Nonanalyticities of thermodynamic functions are studied by adopting an approach based on stationary points of the potential energy. For finite systems, each stationary point is found to cause a nonanalyticity in the microcanonical entropy, and the functional form of this nonanalytic term is derived explicitly. With increasing system size, the order of the nonanalytic term grows, leading to an increasing differentiability of the entropy. It is found that only asymptotically flat stationary points may cause a nonanalyticity that survives in the thermodynamic limit, and this property is used to derive an analytic criterion establishing the existence or absence of phase transitions. We sketch how this result can be employed to analytically compute transition energies of classical spin models.
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