Topology and Phase Transitions II. Theorem on a necessary relation

Details Topology and Phase Transitions II. Theorem on a necessary relation Topology and Phase Transitions II. Theorem on a necessary relation

2005-05-20

arxiv.org/abs/math-ph/0505058v2

Nuclear Physics B782 [PM], 219 (2007).

Physics

Mathematical Physics

21 pages. This second paper follows up paper I archived in math-ph/0505057. Added minor changes: Title, Abstract, Introduction

Scientific paper

10.1016/j.nuclphysb.2007.04.035

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.

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