Theorem on the Origin of Phase Transitions

For physical systems described by smooth, finite-range, and confining microscopic interaction potentials V with continuously varying coordinates, we announce and outline the proof of a theorem that establishes that, unless the equipotential hypersurfaces of configuration space Σ_v={(q₁,…,q_N)∈R^N|V(q₁,…,q_N)=v}, v∈R, change topology at some v_c in a given interval [v₀,v₁] of values v of V, the Helmoltz free energy must be at least twice differentiable in the corresponding interval of inverse temperature (β(v₀),β(v₁)) also in the N→∞ limit. Thus, the occurrence of a phase transition at some β_c=β(v_c) is necessarily the consequence of the loss of diffeomorphicity among the {Σ_v}_v<vc and the {Σ_v}_v>vc, which is the consequence of the existence of critical points of V on Σ_v=vc, that is, points where ∇V=0.