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

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

On the approach to equilibrium of an Hamiltonian chain of anharmonic oscillators

In this note we study the approach to equilibrium of a chain of anharmonic oscillators. We find indications that a sufficiently large system always relaxes to the usual equilibrium distribution. There is no sign of an ergodicity threshold. The time however to arrive to equilibrium diverges when $g \to 0$, $g$ being the anharmonicity.Comment: 8 pages, 5 figure

Semi-Analytic Estimates of Lyapunov Exponents in Lower-Dimensional Systems

Recent work has shown that statistical arguments, seemingly well-justified in higher dimensions, can also be used to derive reasonable, albeit less accurate, estimates of the largest Lyapunov exponent ${\chi}$ in lower-dimensional Hamiltonian systems. This letter explores the detailed assumptions incorporated into these arguments. The predicted values of ${\chi}$ are insensitive to most of these details, which can in any event be relaxed straightforwardly, but {\em can} depend sensitively on the nongeneric form of the auto-correlation function characterising the time-dependence of an orbit. This dependence on dynamics implies a fundamental limitation to the application of thermodynamic arguments to such lower-dimensional systems.Comment: 6 pages, 3 PostScript figure

TuST: from Raw Data to Vehicular Traffic Simulation in Turin

Traffic simulations are becoming a standard way to study urban mobility patterns, to evaluate new traffic policies and to test modern vehicular technologies. For this reason, in recent years, mobility projects pushed towards an increase in the demand of traffic simulators and towards an extension of their area of investigation, aiming at covering a whole city and its suburbs. In this paper we describe the methodology we followed in the creation of a large-scale traffic simulation of a 400-Km^2 area around the Municipality of Turin. Our preliminary results demonstrate that a complete modeling of such a wide tool is possible at the expense of minor simplifications

The Southern Proper Motion Program IV. The SPM4 Catalog

We present the fourth installment of the Yale/San Juan Southern Proper Motion Catalog, SPM4. The SPM4 contains absolute proper motions, celestial coordinates, and (B,V) photometry for over 103 million stars and galaxies between the south celestial pole and -20 deg declination. The catalog is roughly complete to V=17.5 and is based on photographic and CCD observations taken with the Yale Southern Observatory's double-astrograph at Cesco Observatory in El Leoncito, Argentina. The proper-motion precision, for well-measured stars, is estimated to be 2 to 3 mas/yr, depending on the type of second-epoch material. At the bright end, proper motions are on the International Celestial Reference System by way of Hipparcos Catalog stars, while the faint end is anchored to the inertial system using external galaxies. Systematic uncertainties in the absolute proper motions are on the order of 1 mas/yr.Comment: 34 pages, 8 figures, 3 tables; accepted for publication in AJ; note - modified author list and acknowledgements sectio