Phase space geometry and stochasticity thresholds in hamiltonian dynamics

Results of numerical computations of the largest Lyapunov exponent lambda(1)(varepsilon,N) as a function of the energy density varepsilon and the number of particles N are here reported for a Fermi-Pasta-Ulam alpha+beta model. These results show the coexistence at large N of two thresholds: a stochasticity threshold, found before for the alpha model alone, and a strong stochasticity threshold (SST), found before for the beta model alone. Although this coexistence may seem at first sight plausible, it is not obvious a priori that the alpha+beta model superimposes properties of the alpha and beta models independently. The main point of this paper, however, is a geometric characterization of the SST via the mean curvature of the constant energy hypersurfaces in the phase space of the model and the characteristic decay time of its time autocorrelation function tau(c)(varepsilon,N), which correlates with that of lambda(1)(varepsilon,N) for fixed N. This appears to provide important information on the very complicated geometry of the phase space of this simple solidlike model.

Hamiltonian dynamics and geometry of phase transitions in classical XY models

The Hamiltonian dynamics associated with classical, planar, Heisenberg XY models is investigated for two- and three-dimensional lattices. In addition to the conventional signatures of phase transitions, here obtained through time averages of thermodynamical observables in place of ensemble averages, qualitatively different information is derived from the temperature dependence of Lyapunov exponents. A Riemannian geometrization of Newtonian dynamics suggests consideration of other observables of geometric meaning tightly related to the largest Lyapunov exponent. The numerical computation of these observables--unusual in the study of phase transitions--sheds light on the microscopic dynamical counterpart of thermodynamics, also pointing to the existence of some major change in the geometry of the mechanical manifolds at the thermodynamical transition. Through the microcanonical definition of the entropy, a relationship between thermodynamics and the extrinsic geometry of the constant energy surfaces sigma E of phase space can be naturally established. In this framework, an approximate formula is worked out determining a highly nontrivial relationship between temperature and topology of sigma E. From this it can be understood that the appearance of a phase transition must be tightly related to a suitable major topology change of sigma E. This contributes to the understanding of the origin of phase transitions in the microcanonical ensemble.