  In this Letter we pose the question of whether a many-body quantum system
with a full set of conserved quantities can relax to an equilibrium state, and,
if it can, what the properties of such state are. We confirm the relaxation
hypothesis through a thorough ab initio numerical investigation of the dynamics
of hard-core bosons on a one-dimensional lattice. Further, a natural extension
of the Gibbs ensemble to integrable systems results in a theory that is able to
predict the mean values of physical observables after relaxation. Finally, we
show that our generalized equilibrium carries more memory of the initial
conditions than the usual thermodynamic one. This effect may have many
experimental consequences, some of which having already been observed in the
recent experiment on the non-equilibrium dynamics of one-dimensional hard-core
bosons in a harmonic potential [T. Kinoshita, T. Wenger, D. S. Weiss, Nature
(London) 440, 900 (2006)].
