Hierarchy recurrences in local relaxation

Inside a closed many-body system undergoing the unitary evolution, small partition of whole exhibits local relaxation. If total degrees freedom is large but finite number, such relaxation would come across recurrence after certain time, namely, dynamics suddenly appears random well-ordered oscillatory decay process. It found in this paper, among collection N two-level systems (TLSs), one TLS inside has hierarchy structure hiding randomness recurrence: similar recurrences appear periodical way, and later brings stronger than previous one. Both analytical numerical results that we obtained well explains recurrences: population (as an open system) diffuses out regathers back periodically due to finite-size effect bath [the remaining $(\mathrm{N}\ensuremath{-}1)$ TLSs]. We also find correlation entropy, which sums up entropy all n TLSs, approximately exhibit monotonic increase; contrast, each single increases decreases from time n-body keeps constant during evolution.