Solvable model of a self-gravitating system
  We introduce and discuss an effective model of a self-gravitating system
whose equilibrium thermodynamics can be solved in both the microcanonical and
the canonical ensemble, up to a maximization with respect to a single variable.
Such a model can be derived from a model of self-gravitating particles confined
on a ring, referred to as the self-gravitating ring (SGR) model, allowing a
quantitative comparison between the thermodynamics of the two models. Despite
the rather crude approximations involved in its derivation, the effective model
compares quite well with the SGR model. Moreover, we discuss the relation
between the effective model presented here and another model introduced by
Thirring forty years ago. The two models are very similar and can be considered
as examples of a class of minimal models of self-gravitating systems.
