  We here discuss the emergence of Quasi Stationary States (QSS), a universal
feature of systems with long-range interactions. With reference to the
Hamiltonian Mean Field (HMF) model, numerical simulations are performed based
on both the original $N$-body setting and the continuum Vlasov model which is
supposed to hold in the thermodynamic limit. A detailed comparison
unambiguously demonstrates that the Vlasov-wave system provides the correct
framework to address the study of QSS. Further, analytical calculations based
on Lynden-Bell's theory of violent relaxation are shown to result in accurate
predictions. Finally, in specific regions of parameters space, Vlasov numerical
solutions are shown to be affected by small scale fluctuations, a finding that
points to the need for novel schemes able to account for particles
correlations.
