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We consider a situation where an $N$-level system (NLS) is coupledsuccessively to two heat baths with different temperatures without beingnecessarily thermalized and approaches a steady state. For this situation weapply a general Jarzinski-type equation and conclude that heat and entropy isflowing from the hot bath to the cold one. The Clausius relation betweenincrease of entropy and transfer of heat divided by a suitable temperatureassumes the form of two inequalities. Our approach is illustrated by ananalytical example. For the linear regime, i.~e., for small temperaturedifferences between the two heat baths we derive an expression for the heatconduction coefficient.

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