  We study dynamical phase transitions in systems with long-range interactions,
using the Hamiltonian Mean Field (HMF) model as a simple example. These systems
generically undergo a violent relaxation to a quasi-stationary state (QSS)
before relaxing towards Boltzmann equilibrium. In the collisional regime, the
out-of-equilibrium one-particle distribution function (DF) is a
quasi-stationary solution of the Vlasov equation, slowly evolving in time due
to finite $N$ effects. For subcritical energies $7/12<U<3/4$, we exhibit cases
where the DF is well-fitted by a Tsallis $q$-distribution with an index $q(t)$
slowly decreasing in time from $q\simeq 3$ (semi-ellipse) to $q=1$ (Boltzmann).
When the index $q(t)$ reaches a critical value $q_{crit}(U)$, the
non-magnetized (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 energies, we report for the first time the
existence of a magnetized QSS with a very long lifetime.
