The effect of angular momentum conservation in the phase transitions of collapsing systems

The effect of angular momentum conservation in microcanonical thermodynamics is considered. This is relevant in gravitating systems, where angular momentum is conserved and the collapsing nature of the forces makes the microcanonical ensemble the proper statistical description of the physical processes. The microcanonical distribution function with non-vanishing angular momentum is obtained as a function of the coordinates of the particles. As an example, a simple model of gravitating particles, introduced by Thirring long ago, is worked out. The phase diagram contains three phases: for low values of the angular momentum $L$ the system behaves as the original model, showing a complete collapse at low energies and an entropy with a convex intruder. For intermediate values of $L$ the collapse at low energies is not complete and the entropy still has a convex intruder. For large $L$ there is neither collapse nor anomalies in the thermodynamical quantities. A short discussion of the extension of these results to more realistic situations is exposed.