Dynamical Quasi-Stationary States in a system with long-range forces

The Hamiltonian Mean Field model describes a system of N fully-coupled particles showing a second-order phase transition as a function of the energy. The dynamics of the model presents interesting features in a small energy region below the critical point. In particular, when the particles are prepared in a ``water bag’’ initial state, the relaxation to equilibrium is very slow. In the transient time the system lives in a dynamical quasi-stationary state and exhibits anomalous (enhanced) diffusion and L'evy walks. In this paper we study temperature and velocity distribution of the quasi-stationary state and we show that the lifetime of such a state increases with N. In particular when the $N\to \mathrm{\infty }$ limit is taken before the $t\to \mathrm{\infty }$ limit, the results obtained are different from the expected canonical predictions. This scenario seems to confirm a recent conjecture proposed by C.Tsallis.