do not need to describe the non-equilibrium state of the quantum system during step 2. Instead, we need that quantum system’s transmission function, defined in section \[Sect:ScatteringTheory\].

In this article we will [*only*]{} discuss the long-time average of the rates of flows (not the noisy instantaneous flows), and thus will not explicitly indicate the average; so $I_i$, $J_i$ and $\dot S_i$ should be interpreted as $\langle I_i \rangle$, $\langle J_i \rangle$ and $\langle\dot S_i \rangle$.

Entropy production {#Sect:entropy-prod}
==================

There are little known universal relations between efficiency, power and and entropy production, which follow trivially from the laws of thermodynamics [@Cleuren2012]. Consider the lower thermoelectric in Fig. \[Fig:thermocouple\]a (filled circle), with $J_L$ and $J_R$ being steady-state heat currents into it from reservoir $L$ and R. Then the first law of thermodynamics is $$\begin{aligned}
J_R + J_L=P_{\rm gen},
\label{Eq:firstlaw}\end{aligned}$$ where $P_{\rm gen}$ is the electrical power generated. The Clausius relation for the rate of change of total entropy averaged over long times as in Eq. (\[Eq:average-dotS-def\]), is $$\begin{aligned}
\dot S  = -{J_L \over T_L} + {J_L - P_{\rm gen} \over T_R},
\label{Eq:secondlaw}\end{aligned}$$ where we have used Eq. (\[Eq:firstlaw\]) to eliminate $J_R$.

For a heat engine, we take $J_L$ to be positive, 