\!  
{1\over h} \sum_\mu \int_0^\infty {\rm d}{\epsilon}\ \mu{ e^{\operatorname{-}} }(V_R-V_L)\,
{\cal T}^{\mu\mu}_{RL}({\epsilon}) \, \big[f_L^\mu ({\epsilon}) - f_R^\mu ({\epsilon})\big], 
\nonumber \\  
\label{Eq:Pgen}\end{aligned}$$ It is easy to verify that Eqs. (\[Eq:JL\]-\[Eq:Pgen\]) satisfy the first law of thermodynamics, Eq. (\[Eq:firstlaw\]). This theory assumes the quantum system to be relaxation-free, although decoherence is allowed as it does not change the structure of Eqs. (\[Eq:JL\]-\[Eq:Pgen\]). Relaxation is discussed in Section \[Sect:Relax\].

We define the voltage drop as $V=V_R-V_L$. Without loss of generality we take the reference chemical potential to be that of reservoir $L$, so $$\begin{aligned}
V_L=0, \qquad V_R=V,
\label{Eq:def-V}\end{aligned}$$ then $J_L$ and $P_{\rm gen}$ coincide with Eqs. (8,9) in Ref. \[\].

Numerous works have found the properties of thermoelectric systems from their transmission functions, ${\cal T}_{RL}({\epsilon})$. Linear-response examples include Refs. \[\], while nonlinear responses were considered in Refs. \[\], see Refs. \[\] for recent reviews. However, here we do not ask what is the efficiency of a given system, we ask what is the system that would achieve the highest efficiency, and what is this efficiency? This is similar in spirit to Ref. \[\], except that we maximize the efficiency for given power output.

We need to answer this question in the context of 