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.

**Above you may see the evolution of a Poisson distribution of massive particles with periodic boundary conditions. Below the initial condition is represented by a coherently displaced lattice.**

While such simulations constitute a very powerful tool, they lack the valuable guidance that a fuller analytic understanding of the problem would provide. The non-linear dynamics of self-gravitating systems is thus both a fascinating theoretical problem of out of equilibrium statistical mechanics, directly relevant both in the context of cosmology, astrophysics and, more generally, in the physics of systems with long-range interactions.

**Below you may see the evolution of an isolated system of massive particles interacting solely by Newtonian gravity**. This is a paradigmatic problem for astrophysics, cosmology and statistical physics. The underlying open question concerns the relaxation mechanism that drives the system to form structures which seem to be in a sort of equilibrium, as for instance different kind of astrophysical objects such as globular clusters, galaxies, and galaxy clusters.

In a galaxy the two body relaxation time is of order τ2 ≈ 10^(17) years and is much longer than the age of the universe (i.e., ≈ 10^(10) years): for this reason these objects are not in thermal equilibrium. However, they present common features as the luminosity profiles. Much theoretical work has been devoted to study the dynamical model to characterize such profiles and despite the numerical simulations have shown that structures formed in some cases are compatible with observations, the physical origin of these profiles has not been yet clarified from a theoretical point of view. Namely, the problem still remains to explain how to form the shape of density profiles and of velocity distributions of stellar structures like elliptical galaxies and globular clusters that are generally characterized by a dense central core and a dilute halo — where the halo is often featured by a power-law decay of the radial density.

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.

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.