Errico Presutti (Roma): Introduction

Amine Asselah (Marseille) : Exit problem from metastable states for mean field conservative dynamics.

We consider an exclusion particle system with long range mean field type interactions at temperature T. The hydrodynamic limit of such a system is given by an integrodifferential equation with one conservation law on the circle C: it is the gradient flux of the Kac free energy functional F_T.
For T >= 1, any constant function with value m in [-1, +1] is the global minimizer of F_T in the space of functions on C with mean m.
For T < 1, F_T restricted to the space of functions on C with mean m may have several local minima: in particular the constant solution may not be the absolute minimizer of F_T.
We therefore study the long time behavior of the particle system when the initial condition is close to a homogeneous stable state, giving results on the time of exit from (suitable) subsets of its domain of attraction. We follow the Freidlin-Wentzell approach : first, we study in details F_T together with the time asymptotics of the solution of the hydrodynamic equation; then, we study the probability of rare events for the particle system, i.e. Large Deviations from the hydrodynamic limit.

Lorenzo Bertini (Roma): .

Thierry Bodineau (Paris): Phase coexistence for a finite range Kac Ising model.

In the recent years, the Kac Ising model played a pivotal role for the rigorous microscopic derivation of the phase coexistence in three dimensional particle systems. In particular, the basic philosophy of the description of phase segregation in terms of L¹ estimates was initiated in the context of the Kac Ising model.

We first recall the main features of the model (mean field functional, coarse graining, ...) and explain why they fit so well with the L¹ approach of phase coexistence.

In a second part, a recent result on the Wulff construction for long but finite range interaction Kac Ising model will be presented. As a conclusion, some future prospects about the liquid/vapor phase coexistence will be discussed.

Joel L. Lebowitz (Rutgers): Why van der Waals?: Behavior and relevance of systems with long range potentials

Oliver Penrose (Edinburgh): Statistical mechanics of nonlinear elasticity and some related solid-solid phase transitions.

I will discuss how to define free energy in microscopic terms for a deformed elastic solid and the possibility of proving the existence of solid-solid phase transitions in suitable cases.

Livio Triolo (Roma): Free energy functionals and their critical points in the mesoscopic description of phase transitions.

The role of the free energy functional F in two models of phase transition at the mesoscopic level is examined. These models are both Kac-type, namely a ferromagnetic spin system and a particle system in the continuum.

In the mesoscopic limit, i.e. sending the microscopic range of the interaction to infinity and, in the same time, scaling space so that we get a finite range in this limit, we study the interface structure looking at the critical points of the corresponding functionals. An important point in this analysis is the introduction of suitable dynamics which have F as Lyapunov functional.

Milos Zahradník (Praha): Cluster expansions and Pirogov Sinai theory for long range spin systems of Kac Ising type.

We develop the Pirogov Sinai theory for lattice spin systems with interactions of Kac type. There are two essential steps in our approach:

1) We develop cluster expansion (essentially of high temperature type) of suitably defined ``restricted ensembles''. The latter are defined as ensembles of almost constant configurations, with low density perturbations.

2) Representing any configuration of the model as a collection of contours and configurations from restricted ensembles (around the contours) we transcript the problem into the language of an ``abstract Pirogov Sinai model'' with ``matching compatible'' contours and with an ``external cluster field''. Then we describe the appropriate P.S. machinery needed to treat such a problem. Though there is of course some similarity in the ideas, technically our approach is very different from that used in the work of Lebowitz, Mazel and Presutti.

(based on joint work with Anton Bovier)