Left panel of figure: NGC 3310. Right panel: Snapshot of one of our numerical Nbody simulations
The evolution of a very large set of massive particles interacting solely by Newtonian gravity is a paradigmatic problem for the statistical physics of longrange 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 wellposed problem of outofequilibrium statistical mechanics that can be studied through Nbody simulations; more specifically Newton’s gravity belongs to the class of longrange interactions, characterized by an interaction decaying as a powerlaw function of the distance with an exponent smaller than the space dimension. Selfgravitating systems present fundamental problems, that are also common to other longrange 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 shortrange interactions, a distinguished feature of longrange interacting systems, is that instead of relaxing to thermodynamical equilibria through twobody collisions, these systems reach, driven by a meanfield collisionless relaxation dynamics, quasiequilibrium configurations, or quasistationary 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 nonequilibrium 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 wellcontrolled experiments.
For this reason much theoretical and numerical effort has been dedicated to the comprehension of the quasistationary 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 meanfield and noncollisional, can be approached in terms of the VlasovPoisson system of equations: that is, the equation of motion for a self gravitating fluid. Indeed, the Vlasov equation is the noncollisional 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 twobody 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 pointlike 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 VlasovPoisson 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 selfgravitating system may show during its evolution the appearance of longlived 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 nonstationary configurations that show a secular evolution, that is, slow and constant, guided by a noncollisional dynamics. Up to now, secular evolution was considered to be the result of longterm 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 selfgravity 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 (nonlinear and nonstationary) 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 selfgravitating particles with an ellipsoidal shape (which constitutes the simpler geometry than beyond a spherical cloud), with uniform density rotating around their minor semiaxis. 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, nonrotating 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 outofequilibrium 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 substructures, coplanar 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.
References
 Transient spiral arms from far out of equilibrium gravitational evolution, , D. Benhaiem, M. Joyce, F. Sylos Labini, Astrophysical Journal, in the press 2017
 Particle number dependence in the nonlinear evolution of Nbody selfgravitating systems, D. Benhaiem, M. Joyce, F. Sylos Labini, T. Worrakitpoonpon Monthly Notices of the Royal Astronomical Society, 473, 2348 (2017)

Formation of satellites from cold collapse, David Benhaiem, Francesco Sylos Labini, Astronomy and Astrophysics 598, A95 (2017)
Angular momentum generation in cold gravitational collapse, D. Benhaiem, M. Joyce, F. Sylos Labini, T. Worrakitpoonpon, Astronomy and Astrophysics 585, A139 (2016) 
Violent relaxation of ellipsoidal clouds, David Benhaiem, Francesco Sylos Labini, MNRAS 448, 2634–2643 (2015)
On the generation of triaxiality in the collapse of cold spherical selfgravitating systems Francesco Sylos Labini, David Benhaiem, Michael Joyce, MNRAS (June 01, 2015) 449 (4): 44584464  Universal properties of violently relaxed gravitational structures, Francesco Sylos Labini, MNRAS (February 11, 2013) 429 (1): 679687
 Evolution of isolated overdensities as a control on cosmological N body simulations, Michael Joyce, Francesco Sylos Labini, MNRAS (February 21, 2013) 429 (2): 10881101
 Violent and mild relaxation of an isolated selfgravitating uniform and spherical cloud of particles, Francesco Sylos Labini, Mon. Not. R. Astron. Soc. 423, 1610 – 1622 (2012)
 Dynamics of finite and infinite selfgravitating systems with cold quasiuniform initial conditions, M. Joyce, B. Marcos, F. Sylos Labini, Journal of Statistical Mechanics, 2009, P04019
 Energy ejection in the collapse of a cold spherical selfgravitating cloud, M. Joyce, B. Marcos, F. Sylos Labini, Mon.Not.R.Astron.Soc. (2008)