Topology and phase transitions I. Preliminary results

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.