  We provide a detailed discussion of out-of-equilibrium phase transitions in
the Hamiltonian Mean Field (HMF) model in the framework of Lynden-Bell's
statistical theory of the Vlasov equation. For two-levels initial conditions,
the caloric curve $\beta(E)$ only depends on the initial value $f_0$ of the
distribution function. We evidence different regions in the parameter space
where the nature of phase transitions between magnetized and non-magnetized
states changes: (i) for $f_0>0.10965$, the system displays a second order phase
transition; (ii) for $0.109497<f_0<0.10965$, the system displays a second order
phase transition and a first order phase transition; (iii) for
$0.10947<f_0<0.109497$, the system displays two second order phase transitions;
(iv) for $f_0<0.10947$, there is no phase transition. The passage from a first
order to a second order phase transition corresponds to a tricritical point.
The sudden appearance of two second order phase transitions from nothing
corresponds to a second order azeotropy. This is associated with a phenomenon
of phase reentrance. When metastable states are taken into account, the problem
becomes even richer. In particular, we find a new situation of phase
reentrance. We consider both microcanonical and canonical ensembles and report
the existence of a tiny region of ensembles inequivalence. We also explain why
the use of the initial magnetization $M_0$ as an external parameter, instead of
the phase level $f_0$, may lead to inconsistencies in the thermodynamical
analysis.
