  We present the phase diagram, in both the microcanonical and the canonical
ensemble, of the Self-Gravitating-Ring (SGR) model, which describes the motion
of equal point masses constrained on a ring and subject to 3D gravitational
attraction. If the interaction is regularized at short distances by the
introduction of a softening parameter, a global entropy maximum always exists,
and thermodynamics is well defined in the mean-field limit. However, ensembles
are not equivalent and a phase of negative specific heat in the microcanonical
ensemble appears in a wide intermediate energy region, if the softening
parameter is small enough. The phase transition changes from second to first
order at a tricritical point, whose location is not the same in the two
ensembles. All these features make of the SGR model the best prototype of a
self-gravitating system in one dimension. In order to obtain the stable
stationary mass distribution, we apply a new iterative method, inspired by a
previous one used in 2D turbulence, which ensures entropy increase and, hence,
convergence towards an equilibrium state.
