The correlational entropy production during the local relaxation in a many body system with Ising interactions

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.