  Bridging the second law of thermodynamics and microscopic reversible dynamics
has been a longstanding problem in statistical physics. We here address this
problem on the basis of quantum many-body physics, and discuss how the entropy
production saturates in isolated quantum systems under unitary dynamics. First,
we rigorously prove the saturation of the entropy production in the long time
regime, where a total system can be in a pure state. Second, we discuss the
non-negativity of the entropy production at saturation, implying the second law
of thermodynamics. This is based on the eigenstate thermalization hypothesis
(ETH), which states that even a single energy eigenstate is thermal. We also
numerically demonstrate that the entropy production saturates at a non-negative
value even when the initial state of a heat bath is a single energy eigenstate.
Our results reveal fundamental properties of the entropy production in isolated
quantum systems at late times.
