  We consider a situation where an $N$-level system (NLS) is coupled to a heat
bath without being necessarily thermalized. For this situation we derive
general Jarzinski-type equations and conclude that heat and entropy is flowing
from the hot bath to the cold NLS and, vice versa, from the hot NLS to the cold
bath. The Clausius relation between increase of entropy and transfer of heat
divided by a suitable temperature assumes the form of two inequalities which
have already been considered in the literature. Our approach is illustrated by
an analytical example.
