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 ${\Sigma }_v=\{(q_1,\dots ,q_N)\in {\mathbb{R}}^N|V(q_1,\dots ,q_N)=v\}$, $v\in \mathbb{R}$, change topology at some $v_c$ in a given interval $[v_0,v_1]$ of values $v$ of $V$, the Helmoltz free energy must be at least twice differentiable in the corresponding interval of inverse temperature $\mathbf{(}\beta (v_0),\beta (v_1)\mathbf{)}$ also in the $N\to \infty$ limit. Thus, the occurrence of a phase transition at some ${\beta }_c=\beta (v_c)$ is necessarily the consequence of the loss of diffeomorphicity among the $\{{\Sigma }_v{\}}_{v<v_c}$ and the $\{{\Sigma }_v{\}}_{v>v_c}$, which is the consequence of the existence of critical points of $V$ on ${\Sigma }_{v=v_c}$, that is, points where $\nabla V=0$.