Topology and Phase Transitions I. Preliminary Results
Résumé
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 \in R} of the potential function V_N and the configuration space submanifolds enclosed by the \Sigma_{v} defined by {M_v := V_N^{-1} ((-\infty,v])}_{v \in 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 \in R}, as well as of the ${M_v}_{v \in R}, in an arbitrary interval of values for \vb=v/N, the Helmoltz 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.