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 VN(q), among N degrees of freedom, and the associated family of configuration space submanifolds {Mv}v∈R, with Mv={q∈RN|VN(q)⩽v}. On the basis of an analytic relationship between a suitably weighed sum of the Morse indexes of the manifolds {Mv}v∈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∫MvdNq, that is of |∂kS(−)(v)/∂vk|, 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.