From a theoretical point of view, the evolution of a very large set of massive particles interacting solely by Newtonian gravity is a paradigmatic problem for the statistical physics of long range interacting systems finding application in many different areas of astrophysics and cosmology. Some examples are the study of globular clusters, galaxies and gravitational clustering in the expanding universe. Gravitational clustering of mass structures is a well-posed problem of out-of-equilibrium statistical mechanics that can be studied through N-body simulations.

The question of the nature of the equilibrium properties of these core-halo structures is thus relevant both in astrophysics and cosmology and thus one would like to develop a statistical mechanics approach to describe these systems. However, one must consider that, from the point of view of statistical physics, self-gravitating systems present fundamental problems, that are also common to other long-range interacting systems. Indeed, it is well known since the pioneering works of Boltzmann and Gibbs, that systems with a pair potential decaying with an exponent smaller than that of the embedding space, present several fundamental problems that prevent the use of equilibrium statistical mechanics: thermodynamic equilibrium is never reached and the laws of equilibrium thermodynamics do not apply.

Rather these systems reach, driven by a mean-field collisionless relaxation dynamics, quasi-equilibrium configurations, or quasi-stationary state (QSS), whose lifetime diverge with the number of particles N. The formation of QSS is at present one of the most living subjects in non-equilibrium statistical physics and a general theoretical framework is still lacking: it is thus necessary to consider toy models and/or relatively simple systems that can be studied through numerical well-controlled experiments

Approaching the problem in the context of statistical mechanics, as we propose here, it is natural to start by reducing as much as possible the complexity of the analogous astrophysical and cosmological problems: our goal is to identify the main features of the (non-linear) gravitational dynamics, which shape the structures observed in the sky.

These are the main projects on which we have worked over the past few years.