SEMINAIRE MERCREDI 9 AVRIL 2003
14 heures
Salle Séminaire 5
Centre de Physique Théorique
Marseille-Luminy

Freddy Bouchet
Université de Firenze
Dipartimento di energetica "Sergio Stecco"

Title: Study of Equilibrium Fluctuations and Classification of ensemble
inequivalence situations in systems governed by long range
interactions.

Summary: We will first show that physical systems governed by long range
interactions are common, using examples from geophysical fluid
dynamics, plasma physics and self gravitating systems. Such systems
share peculiar properties, as for instance possible statistical
ensemble inequivalence (negative specific heat), or very slow
relaxations toward the equilibrium state. We will address these
questions using only classical thermodynamic ideas (Gibbs averages).
In a first part, we compute explicitely, from the microscopic
dynamics, the diffusion coefficient in a simple model with long range
interactions. In a second part, we propose a classification of
microcanonical and canonical phase transitions and of situations of
ensemble inequivalence in such systems. We further describe these two
parts in the following.

Working with a model system with long range interactions, the HMF
model, we will analytically derive the properties of the stochastic
process describing the macroscopic variable, and a Fokker-Planck
equation describing the statistical evolution of a particle in an
equilibrium bath. Classical approaches (projection techniques or
kinetic models) of such problems allow to obtain such a Fokker-Planck
equation. However, the diffusion coefficient is then expressed in
terms of a formal integral on some Liouvillian operator, and can be
computed explicitly, only in some particular limits. In our case,
combining a perturbative approach for the Hamiltonian dynamics, with
microcanonical averages, we explicitly compute the diffusion
coefficient. We show that the macroscopic variable is a Gaussian non
Markovian process. Its autocorrelation function verify an explicit
integral equation, with memory. We explain why we think that such
results may be easily generalized to the study of the relaxation
toward equilibrium, of out of equilibrium distributions.

In a second part of the talk, we will propose a classification of
situations of statistical ensemble inequivalence and of phase
transitions, for systems with long range interaction. For many of
these systems, large deviation techniques allow to prove that
microcanonical statistical equilibrium is described by the
maximization of an entropy with an energy constraint, whereas the
canonical one is described by the associated free energy minimization.
The solutions of these two variational problems may be different and
lead to ensemble inequivalence. Starting from such variational
problems, we will derive the classification, independently of any
model. This allow to classify the links between microcanonical and
canonical phase transitions. We will also stress that some generic
situations for ensemble inequivalence have not yet been observed in
any systems.