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 epsilon R, with M(v) = {q epsilon R(N) vertical bar V(N)(q) <= v} . On the basis of an analytic relationship between a suitably weighed sum of the Morse indexes of the manifolds {M(v) }(v epsilon 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 integral M(v), d(N)q, that is of broken vertical bar partial derivative(k)S((-))(v)/partial derivative v(k)vertical bar, fork = 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 ofits 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. (c) 2007 Elsevier B.V. All rights reserved.