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 ${\mathit{T}}_{\mathit{c}}$, 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≤${\mathit{T}}_{\mathit{c}}$ can exist if the domain is ‘‘star shaped.’’

• Received 15 February 1994

DOI:https://doi.org/10.1103/PhysRevE.49.3771