  In his pioneering work on negative specific heat, Walter Thirring
in\-tro\-duced a model that is solvable in the microcanonical ensemble. Here,
we give a complete description of the phase-diagram of this model in both the
microcanonical and the canonical ensemble, highlighting the main features of
ensemble inequivalence. In both ensembles, we find a line of first-order phase
transitions which ends in a critical point. However, neither the line nor the
point have the same location in the phase-diagram of the two ensembles. We also
show that the microcanonical and canonical critical points can be analytically
related to each other using a Landau expansion of entropy and free energy,
respectively, in analogy with what has been done in [O. Cohen, D. Mukamel, J.
Stat. Mech., P12017 (2012)]. Examples of systems with certain symmetries
restricting the Landau expansion have been considered in this reference, while
no such restrictions are present in Thirring's model. This leads to a phase
diagram that can be seen as a prototype for what happens in systems of
particles with kinematic degrees of freedom dominated by long-range
interactions.
