Jean-Pierre Eckmann (Genève): Non-equilibrium statistical mechanics of a chain coupled to two heat baths

Giovanni Gallavotti (Roma): Une revue de l'hypothèse chaotique et l'équivalence des ensembles hors équilibre

L'hypothèse chaotique permet d'identifier la distributuion de probabilité à utiliser pour calculer les moyennes des observables dans les états stationnaires hors équilibre. Quelques applications ont été proposées mais elles concernent des systèmes réversibles. Pour développer la théorie il semble utile de comprendre comment traiter des systèmes à évolution irreversible: on examinera des conjectures qui suggèrent qu'il y ait équivalence entre évolutions réversibles et irréversibles (bien entendu sous certaines conditions). On va suggérer que hors équilibre les équations du mouvement soient parmi les quantités qui caracterisent un ensemble statistique et que la dite équivalence est une extension des équivalences bien connues à l'équilibre (par exemple équivalence canonique-grand canonique).

Joel Lebowitz (Rutgers): Microscopic origin of macroscopic patterns

I will give an overview of the origins, formation and time evolution of patterns observed in various physical systems and in computer simulations of idealized model systems. The emphasis will be on the analysis of the dynamics leading to structures exhibiting scaling type of behavior in space/time. Examples include phase-segregation phenomena in quenched binary alloys or fluids and diffusion controlled chemical reactions. Both deterministic and stochastic dynamics with random or regular initial conditions will be considered.

Christian Maes (Leuven): On the positivity of entropy production

The mean entropy production is defined as the relative entropy between the steady state and its time-reversal for open systems in non-equilibrium conditions. This definition assumes a dynamic reversibility which is linked with a Gibbsian hypothesis for the path space measure on the space-time trajectories. Microscopic reversibility (detailed balance) makes the mean entropy production zero but the inverse implication is much harder to establish in the thermodynamic limit due to the possibility of spontaneous breaking of time-reversal symmetry. We discuss the problem via examples giving special attention to certain non-mechanical issues involving the choice of dynamical variables and time-reversal transformation (joint work with F. Redig).

Sylvie Méléard (Nanterre): Probabilistic interpretation of some Boltzmann equations and approximating stochastic particle systems

We present a probabilistic interpretation of some Boltzmann equations, as the evolution equations satisfied by the time-marginals of the law of a nonlinear Markov process. We use this interpretation to construct some simple or binary mean-field interacting n-particle systems, which satisfy a strong propagation of chaos result: the law of each subsystem of fixed size k converges in variation norm to the k-fold product of a law of which the time-marginals are solution of the Boltzmann equation. This convergence result allows us to obtain a particle algorithm simulating the solutions of these Boltzmann equations. Our approach can also allow us to prove some existence results or to give a sense to a uniqueness of solutions. It can be developped in many different situations: mollified Boltzmann equations with cutoff, Boltzmann equations with cutoff and small initial data, Maxwellian Boltzmann equations without cutoff.

Claude-Alain Pillet (Toulon): Towards a spectral theory of open systems out of equilibrium ?

As recent developments have shown, spectral theory is a very effective tool in the study of classical and quantum dynamical systems near thermal equilibrium. How to use these techniques in non-equilibrium situations is however not so clear. I will present some ideas in this direction and a few, very simple, illustrative examples.