Incomplete equilibrium in long-range interacting systems

We use a Hamiltonian dynamics to discuss the statistical mechanics of long-lasting quasi-stationary states particularly relevant for long-range interacting systems. Despite the presence of an anomalous single-particle velocity distribution, we find that the Central Limit Theorem implies the Boltzmann expression in Gibbs’ $\mathrm{\Gamma }$-space. We identify the nonequilibrium sub-manifold of $\mathrm{\Gamma }$-space characterizing the anomalous behavior and show that by restricting the Boltzmann-Gibbs approach to this sub-manifold we obtain the statistical mechanics of the quasi-stationary states.