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.
Ground states of interacting QFTs are non-gaussian states, i.e. their connected n-point correlation functions do not vanish for n>2, in contrast to the free QFT case. We show that when the ground state of an interacting QFT evolves under a free massi
ve QFT for a long time (a scenario that can be realised by a Quantum Quench), the connected correlation functions decay and all local physical observables equilibrate to values that are given by a gaussian density matrix that keeps memory only of the two-point initial correlation function. The argument hinges upon the fundamental physical principle of cluster decomposition, which is valid for the ground state of a general QFT. An analogous result was already known to be valid in the case of d=1 spatial dimensions, where it is a special case of the so-called Generalised Gibbs Ensemble (GGE) hypothesis, and we now generalise it to higher dimensions. Moreover in the case of massless free evolution, despite the fact that the evolution may not lead to equilibration but unbounded increase of correlations with time instead, the GGE gives correctly the leading order asymptotic behaviour of correlation functions in the thermodynamic and large time limit. The demonstration is performed in the context of bosonic relativistic QFT, but the arguments apply more generally.
