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, 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 give an analytical estimate of the largest Lyapunov exponent by means of a geometric model independent of the dynamics. 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 exhibit a singular behavior at the phase transition point, which can be qualitatively reproduced using geometric models. In these models the origin of the singular behavior of the curvature fluctuations appears to be caused by a topological transition in configuration space. 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.Comment: REVTeX, 81 pages, 36 ps/eps figures (some low-quality figures to save space); review article submitted to Physics Report

Dynamics of Oscillator Chains

“…The scalings of λ with ε can now be determined, analytically, by considering the geometry of the phase space near equipartition. Making suitable assumptions about the geometry of mechanical manifolds, the scaling of λ with ε, both below and above the SST transition and the value of ε at the transition has been theoretically calculated in agreement with numerical findings [42,60,61]. Although the method was developed to understand the FPU-β scaling, it is applicable to oscillator chains with various force laws, as can be found in the referenced works.…”

“…The last integral indicates that if the configuration space M of a system with N degrees of freedom is given a proper Riemannian structure by introducing the metric [42,145] …”

“…However, the advent of computers has been of invaluable help. As a matter of fact, during the last few years an interplay between analytic methods and numerical simulation has made it possible to overcome the difficulties, showing that the Riemannian geometric approach can be applied to dynamical systems of interest to statistical mechanics, field theory, and condensed matter physics [42]. This has extended the domain of application of geometric techniques, and has also introduced a new point of view about the origin of chaos in Hamiltonian systems, as well as new methods to describe and understand it, "new" in a sense that will be made clear in Sect.…”

“…Particularly noticeable is the fact that the analytic values of λ check with the numerical ones within errors of few percent in a range of six orders of magnitude both in ε and λ, with no adjustable parameters. The analysis follows [60] and has been reviewed in [42].…”

“…2.8.2. A complete exposition of the geometric method and its specific application to the FPU-β has been given in a review [42]. From the definition in (2.5), the Lyapunov exponent is determined by a long-time average, and is thus applicable from any initial condition.…”