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

Abstract

We study dynamical phase transitions in systems with long-range interactions, using the Hamiltonian mean field model as a simple example. These systems generically undergo a violent relaxation to a quasistationary state (QSS) before relaxing towards Boltzmann equilibrium. In the collisional regime, the out-of-equilibrium one-particle distribution function (DF) is a quasistationary solution of the Vlasov equation, slowly evolving in time due to finite- N effects. For subcritical energy densities, we exhibit cases where the DF is well fitted by a Tsallis q distribution with an index q(t) slowly decreasing in time from q approximately = 3 (semiellipse) to q=1 (Boltzmann). When the index q(t) reaches an energy-dependent critical value q_(crit) , the nonmagnetized (homogeneous) phase becomes Vlasov unstable and a dynamical phase transition is triggered, leading to a magnetized (inhomogeneous) state. While Tsallis distributions play an important role in our study, we explain this dynamical phase transition by using only conventional statistical mechanics. For supercritical energy densities, we report the existence of a magnetized QSS with a very long lifetime.