The Vlasov equation and the Hamiltonian Mean-Field model

We show that the quasi-stationary states observed in the $N$-particle dynamics of the Hamiltonian Mean-Field (HMF) model are nothing but Vlasov stable homogeneous (zero magnetization) states. There is an infinity of Vlasov stable homogeneous states corresponding to different initial momentum distributions. Tsallis $q$-exponentials in momentum, homogeneous in angle, distribution functions are possible, however, they are not special in any respect, among an infinity of others. All Vlasov stable homogeneous states lose their stability because of finite $N$ effects and, after a relaxation time diverging with a power-law of the number of particles, the system converges to the Boltzmann-Gibbs equilibrium.