Hamiltonian dynamics and geometry of phase transitions in classical XY models

The Hamiltonian dynamics associated with classical, planar, Heisenberg XY m odels 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 depend ence of Lyapunov exponents. A Riemannian geometrization of Newtonian dynami cs suggests consideration of other observables of geometric meaning tightly related to the largest Lyapunov exponent. The numerical computation of the se 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 geome try of the constant energy surfaces Sigma(E) of phase space can be naturall y established. In this framework, an approximate formula is worked out dete rmining a highly nontrivial relationship between temperature and topology o f Sigma(E). From this it can be understood that the appearance of a phase t ransition must be tightly related to a suitable major topology change of Si gma(E). This contributes to the understanding of the origin of phase transi tions in the microcanonical ensemble.