MARK KAC SEMINAR

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# Archive 2003 - 2004

 The seminar of 2003-2004: Main speaker
 October 10 Main speaker 2003-2004: Our main speaker for this year is: Charles-Edouard Pfister  On the Nature of Isotherms at First Order Phase Transitions.  See lecture notes in PDF-format (Institute of Analysis and Scientific Computing, School of Mathematics, EPF-L, CH-1015 Lausanne) The first theory of condensation originated with the celebrated equation of state of van der Waals.  (p+a*v^{-2})(v-b)=RT.  When complemented with the Maxwell Construction (``equal area rule'') it leads to isotherms describing general characteristics of the liquid-vapor equilibrium. The isotherms obtained with the van der Waals-Maxwell Theory have a very simple analytic structure: they are analytic in a pure phase and have analytic continuations along the liquid and gas branches, through the transition points. These analytic continuations were originally interpreted as describing the pressure of metastable states. The theoretical question of knowing whether the predictions of the van der Waals Theory can be derived from first principles of Statistical Mechanics remained an open important problem during a large part of the twentieth century. The first rigorous result was the study of Isakov [3] on the Ising model, which confirmed the impossibility of an analytic continuation at low enough temperatures. It was later generalized in [4].  I shall present recent results about this question, which have been obtained by Sacha Friedli and myself. In [1] we consider 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 first order phase transition points. In [2] we consider Ising models with Kac potentials Jγ(x)=γ^dφ(γ x), in the limit when γ tends to 0. Our analysis exhibits a crossover between the non-analytic behaviour of finite range models (γ>0) and analyticity in the mean field limit for γ to 0. The first lecture will be devoted to the history of the problem,  to a precise formulation of the results, as well as an exposition of Pirogov-Sinai Theory, which is the framework in which they are established.  The second lecture will be devoted to a detailed  proof of Isakov's theorem and its generalization [1].  In the last lecture the  results concerning the Kac limit γ to 0 will be presented [2].  I shall conclude with a discussion of important open problems. [1] Friedli S., Pfister C.-E., On the Singularity of the Free Energy at First Order Phase Transition, to appear in Commun. Math. Phys.  [2]  Friedli S., Pfister C.-E., Non-Analyticity and the van der Waals limit, to appear in J. Stat. Phys.  [3] Isakov S.N., Nonanalytic Features of the First Order Phase Transition in the Ising Model}, Commun. Math. Phys. 95, 427-443, (1984).   [4]  Isakov S.N., Phase Diagrams and Singularity at the Point of a Phase Transition of the First Kind in Lattice Gas Models, Teoreticheskaya i Matematicheskaya Fizika, 71, 426-440, (1987). November  7 November 28 February  6 March 5 March 26 May 7 V. Sidoravicius  F. Merkl