Phase Transitions and Topology: Necessity Theorems

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.