  Systems with long-range interactions display a short-time relaxation towards
Quasi Stationary States (QSS) 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) QSS. 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 pseudo-conservation 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 precedently unaccessible feature of the QSS, the
multiple bumps in the velocity profile, is resolved by our new approach.
