\rangle = \langle \dot S \rangle +\langle \dot S' \rangle$, where $\langle \dot S \rangle$ is the average rate of change of entropy associated with the lower thermoelectric in Fig. (\[Fig:thermocouple\]), while $\langle \dot S' \rangle$ is that for the upper thermoelectric. Then $$\begin{aligned}
\langle \dot S\rangle = \langle \dot S_L \rangle + \langle \dot S_R \rangle 
 =  -{\langle J_{\rm L} \rangle \big/ T_L} \,-\, {\langle J_{\rm R} \rangle \big/ T_R}\,,
\label{Eq:average-dotS-def}\end{aligned}$$ while $\langle \dot S'\rangle$ is the same with $J_L,J_R,T_R$ replaced by $J_{L'},J_{R'},T_{R'}$. We neglect the entropy of the thermoelectrics and load, by assuming their initial and final state are the same. This will be the case if they are small compared to the reservoirs, so their initial and final states a simply given by the temperature $T_R$.

The nonlinear scattering theory in Ref. \[\] captures long-time average currents (usually called the DC response in electronics), such as electrical current $\langle I_i \rangle$ and heat current $\langle J_i \rangle$, see references in Section \[Sect:ScatteringTheory\]. It is believed to be exact for non-interacting particles, and also applies when interactions can be treated in a mean-field approximation (see again section \[Sect:ScatteringTheory\]). A crucial aspect of the scattering theory is that we 