Stability criteria of the Vlasov equation and quasi-stationary states of the HMF model (original) (raw)

2004, Physica A-statistical Mechanics and Its Applications

We perform a detailed study of the relaxation towards equilibrium in the Hamiltonian Mean-Field model, a prototype for long-range interactions in N -particle dynamics. In particular, we point out the role played by the inÿnity of stationary states of the associated N → ∞ Vlasov dynamics. In this context, we derive a new general criterion for the stability of any spatially homogeneous distribution, and compare its analytical predictions with numerical simulations of the Hamiltonian, ÿnite N , dynamics. We then propose, and verify numerically, a scenario for the relaxation process, relying on the Vlasov equation. When starting from a nonstationary or a Vlasov unstable stationary state, the system shows initially a rapid convergence towards a stable stationary state of the Vlasov equation via nonstationary states: we characterize numerically this dynamical instability in the ÿnite N system by introducing appropriate indicators. This ÿrst step of the evolution towards Boltzmann-Gibbs equilibrium is followed by a slow quasi-stationary process, that proceeds through di erent stable stationary states of the Vlasov equation. If the ÿnite N system is initialized in a Vlasov stable homogeneous state, it remains trapped in a quasi-stationary state for times that increase with the nontrivial power law N 1:7 . Single particle momentum distributions in such a quasi-stationary regime do not have power-law tails, and hence cannot be ÿtted by the q-exponential distributions derived from Tsallis statistics.