Relaxation times of unstable states in systems with long range interactions

We consider several models with long range interactions evolving via Hamiltonian dynamics. The microcanonical dynamics of the basic Hamiltonian mean field (HMF) model and perturbed HMF models with either global anisotropy or an on-site potential are studied both analytically and numerically. We find that, in the magnetic phase, the initial zero magnetization state remains stable above a critical energy and is unstable below it. In the dynamically stable state, these models exhibit relaxation timescales that increase algebraically with the number N of particles, indicating the robustness of the quasistationary state seen in previous studies. In the unstable state, the corresponding timescale increases logarithmically in N.