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Systems with long-range interactions display a short-time relaxation towardsQuasi Stationary States (QSS) whose lifetime increases with the system size. Inthe paradigmatic Hamiltonian Mean-field Model (HMF) out-of-equilibrium phasetransitions are predicted and numerically detected which separate homogeneous(zero magnetization) and inhomogeneous (nonzero magnetization) QSS. In theformer regime, the velocity distribution presents (at least) two large,symmetric, bumps, which cannot be self-consistently explained by resorting tothe conventional Lynden-Bell maximum entropy approach. We propose a generalizedmaximum entropy scheme which accounts for the pseudo-conservation of additionalcharges, the even momenta of the single particle distribution. These latter areset to the asymptotic values, as estimated by direct integration of theunderlying 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 aninfinite number of conserved charges. The agreement between theory andsimulations is satisfying, both above and below the out of equilibriumtransition threshold. A precedently unaccessible feature of the QSS, themultiple bumps in the velocity profile, is resolved by our new approach.

Generalized maximum entropy approach to quasi-stationary states in long range systems