Steepest-descent technique and stellar equilibrium statistical mechanics. I - Newtonian clusters in a box

A microcanonical statistical-mechanics formulation which produces an exact functional integral expression for entropy is used to study a model of star clusters in which the addition of certain constraints makes possible the existence of thermodynamic equilibrium configurations corresponding to slowly evolving stages of a real star cluster. In the analysis, stellar evaporation is eliminated by confining the stars to a finite volume, and tightly bound subclustering is eliminated by adopting an appropriate short-distance cutoff for the gravitational interaction between particles. The Gibbs microcanonical entropy of a confined cluster is transformed to a functional integral, this integral is evaluated by steepest-descent methods, and the mean field entropy is determined from the saddle-point value. Thermodynamic and local-fluctuation stability conditions are obtained by analyzing quadratic fluctuations. An evaluation of the stability conditions for spherically symmetric clusters in a rigid sphere relative to arbitrary fluctuations about the mean field shows that only clusters with almost uniform density are thermodynamically stable.