Thermodynamics of a two-dimensional self-gravitating system

The mean field thermodynamics of a system of N gravitationally interacting particles confined in some bounded plane domain {Ω} 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 {∂}{Ω} or to an imposed external pressure (in which case {∂}{Ω} 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 {∂}{Ω} rigid or free, respectively) exists and is unique if and only if T exceeds a critical value Tc, and (ii) for a thermally insulated system, a unique globally stable (maximum entropy) equilibrium exists for any value of the energy (rigid {∂}{Ω}) or of the enthalpy (free {∂}{Ω}). 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{≤}Tc can exist if the domain is ``star shaped.''