Topology and phase transitions I. Preliminary results

Year: 2007

Authors: Franzosi R., Pettini M., Spinelli. L.

Autors Affiliation: [1] Dipartimento di Fisica dell’Università di Firenze, Via G. Sansone 1, I-50019 Sesto Fiorentino, Italy
[1] C.N.R. – Istituto Nazionale per la Fisica della Materia, Firenze, Italy
[2] Istituto Nazionale di Astrofisica – Osservatorio Astrofisico di Arcetri, Largo E. Fermi 5, 50125 Firenze, Italy
d Istituto Nazionale di Fisica Nucleare, Sezione di Firenze, Italy
[3] Centre de Physique Théorique du C.N.R.S., Luminy Case 907, F-13288 Marseille cedex 9, France

Abstract: In this first paper, we demonstrate a theorem that establishes a first step toward proving a necessary topological condition for the occurrence of first- or second-order phase transitions: we prove that the topology of certain submanifolds of configuration space must necessarily change at the phase transition point. The theorem applies to smooth, finite-range and confining potentials V bounded below, describing systems confined in finite regions of space with continuously varying coordinates. The relevant configuration space submanifolds are both the level sets {Sigma(v) := V(N)(-1) (v) v epsilon R} of the potential function V(N) and the configuration space submanifolds enclosed by the Sigma v, defined by {Mv := V(N)(-1) ((-infinity, v])}(v epsilon R), which are labeled by the potential energy value v, and where N is the number of degrees of freedom. The proof of the theorem proceeds by showing that, under the assumption of diffeomorphicity of the equipotential hypersurfaces {Sigma v}(v epsilon R) as well as of the {Mv}(v epsilon R) in an arbitrary interval of values for v = v/N, the Helmholtz free energy is uniformly convergent in N to its thermodynamic limit, at least within the class of twice differentiable functions, in the corresponding interval of temperature. This preliminary theorem is essential to prove another theorem-in paper II-which makes a stronger statement about the relevance of topology for phase transitions. (c) 2007 Elsevier B.V. All rights reserved.

Journal/Review: NUCLEAR PHYSICS B

Volume: 782 (3)      Pages from: 189  to: 218

KeyWords: Phase transitions; Statistical mechanics; Topology
DOI: 10.1016/j.nuclphysb.2007.04.025

Citations: 37
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