  Although the Vlasov equation is used as a good approximation for a
sufficiently large $N$, Braun and Hepp have showed that the time evolution of
the one particle distribution function of a $N$ particle classical Hamiltonian
system with long range interactions satisfies the Vlasov equation in the limit
of infinite $N$. Here we rederive this result using a different approach
allowing a discussion of the role of inter-particle correlations on the system
dynamics. Otherwise for finite N collisional corrections must be introduced.
This has allowed the a quite comprehensive study of the Quasi Stationary States
(QSS) but many aspects of the physical interpretations of these states remain
unclear. In this paper a proper definition of timescale for long time evolution
is discussed and several numerical results are presented, for different values
of $N$. Previous reports indicates that the lifetimes of the QSS scale as
$N^{1.7}$ or even the system properties scales with $\exp(N)$. However,
preliminary results presented here shows indicates that time scale goes as
$N^2$ for a different type of initial condition. We also discuss how the form
of the inter-particle potential determines the convergence of the $N$-particle
dynamics to the Vlasov equation. The results are obtained in the context of
following models: the Hamiltonian Mean Field, the Self Gravitating Ring Model,
and a 2-D Systems of Gravitating Particles. We have also provided information
of the validity of the Vlasov equation for finite $N$, i. e.\ how the dynamics
converges to the mean-field (Vlasov) description as $N$ increases and how
inter-particle correlations arise.
