strong chaos at the same value of the energy density that the crossover
value of largest Lyapunov exponent. Implications of this result
are discussed.Comment: 6 pages, 5 figures, revtex4 styl
Geometric dynamical observables in rare gas crystals
We present a detailed description of how a differential geometric approach to
Hamiltonian dynamics can be used for determining the existence of a crossover
between different dynamical regimes in a realistic system, a model of a rare
gas solid. Such a geometric approach allows to locate the energy threshold
between weakly and strongly chaotic regimes, and to estimate the largest
Lyapunov exponent. We show how standard mehods of classical statistical
mechanics, i.e. Monte Carlo simulations, can be used for our computational
purposes. Finally we consider a Lennard Jones crystal modeling solid Xenon. The
value of the energy threshold turns out to be in excellent agreement with the
numerical estimate based on the crossover between slow and fast relaxation to
equilibrium obtained in a previous work by molecular dynamics simulations.Comment: RevTeX, 19 pages, 6 PostScript figures, submitted to Phys. Rev.
Phase transitions as topology changes in configuration space: an exact result
The phase transition in the mean-field XY model is shown analytically to be
related to a topological change in its configuration space. Such a topology
change is completely described by means of Morse theory allowing a computation
of the Euler characteristic--of suitable submanifolds of configuration
space--which shows a sharp discontinuity at the phase transition point, also at
finite N. The present analytic result provides, with previous work, a new key
to a possible connection of topological changes in configuration space as the
origin of phase transitions in a variety of systems.Comment: REVTeX file, 5 pages, 1 PostScript figur
Geometry of dynamics, Lyapunov exponents and phase transitions
The Hamiltonian dynamics of classical planar Heisenberg model is numerically
investigated in two and three dimensions. By considering the dynamics as a
geodesic flow on a suitable Riemannian manifold, it is possible to analytically
estimate the largest Lyapunov exponent in terms of some curvature fluctuations.
The agreement between numerical and analytical values for Lyapunov exponents is
very good in a wide range of temperatures. Moreover, in the three dimensional
case, in correspondence with the second order phase transition, the curvature
fluctuations exibit a singular behaviour which is reproduced in an abstract
geometric model suggesting that the phase transition might correspond to a
change in the topology of the manifold whose geodesics are the motions of the
system.Comment: REVTeX, 10 pages, 5 PostScript figures, published versio
Geometric Approach to Lyapunov Analysis in Hamiltonian Dynamics
As is widely recognized in Lyapunov analysis, linearized Hamilton's equations
of motion have two marginal directions for which the Lyapunov exponents vanish.
Those directions are the tangent one to a Hamiltonian flow and the gradient one
of the Hamiltonian function. To separate out these two directions and to apply
Lyapunov analysis effectively in directions for which Lyapunov exponents are
not trivial, a geometric method is proposed for natural Hamiltonian systems, in
particular. In this geometric method, Hamiltonian flows of a natural
Hamiltonian system are regarded as geodesic flows on the cotangent bundle of a
Riemannian manifold with a suitable metric. Stability/instability of the
geodesic flows is then analyzed by linearized equations of motion which are
related to the Jacobi equations on the Riemannian manifold. On some geometric
setting on the cotangent bundle, it is shown that along a geodesic flow in
question, there exist Lyapunov vectors such that two of them are in the two
marginal directions and the others orthogonal to the marginal directions. It is
also pointed out that Lyapunov vectors with such properties can not be obtained
in general by the usual method which uses linearized Hamilton's equations of
motion. Furthermore, it is observed from numerical calculation for a model
system that Lyapunov exponents calculated in both methods, geometric and usual,
coincide with each other, independently of the choice of the methods.Comment: 22 pages, 14 figures, REVTeX
Kinematics in Kapteyn's Selected Area 76: Orbital Motions Within the Highly Substructured Anticenter Stream
We have measured the mean three-dimensional kinematics of stars in Kapteyn's
Selected Area (SA) 76 (l=209.3, b=26.4 degrees) that were selected to be
Anticenter Stream (ACS) members on the basis of their radial velocities, proper
motions, and location in the color-magnitude diagram. From a total of 31 stars
ascertained to be ACS members primarily from its main sequence turnoff, a mean
ACS radial velocity (derived from spectra obtained with the Hydra multi-object
spectrograph on the WIYN 3.5m telescope) of V_helio = 97.0 +/- 2.8 km/s was
determined, with an intrinsic velocity dispersion sigma_0 = 12.8 \pm 2.1 km/s.
