We derive the global phase diagram of a self-gravitating N -body system enclosed in a finite threedimensional spherical volume V as a function of total energy and angular momentum, employing a microcanonical mean-field approach. At low angular momenta (i.e. for slowly rotating systems) the known collapse from a gas cloud to a single dense cluster is recovered. At high angular momenta, instead, rotational symmetry can be spontaneously broken and rotationally asymmetric structures (double clusters) appear.PACS numbers: 05.20.-y, 04.40.-b The statistical equilibrium properties of systems of particles interacting via long-range forces (the so-called non-extensive systems) are currently the subject of intense research, both for their highly non-trivial thermodynamics (displaying such features as negative heat capacities [1,2]) and for the considerable conceptual and technical difficulties they present. It is known that the long-range nature of the potential makes the canonical ensemble inadequate for describing their statics [3,4,5], because the usual thermodynamic limit, where (number of particles) N → ∞ and (volume) V → ∞ while intensive variables are kept fixed, does not exist. A central issue is hence whether phase transitions and other conventional statistical phenomena are possible in nonextensive systems [6].Among non-extensive systems, self-gravitating gases, i.e. systems of classical particles subject to mutual gravitation, have deserved the most attention. Their usual static description is based on the microcanonical ensemble [5]. In this framework, the key problem is finding the most probable equilibrium configuration of a selfgravitating gas enclosed in a finite 3-dimensional box of volume V as a function of the conserved quantities (integrals of motion), the simplest (but possibly not the only relevant ones) being the total energy E and the total angular momentum L.Dynamical methods [7] based on fluid-mechanics techniques suggest (see e.g. [8]) that upon increasing the ratio between rotational and gravitational energy, the stationary distribution can change from a single dense cluster to a double cluster, and that other structures such as disks and rings might appear.On the other hand, so far static theories could not recover the richness of the dynamical picture. Taking the total energy as the only control parameter (see [5] for a review and [9, 10, 11] for more recent work and references) after removing the rotational symmetry artificially e.g. by constraining the system into a non-spherical box, a "collapse" transition has been found [1], where, as the energy (temperature) is lowered, the equilibrium configuration changes from a homogeneous cloud to a dense cluster lying in an almost void background, with an intermediate "transition" regime characterized by negative specific heat. Despite some attempts [12], a detailed static theory embodying angular momentum is lacking.In this work, building essentially on [8,13], we aim at bridging the gap between the static and the dynamical approaches by analyz...
