Out-of-equilibrium tricritical point in a system with long-range interactions

Systems with long-range interactions display a short-time relaxation towards Quasi Stationary States (QSSs) whose lifetime increases with system size. With reference to the Hamiltonian Mean Field (HMF) model, we here show that a maximum entropy principle, based on Lynden-Bell's pioneering idea of "violent relaxation", predicts the presence of out-of-equilibrium phase transitions separating the relaxation towards homogeneous (zero magnetization) or inhomogeneous (non zero magnetization) QSSs. When varying the initial condition within a family of "water-bags" with different initial magnetization and energy, first and second order phase transition lines are found that merge at an out--of--equilibrium tricritical point. Metastability is theoretically predicted and numerically checked around the first-order phase transition line.