THERMODYNAMICS OF A 2-DIMENSIONAL SELF-GRAVITATING SYSTEM

The mean field thermodynamics of a system of N gravitationally interacting particles confined in some bounded plane domain OMEGA is considered in the four possible situations corresponding to the following two pairs of alternatives: (a) Confinement is due either to a rigid circular wall partial derivative OMEGA or to an imposed external pressure (in which case partial derivative OMEGA is a free boundary). (b) The system is either in contact with a thermal bath at temperature T, or it is thermally insulated. It is shown in particular that (i) for a system at given temperature T, a globally stable equilibrium (minimum free energy or minimum free enthalpy state for partial derivative OMEGA rigid or free, respectively) exists and is unique if and only if T exceeds a critical value T(c), and (ii) for a thermally insulated system, a unique globally stable (maximum entropy) equilibrium exists for any value of the energy (rigid partial derivative OMEGA) or of the enthalpy (free partial derivative OMEGA). The case of a system confined in a domain of arbitrary shape is also discussed. Bounds on the free energy and the entropy are derived, and it is proven that no isothermal equilibrium (stable or unstable) with a temperature T less-than-or-equal-to T(c), can exist if the domain is ''star shaped.''