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Out-of-equilibrium phase transitions in the HMF model: a closer look

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 $β(E)$ 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.