The time evolution of the one-particle distribution function of an $N$-particle classical Hamiltonian system with long-range interactions satisfies the Vlasov equation in the limit of infinite $N$. In this paper we present a new derivation of this result using a different approach allowing a discussion of the role of interparticle correlations on the system dynamics. Otherwise for finite N collisional corrections must be introduced. This has allowed a quite comprehensive study of the quasistationary states (QSSs) though many aspects of the physical interpretations of these states still remain unclear. In this paper a proper definition of time scale for long time evolution is discussed, and several numerical results are presented for different values of $N$. Previous reports indicate that the lifetimes of the QSS scale as $N^{1.7}$ or even the system properties scale with $exp\left(N\right)$. However, preliminary results presented here indicates that time scale goes as $N^2$ for a different type of initial condition. We also discuss how the form of the interparticle potential determines the convergence of the $N$-particle dynamics to the Vlasov equation. The results are obtained in the context of the following models: the Hamiltonian mean field, the Self-gravitating ring model, and one- and two-dimensional systems of gravitating particles. We have also provided information of the validity of the Vlasov equation for finite $N$.

• Received 13 May 2013
• Revised 23 July 2013

DOI:https://doi.org/10.1103/PhysRevE.89.032116