f_R^\mu ({\epsilon})\big],
\nonumber \\
\label{Eq:JL}
\\
J_R  \! &=& \!
{1 \over h} \sum_\mu \int_0^\infty {\rm d}{\epsilon}\, ({\epsilon}-  \mu{ e^{\operatorname{-}} }V_R) \,  
{\cal T}^{\mu\mu}_{RL}({\epsilon})   \, \big[f_R^\mu ({\epsilon}) - f_L^\mu ({\epsilon})\big],
\nonumber \\
\label{Eq:JR}\end{aligned}$$ where ${ e^{\operatorname{-}} }$ is the electron charge (${ e^{\operatorname{-}} }<0$), so ${ e^{\operatorname{-}} }V_i$ is the chemical potential of reservoir $i$ measured from the reference chemical potential (${\epsilon}=0$). The sum is over $\mu=1$ for “electron” states (full states above the reference chemical potential), and $\mu=-1$ for “hole” states (empty states below that chemical potential). The Fermi function for particles entering from reservoir $j$, is $$\begin{aligned}
f_j^\mu({\epsilon}) = \left(1+\exp\left[({\epsilon}- \mu { e^{\operatorname{-}} }V_j)\big/ ({k_{\rm B}}T_j) \right] \right)^{-1}.
\label{Eq:Fermi}\end{aligned}$$ The transmission function, ${\cal T}^{\nu\mu}_{ij}({\epsilon})$, is the probability that a particle $\mu$ with energy ${\epsilon}$ entering the quantum system from reservoir $j$ will exit into reservoir $i$ as a particle $\nu$ with energy ${\epsilon}$. We only allow $\nu=\mu$ here, since we do not consider electron to hole scattering within the quantum system (only common when superconductors are present). Interactions mean that ${\cal T}^{\mu\mu}_{RL}({\epsilon})$, is a [*self-consistently*]{} determined function of $T_L$, $T_R$ $V_L$ and $V_R$.

The system generates power $P_{\rm gen} = (V_R-V_L) I_L$, so $$\begin{aligned}
P_{\rm gen} \! &=& 