  We investigate the equilibration and thermalization properties of quantum
systems interacting with a finite dimensional environment. By exploiting the
concept of time averaged states, we introduce a completely positive map which
allows to describe in a quantitative way the dependence of the equilibrium
state on the initial condition. Our results show that the thermalization of
quantum systems is favored if the dynamics induces small system-environment
correlations, as well as small changes in the environment, as measured by the
trace distance.
