Equilibration of quantum systems and subsystems

We unify two recent results concerning equilibration in quantum theory. We first generalize a proof of Reimann (2008 Phys. Rev. Lett. 101 190403), that the expectation value of 'realistic' quantum observables will equilibrate under very general conditions, and discuss its implications for the equilibration of quantum systems. We then use this to re-derive an independent result of Linden et al (2009 Phys. Rev. E 79 061103), showing that small subsystems generically evolve to an approximately static equilibrium state. Finally, we consider subspaces in which all initial states effectively equilibrate to the same state.