continuum thermodynamics[@Kern1975; @Day1977; @book:irreversible-thermodyn]. Here the situation is even more difficult, since the electrons within the quantum systems (circles in Fig. \[Fig:thermocouple\]) are not at equilibrium, and so their temperature cannot be defined. Thus, it is crucial to specify the logic which leads to our definitions of temperature, heat flow and entropy flow.

Our definition of heat flow originated in Refs. \[\], the rate of change of entropy is then found using the Clausius relation [@Footnote:Sivan-Imry] (see below). To explain these quantities and show they are unambiguous, we consider the following three step procedure for a heat engine. An analogue procedure works for a refrigerator.

-   [**Step 1.**]{} Reservoir $L$ is initially decoupled from the rest of the circuit (see Fig. \[Fig:Unique\]a), has internal heat energy $Q_L^{(0)}$, and is in internal equilibrium at temperature $T_L^{(0)}$. The rest of the circuit is in equilibrium at temperature $T_R^{(0)}$ with internal heat energy $Q_R^{(0)}$. The internal heat energy is the total energy of the reservoir’s electron gas minus the energy which that gas would have in its ground-state. As such, the internal energy can be written as a sum over electrons and holes, with an electron at energy ${\epsilon}$ above the reservoir’s chemical 