  Isolated quantum systems follow the unitary evolution, which guarantees the
full many body state always keeps a constant entropy as its initial one. In
comparison, the local subsystems exhibit relaxation behavior and evolve towards
certain steady states, which is called the local relaxation. Here we consider
the local dynamics of finite many body system with Ising interaction. In both
strong and weak coupling situations, the local observables exhibit similar
relaxation behavior as the macroscopic thermodynamics; due to the finite size
effect, recurrence appears after a certain typical time. Especially, we find
that the total correlation of this system approximately exhibits a monotonic
increasing envelope in both strong and weak coupling cases, which corresponds
to the irreversible entropy production in the standard macroscopic
thermodynamics. Moreover, the possible maximum of such total correlation
calculated under proper constraints also coincides well with the exact result
of time dependent evolution.
