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 {Σv:=VN−1(v)}v∈R of the potential function VNVN and the configuration space submanifolds enclosed by the ΣvΣv defined by {Mv:=VN−1((−∞,v])}v∈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 {Σv}v∈R{Σv}v∈R, as well as of the {Mv}v∈R{Mv}v∈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.