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; more specifically Newton’s gravity belongs to the class of long-range interactions, characterized by an interaction decaying as a power-law function of the distance with an exponent smaller than the space dimension. **Self-gravitating systems** present fundamental problems, that are also common to other long-range interacting systems: 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.

Differently from systems with short-range interactions, a distinguished feature of long-range interacting systems, is that instead of relaxing to thermodynamical equilibria through two-body collisions, these systems reach, driven by a mean-field collisionless relaxation dynamics, quasi-equilibrium configurations, or **quasi-stationary state** (QSS), whose lifetime diverges 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 for the understanding of their statistical properties 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.

For this reason much theoretical and numerical effort has been dedicated to the comprehension of the **quasi-stationary properties** of the mass distributions resulting from the gravitational evolution. The QSS, having different properties from thermodynamical equilibrium ones, can be described in terms of time average of global quantities: notably they are characterized by virial equilibrium, i.e. by a relation that relates the average over time of the total kinetic energy of a stable system consisting of N particles, bound by potential forces, with that of the total potential energy. From a theoretical point of view, the description of such a stationary configuration, where dynamics is mean-field and non-collisional, can be approached in terms of the **Vlasov-Poisson system** of equations: that is, the equation of motion for a self gravitating fluid. Indeed, the Vlasov equation is the non-collisional Boltzmann equation, describing time evolution of the distribution function of the fluid consisting of massive particles with gravitational interaction, and the Poisson equation, describing the gravitational field generated by such a mass distribution. In most astrophysical system of interest two-body relaxation occur on a time scale longer than the Hubble time and thus can be neglected.

If a gravitational (i.e., a stellar) system may be considered to be **collisionless**, then we obtain a good approximation to the orbit of any star by calculating the orbit that it would have if the system’s mass were smoothly distributed in space rather than concentrated into nearly point-like stars. Eventually, the true orbit deviates significantly from this model orbit, but in systems with more than a few thousand stars, the deviation is small. In this situation one looks for stationary solution of the Vlasov-Poisson equations, that are valid if the system is in a perfect stationary equilibrium and collisions can be neglected. Models derived in these approximations represent the main tool to compare dynamic stellar or galactic theory with observations.

However, it has been recently noticed that, before reaching a truly QSS, an isolated self-gravitating system may show during its evolution the appearance of **long-lived transients** (LLT), that are far out from equilibrium and whose lifetime can be large compared to the system characteristic time, but shorter than the collisional relaxation time scale. These non-stationary configurations that show a secular evolution, that is, slow and constant, guided by a non-collisional dynamics. Up to now, secular evolution was considered to be the result of long-term interactions between a stellar system (such as a galaxy) and its surrounding environment or being induced by internal processes (such as the action of the spiral arms or of the bar). From an astrophysical point of view, the most easily recognizable example of **secular evolution** in galaxies is the formation of stars in the spiral arms. In this context, therefore, attention has focused on the evolution due to other phenomena and interactions than self-gravity itself, while our goal is to study the secular evolution that occurs in far from equilibrium self- gravitational system.

Given the complexity of these systems dynamics, of the spatial configurations formed and of the associated velocity fields, it is necessary to first study numerically simple systems and then gradually to increase the degree of complexity of the problem. In other words, following an approach to the problem **inspired by statistical mechanics**, we reduce the complexity of the astrophysical problem to try to identify the main characteristics of the gravitational (non-linear and non-stationary) dynamics that form the structures observed in the sky. In this context, we have recently considered the spatial configurations and the associated structure of the transient velocity fields that are formed in the case of a very simple initial class of conditions represented by isolated clouds of self-gravitating particles with an ellipsoidal shape (which constitutes the simpler geometry than beyond a spherical cloud), with uniform density rotating around their minor semi-axis. The motivation of this study is that we have shown that the combination of an initial rotational motion with gravitational collapse very naturally leads to transients characterized by very interesting spatial configurations and velocity fields.

Indeed, when the initial state deviates from the spherical symmetry and rigidly rotates, its collapse gives rise to a dense core surrounded by a diluted halo, as happens when the initial conditions are spherical, non-rotating and quite cold. However, in this case there is a qualitatively new and surprising feature of the evolved mass distribution at sufficiently long times after the gravitational collapse: in many cases we observe a spatial configuration of the mass that resembles qualitatively that of **spiral galaxies**. This spiral structure, which is an intrinsic result of the dynamics out-of-equilibrium after the collapse of the system, is progressively stretched over time by the predominantly radial movements of the outermost particles. Therefore there is a coherent rotational motion in an extended region surrounding the core, which is triaxial and has a dispersion of the isotropic velocity, while the most external regions of the system are dominated by radial motions. Moreover, different types of sub-structures, co-planar with spiral arms, are formed during the time evolution of the system and show relatively long life times.

From an observational point of view, a key question concerns the comparison, not only of the spatial configurations of our models with the morphological properties of galaxies, but also a detailed comparison of the model velocity fields with the velocity fields of external galaxies and the stars of our galaxy.