Topology and Phase Transitions II. Theorem on a necessary relation
  In this second paper, we prove a necessity Theorem about the topological origin of phase transitions. We consider physical systems described by smooth microscopic interaction potentials V_N(q), among N degrees of freedom, and the associated family of configuration space submanifolds {M_v}_{v \in R}, with M_v={q \in R^N | V_N(q) \leq v}. On the basis of an analytic relationship between a suitably weighed sum of the Morse indexes of the manifolds {M_ v}_{v \in R} and thermodynamic entropy, the Theorem states that any possible unbound growth with N of one of the following derivatives of the configurational entropy S^{(-)}(v)=(1/N) \log \int_{M_v} d^Nq, that is of |\partial^k S^{(-)}(v)/\partial v^k|, for k=3,4, can be entailed only by the weighed sum of Morse indexes.   Since the unbound growth with N of one of these derivatives corresponds to the occurrence of a first or of a second order phase transition, and since the variation of the Morse indexes of a manifold is in one-to-one correspondence with a change of its topology, the Main Theorem of the present paper states that a phase transition necessarily stems from a topological transition in configuration space. The proof of the Theorem given in the present paper cannot be done without Main Theorem of paper I. 
Roberto Franzosi (1) and Marco Pettini (2) ((1) Dipartimento di   Fisica, CNR-INFM, Universita' di Firenze, Italy, (2) INAF ...