Out-of-equilibrium phase transitions in the Hamiltonian mean-field 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-level 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 the phase transitions between magnetized and nonmagnetized states changes: (i) For f(0) > 0.109 65, the system displays a second-order phase transition; (ii) for 0.109 497 < f(0) < 0.109 65, the system displays a second-order phase transition and a first-order phase transition; (iii) for 0.109 47 < f(0) < 0.109 497, the system displays two second-order phase transitions; and (iv) for f(0) < 0.109 47, 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 another situation of phase reentrance. We consider both microcanonical and canonical ensembles and report the existence of a tiny region of ensemble 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. Finally, we mention different causes of incomplete relaxation that could be a limitation to the application of Lynden-Bell's theory.