potential (or a hole at energy ${\epsilon}$ below that chemical potential) contributing ${\epsilon}$ to this internal heat energy. The initial entropies are then $S^{(0)}_i = Q^{(0)}_i \big/ T^{(0)}_i$ for $i=L,R$.

-   [**Step 2.**]{} We connect reservoir $L$ to the rest of the circuit ( (see Fig. \[Fig:Unique\]b) and leave it connected for a long time $t_{\rm expt}$. While we assume $t_{\rm expt}$ is long, we also assume that the reservoirs are all large enough that the energy distributions within them change very little during time $t_{\rm expt}$. Upon connecting the circuit elements, we assume a transient response during a time $t_{\rm trans}$, after which the circuit achieves a steady-state. We ensure that $t_{\rm expt}\gg t_{\rm trans}$, so the physics is dominated by this steady-state. Even then the flow will be noisy [@Blanter-Buttiker] due to the fact electrons are discrete with probabilistic dynamics. So we also ensure that $t_{\rm expt}$ is much longer than the noise correlation time, so that the noise in the currents is negligible compared to the average currents.

-   [**Step 3.**]{} After the time $t_{\rm expt}$, we disconnect reservoir $L$ from the rest of the circuit. Again, there will be a transient response, however we assume that 