Thermodynamic equilibrium under long-range forces. II. Gravity and other models

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Abstract
The equilibrium configuration of several models with long-range interaction is discussed by maximizing the entropy S with the fields constrained by the field equations. The calculations include a scalar field model (with Newtonian gravity as a special case) and general relativity, as well as a model in which a field is coupled to a nonconserved charge. The results illustrate the connection between temperature (particle number density) -dependent terms in the field equations and the appearance of nonuniform temperature (chemical potential) in equilibrium. The model with the nonconserved charge illustrates the likelihood of ‘‘miraculous’’ cancellations in phenomenological coarse-grain theories.