Geometry and Topology in Hamiltonian Dynamics and Statistical Mechanics

This book reports on an unconventional explanation of the origin of chaos in Hamiltonian dynamics and on a new theory of the origin of thermodynamic phase transitions. The mathematical concepts and methods used are borrowed from Riemannian geometry and from elementary differential topology, respectively. The new approach proposed also unveils deep connections between the two mentioned topics.

In this first chapter we will give an outline of some fundamental elements of statistical mechanics, of Hamiltonian dynamics, and of the relationship between them.

The general problem of statistical physics is the following. Given a collection–in general a large collection–of atoms or molecules, given the interaction laws among the constituents of this collection of particles, and given the dynamical evolution laws, how can we predict the macroscopic physical properties of the matter composed of these atoms or molecules?

The problem of integrability in classical mechanics has been a seminal one. Motivated by celestial mechanics, it has stimulated a wealth of analytical methods and results. For example, as we have discussed in Chapter 2, the weaker requirement of only approximate integrability over finite times, or the existence of integrable regions in the phase space of a globally nonintegrable system, has led to the development of classical perturbation theory, with all its important achievements. However, deciding whether a given Hamiltonian system is globally integrable still remains a difficult task, for which a general constructive framework is lacking.

The purpose of the present chapter is to describe in some detail how it is possibile, using the Jacobi–Levi-Civita equation for geodesic spread as the main tool, to reach a twofold objective: first, to obtain a deeper understanding of the origin of chaos in Hamiltonian systems, and second, to obtain quantitative information on the “strength” of chaos in these systems.

In the previous chapter we have reported results of numerical simulations for the fluctuations of observables of a geometric nature (e.g., configurationspace curvature fluctuations) related to the Riemannian geometrization of the dynamics in configuration space.1 These quantities have been computed, using time averages, for many different models undergoing continuous phase transitions, namely φ⁴ lattice models with discrete and continuous symmetries and XY models. In particular, when plotted as a function of either the temperature or the energy, the fluctuations of the curvature have an apparently singular behavior at the transition point. Moreover, we have seen that the presence of a singularity in the statistical-mechanical fluctuations of the curvature at the transition point has been proved analytically for the mean-field XY model.

In the preceding chapter we have seen that configuration-space topology is suspected to play a significant role in the emergence of phase transition phenomena. We have summarized all the clues in the form of a working hypothesis that we called the topological hypothesis. Then this has been given strong support by a direct numerical investigation of the topological changes of configuration space of 1D and 2D lattice φ⁴ models. This conjecture stems from the peculiar energy density patterns of the largest Lyapunov exponent at phase transition points. In fact, Lyapunov exponents are closely related to configuration space geometry, which, in turn, can be strongly influenced by topology. However, there is another argument, independent of the Riemannian geometrization of Hamiltonian dynamics, that suggests how to make another link between Lyapunov exponents and topology.

In the preceding chapters, we discussed the conceptual development that, starting from the Riemannian theory of Hamiltonian chaos, led us first to conjecture the involvement of topology in phase transition phenomena— formulating what we called the topological hypothesis—and then provided both indirect and direct numerical evidence of this conjecture. The present chapter contains a major leap forward: the rigorous proof that topological changes of equipotential hypersurfaces of configuration space—and of the regions of con- figuration space bounded by them—are a necessary condition for the appearance of thermodynamic phase transitions. This is obtained for a wide class of potential functions of physical relevance, and for first- and second-order phase transitions. However, long-range interactions, nonsmooth potentials, unbound configuration spaces, “exotic” and higher-order phase transitions, are not encompassed by the theorems given below and are still open problems deserving further work. For this reason, and mainly because we do not yet know precisely what kinds of topological changes entail a phase transition, we give in what follows the details of the proofs, making the presentation of the content of this chapter rather formal. We deem it useful to provide these details in order, we hope, to inspire and stimulate the interested reader to cope with these challenging tasks.

The preceding chapter contains a major theoretical achievement: the unbounded growth with N of certain thermodynamic observables, eventually leading to singularities in the N → ∞limit, which are used to define the occurrence of an equilibrium phase transition, is necessarily due to appropriate topological transitions in configuration space. The relevance of topology is made especially clear by the explicit dependence of thermodynamic configurational entropy on a weighed sum of Morse indexes of configuration-space submanifolds, a relation that, loosely speaking, has some analogy with the Yang–Lee “circle theorem,” which relates thermodynamic observables to a fundamental mathematical object in the Yang–Lee theory of phase transitions: the angular distribution of the zeros of the grand-partition function on a circle in the complex fugacity plane.

In fact, the Riemannian theory of Hamiltonian chaos, though still formulated at a somewhat primitive level (in that it does not yet include the role of nontrivial topology of the mechanical manifolds), provides a natural explanation of the origin of the chaotic instability of classical dynamics, substantially in the absence of competing theories.