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Mesoscopic thermoelectric heat engine is much anticipated as a device thatallows us to utilize with high efficiency wasted heat inaccessible byconventional heat engines. However, the derivation of the heat current in thisengine seems to be either not general or described too briefly, eveninappropriate in some cases. In this paper, we give a clear-cut derivation ofthe heat current of the engine with suitable assumptions beyond thelinear-response regime. It resolves the confusion in the definition of the heatcurrent in the linear-response regime. After verifying that we can constructthe same formalism as that of the cyclic engine, we find the following twointeresting results within the Landauer-B\"uttiker formalism: the efficiency ofthe mesoscopic thermoelectric engine reaches the Carnot efficiency if and onlyif the transmission probability is finite at a specific energy and zerootherwise; the unitarity of the transmission probability guarantees the secondlaw of thermodynamics, invalidating Benenti et al.'s argument in thelinear-response regime that one could obtain a finite power with the Carnotefficiency under a broken time-reversal symmetry. These results demonstrate howquantum mechanics constraints thermodynamics.

Thermodynamics of the mesoscopic thermoelectric heat engine beyond the linear-response regime