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We study dynamical phase transitions in systems with long-range interactions,using the Hamiltonian Mean Field (HMF) model as a simple example. These systemsgenerically undergo a violent relaxation to a quasi-stationary state (QSS)before relaxing towards Boltzmann equilibrium. In the collisional regime, theout-of-equilibrium one-particle distribution function (DF) is aquasi-stationary solution of the Vlasov equation, slowly evolving in time dueto finite $N$ effects. For subcritical energies $7/12<U<3/4$, we exhibit caseswhere the DF is well-fitted by a Tsallis $q$-distribution with an index $q(t)$slowly decreasing in time from $q\simeq 3$ (semi-ellipse) to $q=1$ (Boltzmann).When the index $q(t)$ reaches a critical value $q_{crit}(U)$, thenon-magnetized (homogeneous) phase becomes Vlasov unstable and a dynamicalphase transition is triggered, leading to a magnetized (inhomogeneous) state.While Tsallis distributions play an important role in our study, we explainthis dynamical phase transition by using only conventional statisticalmechanics. For supercritical energies, we report for the first time theexistence of a magnetized QSS with a very long lifetime.

Dynamical phase transitions in long-range Hamiltonian systems and Tsallis distributions with a time-dependent index