Theorem on the Origin of Phase Transitions

For physical systems described by smooth, finite-range, and confining microscopic interaction potentials [Formula presented] with continuously varying coordinates, we announce and outline the proof of a theorem that establishes that, unless the equipotential hypersurfaces of configuration space [Formula presented], [Formula presented], change topology at some [Formula presented] in a given interval [Formula presented] of values [Formula presented] of [Formula presented], the Helmoltz free energy must be at least twice differentiable in the corresponding interval of inverse temperature [Formula presented] also in the [Formula presented] limit. Thus, the occurrence of a phase transition at some [Formula presented] is necessarily the consequence of the loss of diffeomorphicity among the [Formula presented] and the [Formula presented], which is the consequence of the existence of critical points of [Formula presented] on [Formula presented], that is, points where [Formula presented]. © 2004 The American Physical Society.