Fuite de la métastabilité pour dynamiques stochastiques conservatives

We study the escape from metastability for a gas of particles under the conservative Kawasaki dynamics at low temperature and inside a two-dimensional box with exponentially large volume in the inverse temperature. We first analyse the typical trajectories followed by the system, in the local version of the model, during the first transition between metastability and stability. We describe geometrically the configurations along these typical trajectories and we show that the whole evolution goes, with very large probability, from “quasisquares” to larger “quasi-squares” and that the growing cluster of this nucleation process “wanders” around the box which it is contained in. In addition, along these trajectories, the fluctuations of the dimensions of the cluster are bounded: if a rectangle l×L circumscribes one of these clusters then L−l 1+2pL. We show that fluctuations of this order cannot be neglected: they take place with probabilities that are “non exponentially small” in . As a consequence the process is very different from the typical nucleation of the non-conservative Glauber dynamics, especially as far as the supercritical part is concerned. Then we prove a property of planar random walks which allows to extend the results obtained for the local version of the model to the original Kawasaki dynamics. We give a lower bound of the non-collision probability in a long tine T for a system of n random walks with fixed obstacles. By “collision” we mean collision with the fixed obstacles as well between the particles themselves. We explain how this property allows to describe in terms of “Quasi Random Walks” a rarefied gas of particles under the Kawasaki dynamics. On the basis of these results we can predict the main features of the escape from metastability for the original Kawasaki dynamics.