Dynamics and physical interpretation of quasi-stationary states in systems with long-range interactions

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.