Topology and Phase Transitions II. Theorem on a necessary relation
| dc.creator | Franzosi, Roberto | |
| dc.creator | Pettini, Marco | |
| dc.date | 2005-05-20 | |
| dc.date | 2006-04-18 | |
| dc.date.accessioned | 2026-07-07T08:28:50Z | |
| dc.date.available | 2026-07-07T08:28:50Z | |
| dc.description | 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. | |
| dc.description | 21 pages. This second paper follows up paper I archived in math-ph/0505057. Added minor changes: Title, Abstract, Introduction | |
| dc.identifier | https://arxiv.org/abs/math-ph/0505058 | |
| dc.identifier | http://arxiv.org/abs/math-ph/0505058 | |
| dc.identifier | Nuclear Physics B782 [PM], 219 (2007). | |
| dc.identifier | doi:10.1016/j.nuclphysb.2007.04.035 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/137762 | |
| dc.subject | Mathematical Physics | |
| dc.subject | Statistical Mechanics | |
| dc.subject | High Energy Physics - Theory | |
| dc.subject | 82B26; 82B03; 82B05; 58E05 | |
| dc.title | Topology and Phase Transitions II. Theorem on a necessary relation | |
| dc.type | text |