On the non-analytic behaviour of thermodynamic potentials at first order phase transitions

This work is devoted to the study of the analyticity properties of thermodynamic potentials (free energy, pressure) for classical lattice systems at low temperature. The central topic of our analysis, in this framework, is to show rigorously the absence of analytic continuation at points of first order phase transition. Our first result applies to the general class of two phase models considered in the Theory of Pirogov-Sinai. The analysis reveals that the Peirls condition, which is the basic hypothesis of the theory, suffices to show the absence of analytic continuation of the pressure at the transition point. In a second part, we study a particular two body potential, of the form γdJ(γx), where γ > 0 is a small parameter and J a function with bounded support (in the limit γ —> 0, this potential gives a rigorous justification of the "equal area rule" of the van der Waals-Maxwell Theory). For all small strictly positive values of the parameter γ, we show that the free energy has no analytic continuation at the transition points. These results confirm early conjectures stating that the finiteness of the range of interaction is responsible for the presence of singularities in the thermodynamic potentials.