Topology and phase transitions II. Theorem on a necessary relation

Abstract

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 ∈ R , with M v = { q ∈ R N | 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 ∈ 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 ∫ M v d N q , that is of | ∂ k S ( − ) ( v ) / ∂ 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.