Generalized maximum entropy approach to quasistationary states in long-range systems

Systems with long-range interactions display a short-time relaxation towards quasistationary states (QSSs) whose lifetime increases with the system size. In the paradigmatic Hamiltonian mean-field model (HMF) out-of-equilibrium phase transitions are predicted and numerically detected which separate homogeneous (zero magnetization) and inhomogeneous (nonzero magnetization) QSSs. In the former regime, the velocity distribution presents (at least) two large, symmetric bumps, which cannot be self-consistently explained by resorting to the conventional Lynden-Bell maximum entropy approach. We propose a generalized maximum entropy scheme which accounts for the pseudoconservation of additional charges, the even momenta of the single-particle distribution. These latter are set to the asymptotic values, as estimated by direct integration of the underlying Vlasov equation, which formally holds in the thermodynamic limit. Methodologically, we operate in the framework of a generalized Gibbs ensemble, as sometimes defined in statistical quantum mechanics, which contains an infinite number of conserved charges. The agreement between theory and simulations is satisfying, both above and below the out-of-equilibrium transition threshold. A previously unaccessible feature of the QSSs, the multiple bumps in the velocity profile, is resolved by our approach.