Establishment of statistical equilibrium in a quantum chain of harmonic oscillators

A quantum chain of $N$ harmonic oscillators is considered using perturbation theory methods. The author derives the density matrix dynamics of a subsystem containing $N'$ oscillators when the whole chain relaxes from some initial pure state. The Gibbs distribution is shown to be the time-average thermodynamical limit of the relevant statistical operator. The presence of a weak anharmonic interaction in the system makes the density matrix deviations from the canonical distribution extremely rare.