Stationary ensemble approximations of dynamic quantum states: Optimizing the generalized Gibbs ensemble

In this work we investigate the nonequilibrium dynamics of closed quantum systems. In particular we focus on the stationary properties of integrable systems. Here we show how a generalized Gibbs ensemble (GGE) can be constructed as the best approximation to the time-dependent density matrix. Our procedure allows for a systematic construction of the GGE by a constrained minimization of the distance between the latter and the true state. Moreover, we show that the entropy of the GGE is a direct measure for the quality of the approximation. We apply our method to a quenched hard core Bose gas. Whereas a correlated GGE properly describes all stationary nonlocal correlations, a simple harmonic GGE is sufficient to completely describe reduced local states.