Stochastic thermodynamics of a finite quantum system coupled to two heat baths

We consider a situation where an $N$-level system (NLS) is coupled successively to two heat baths with different temperatures without being necessarily thermalized and approaches a steady state. For this situation we apply a general Jarzinski-type equation and conclude that heat and entropy is flowing from the hot bath to the cold one. The Clausius relation between increase of entropy and transfer of heat divided by a suitable temperature assumes the form of two inequalities. Our approach is illustrated by an analytical example. For the linear regime, i.~e., for small temperature differences between the two heat baths we derive an expression for the heat conduction coefficient.