Vlasov analysis of relaxation and meta-equilibrium

The Hamiltonian Mean-Field model (HMF), an inertial XY ferromagnet with infinite-range interactions, has been extensively studied in the last few years, especially due to its long-lived meta-equilibrium states, which exhibit a series of anomalies, such as, breakdown of ergodicity, anomalous diffusion, aging, and non-Maxwell velocity distributions. The most widely investigated meta-equilibrium states of the HMF arise from special (fully magnetized) initial conditions that evolve to a spatially homogeneous state with well defined macroscopic characteristics and whose lifetime increases with the system size, eventually reaching equilibrium. These meta-equilibrium states have been observed for specific energies close below the critical value 0.75, corresponding to a ferromagnetic phase transition, and disappear below a certain energy close to 0.68. In the thermodynamic limit, the $\mu$-space dynamics is governed by a Vlasov equation. For finite systems this is an approximation to the exact dynamics. However, it provides an explanation, for instance, for the violent initial relaxation and for the disappearance of the homogeneous states at energies below 0.68.