The mean absolute proper motions of these 31 ACS members are mu_alpha cos
(delta) = -1.20 +/- 0.34 mas/yr and mu_delta = -0.78 \pm 0.36 mas/yr. At a
distance to the ACS of 10 \pm 3 kpc, these measured kinematical quantities
produce an orbit that deviates by ~30 degrees from the well-defined swath of
stellar overdensity constituting the Anticenter Stream in the western portion
of the Sloan Digital Sky Survey footprint. We explore possible explanations for
this, and suggest that our data in SA 76 are measuring the motion of a
kinematically cold sub-stream among the ACS debris that was likely a fragment
of the same infalling structure that created the larger ACS system. The ACS is
clearly separated spatially from the majority of claimed Monoceros ring
detections in this region of the sky; however, with the data in hand, we are
unable to either confirm or rule out an association between the ACS and the
poorly-understood Monoceros structure.Comment: Accepted to ApJ. 48 pages, 20 figures, preprint forma
Topological conditions for discrete symmetry breaking and phase transitions
In the framework of a recently proposed topological approach to phase
transitions, some sufficient conditions ensuring the presence of the
spontaneous breaking of a Z_2 symmetry and of a symmetry-breaking phase
transition are introduced and discussed. A very simple model, which we refer to
as the hypercubic model, is introduced and solved. The main purpose of this
model is that of illustrating the content of the sufficient conditions, but it
is interesting also in itself due to its simplicity. Then some mean-field
models already known in the literature are discussed in the light of the
sufficient conditions introduced here
Scaling laws for the largest Lyapunov exponent in long-range systems: A random matrix approach
We investigate the laws that rule the behavior of the largest Lyapunov
exponent (LLE) in many particle systems with long range interactions. We
consider as a representative system the so-called Hamiltonian alpha-XY model
where the adjustable parameter alpha controls the range of the interactions of
N ferromagnetic spins in a lattice of dimension d. In previous work the
dependence of the LLE with the system size N, for sufficiently high energies,
was established through numerical simulations. In the thermodynamic limit, the
LLE becomes constant for alpha greater than d whereas it decays as an inverse
power law of N for alpha smaller than d. A recent theoretical calculation based
on Pettini's geometrization of the dynamics is consistent with these numerical
results (M.-C. Firpo and S. Ruffo, cond-mat/0108158). Here we show that the
scaling behavior can also be explained by a random matrix approach, in which
the tangent mappings that define the Lyapunov exponents are modeled by random
simplectic matrices drawn from a suitable ensemble.Comment: 5 pages, no figure
On a microcanonical relation between continuous and discrete spin models
A relation between a class of stationary points of the energy landscape of
continuous spin models on a lattice and the configurations of a Ising model
defined on the same lattice suggests an approximate expression for the
microcanonical density of states. Based on this approximation we conjecture
that if a O(n) model with ferromagnetic interactions on a lattice has a phase
transition, its critical energy density is equal to that of the n = 1 case,
i.e., a system of Ising spins with the same interactions. The conjecture holds
true in the case of long-range interactions. For nearest-neighbor interactions,
numerical results are consistent with the conjecture for n=2 and n=3 in three
dimensions. For n=2 in two dimensions (XY model) the conjecture yields a
prediction for the critical energy of the Berezinskij-Kosterlitz-Thouless
transition, which would be equal to that of the two-dimensional Ising model. We
discuss available numerical data in this respect.Comment: 5 pages, no figure
Proper Motions in Kapteyn Selected Area 103: A Preliminary Orbit for the Virgo Stellar Stream
We present absolute proper motions in Kapteyn Selected Area (SA) 103. This
field is located 7 degrees west of the center of the Virgo Stellar Stream (VSS,
Duffau et al. 2006), and has a well-defined main sequence representing the
stream. In SA 103 we identify one RR Lyrae star as a member of the VSS
according to its metallicity, radial velocity and distance. VSS candidate
turnoff stars and subgiant stars have proper motions consistent with that of
the RR Lyrae star. The 3D velocity data imply an orbit with a pericenter of 11
kpc and an apocenter of ~90 kpc. Thus, the VSS comprises tidal debris found
near the pericenter of a highly destructive orbit. Examining the six globular
clusters at distances larger than 50 kpc from the Galactic center, and the
proposed orbit of the VSS, we find one tentative association, NGC 2419. We
speculate that NGC 2419 is possibly the nucleus of a disrupted system of which
the VSS is a part.Comment: ApJL accepte
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