Geometric approach to Hamiltonian dynamics and statistical mechanics

Abstract This paper is a review of results which have been recently obtained by applying mathematical concepts drawn, in particular, from differential geometry and topology, to the physics of Hamiltonian dynamical systems with many degrees of freedom of interest for statistical mechanics. The first part of the paper concerns the applications of methods used in classical differential geometry to study the chaotic dynamics of Hamiltonian systems. Starting from the identity between the trajectories of a dynamical system and the geodesics in its configuration space, when equipped with a suitable metric, a geometric theory of chaotic dynamics can be developed, which sheds new light on the origin of chaos in Hamiltonian systems. In fact, it appears that chaos can be induced not only by negative curvatures, as was originally surmised, but also by positive curvatures, provided the curvatures are fluctuating along the geodesics. In the case of a system with a large number of degrees of freedom it is possible to approximate the chaotic instability behaviour of the dynamics by means of a geometric model independent of the dynamics, which allows then an analytical estimate of the largest Lyapunov exponent in terms of the averages and fluctuations of the curvature of the configuration space of the system. In the second part of the paper the phenomenon of phase transitions is addressed and it is here that topology comes into play. In fact, when a system undergoes a phase transition, the fluctuations of the configuration-space curvature, when plotted as a function of either the temperature or the energy of the system, exhibit a singular behaviour at the phase transition point, which can be qualitatively reproduced using geometric models. In these models the origin of the singular behaviour of the curvature fluctuations appears to be caused by a topological transition in configuration space, which corresponds to the phase transition of the physical system. This leads us to put forward a topological hypothesis (TH). The content of the TH is that phase transitions would be related at a deeper level to a change in the topology of the configuration space of the system. We will illustrate this on a simple model, the mean-field XY model, where the TH can be checked directly and analytically. Since this model is of a rather special nature, namely a mean-field model with infinitely ranged interactions, we discuss other more realistic (non-mean-field-like) models, which cannot be solved analytically, but which do supply direct supporting evidence for the TH via numerical simulations.