Effect of angular momentum on equilibrium properties of a self-gravitating system

The microcanonical properties of a two-dimensional system of N classical particles interacting via a smoothed Newtonian potential, as a function of the total energy E and the total angular momentum L, are discussed. The two first moments of the distribution of the linear momentum of a given particle at a fixed position show that (a) on average the system rotates like a solid body and (b) the velocity dispersion is a function of the distance from the center. In order to estimate suitable observables, a numerical method based on an importance sampling algorithm is presented. The entropy surface S shows a negative specific heat capacity region at fixed L for all L. Observables probing the average mass distribution are used to understand the link between thermostatistical properties and the spatial distribution of particles. In order to define a phase in a nonextensive system, we introduce a more general observable than that proposed by Gross and Votyakov [Eur. Phys. J. B 15, 115 (2000)]. This observable is the sign of the largest eigenvalue of the Hessian matrix of the entropy surface. If it is negative then the system is in a pure (single) phase; if it is positive then the system undergoes a first order phase transition. At large E the gravitational system is in a homogeneous gas phase. At low E there are several collapse phases. At L=0 there is a single-cluster phase and for L not equal 0 there are several phases with two clusters. The relative size of the clusters depends on L. All these pure phases are separated by a first order phase transition region. Signals of critical behavior emerge at several points of the parameter space (E,L). We also show that a huge loss of information appears if we treat the system as a function of the intensive parameters. Besides the known nonequivalence at first order phase transitions, the pure phases with two clusters of different sizes are not accessible to the canonical ensemble. Moreover, for a particular choice of intensive parameters introduced in this paper, there exist in the microcanonical ensemble some values of those intensive parameters for which the corresponding canonical ensemble does not exist, i.e., the partition sum diverges.