  We study thermalization in many-body quantum systems locally coupled to an
external bath. It is shown that quantum chaotic systems do thermalize, that is,
they exhibit relaxation to an invariant ergodic state which, in the bulk, is
well approximated by the grand canonical state. Moreover, the resulting ergodic
state in the bulk does not depend on the details of the baths. On the other
hand, for integrable systems the invariant state does depend on the bath and is
different from the grand canonical state.
