On the Singularity of the Free Energy at a First Order Phase Transition

At first order phase transition the free energy does not have an analytic continuation in the thermodynamical variable, which is conjugate to an order parameter for the transition. This result is proved at low temperature for lattice models with finite range interaction and two periodic ground-states, under the only condition that they satisfy the Peierls conditio

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

Rigorous Analysis of Singularities and Absence of Analytic Continuation at First Order Phase Transition Points in Lattice Spin Models

We report about two new rigorous results on the non-analytic properties of thermodynamic potentials at first order phase transition. The first one is valid for lattice models ($d\geq 2$) with arbitrary finite state space, and finite-range interactions which have two ground states. Under the only assumption that the Peierls Condition is satisfied for the ground states and that the temperature is sufficiently low, we prove that the pressure has no analytic continuation at the first order phase transition point. The second result concerns Ising spins with Kac potentials $J_\gamma(x)=\gamma^d\phi(\gamma x)$, where $0<\gamma<1$ is a small scaling parameter, and $\phi$ a fixed finite range potential. In this framework, we relate the non-analytic behaviour of the pressure at the transition point to the range of interaction, which equals $\gamma^{-1}$. Our analysis exhibits a crossover between the non-analytic behaviour of finite range models ($\gamma>0$) and analyticity in the mean field limit ($\gamma\searrow 0$). In general, the basic mechanism responsible for the appearance of a singularity blocking the analytic continuation is that arbitrarily large droplets of the other phase become stable at the transition point.Comment: 4 pages, 2 figure

Statistical Mechanics of Lattice Systems: a Concrete Mathematical Introduction

This motivating textbook gives a friendly, rigorous introduction to fundamental concepts in equilibrium statistical mechanics, covering a selection of specific models, including the Curie–Weiss and Ising models, the Gaussian free field, O(n) models, and models with Kać interactions. Using classical concepts such as Gibbs measures, pressure, free energy, and entropy, the book exposes the main features of the classical description of large systems in equilibrium, in particular the central problem of phase transitions. It treats such important topics as the Peierls argument, the Dobrushin uniqueness, Mermin–Wagner and Lee–Yang theorems, and develops from scratch such workhorses as correlation inequalities, the cluster expansion, Pirogov–Sinai Theory, and reflection positivity. Written as a self-contained course for advanced undergraduate or beginning graduate students, the detailed explanations, large collection of exercises (with solutions), and appendix of mathematical results and concepts also make it a handy reference for researchers in related areas