photon carrying heat in parallel to the electrons. Section \[Sect:Relax\] treats relaxation within the quantum system.

Comments on existing literature {#Sect:literature}
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There is much interest in using thermoelectric effects to cool fermionic atomic gases [@Grenier2012; @Brantut-Grenier-et-al2013; @Grenier2014], which are hard to cool via other methods. This physics is extremely similar to that in this work, but there is a crucial difference. For the electronic systems that we consider, we can assume the temperatures to be much less than the reservoir’s Fermi energy, and so take all electrons to have the same Fermi wavelength. In contrast, fermionic atomic gases have temperatures of order the Fermi energy, so the high-energy particles in a reservoir have a different wavelength from the low-energy ones. Thus, our results do not apply to atomic gases, although our methodology does[@Grenier2014].

Nonlinear systems and the figure of merit $ZT$ {#Sect:nonlinear+ZT}
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Engineers commonly state that wide-ranging applications for thermoelectrics would require them to have a dimensionless figure of merit, $ZT$, greater than three. This dimensionless figure of merit is a dimensionless combination of the linear-response coefficients [@books] $ZT= TGS^2/\Theta$, for temperature $T$, Seebeck coefficient $S$, electrical conductance $G$, and thermal conductance $\Theta$ . Yet for us, $ZT$ 