In this subsection, we show that the transport matrix $L$ is real positive semi-definite. In other words, we show that the entropy production rate is non-negative: $$\label{entr prod}
\sigma_s = \sum_{i,j=1}^{2N+4}  L_{ij} V_i V_j \geq 0~,$$ where the thermodynamic forces are $$V_i = \left\{ \begin{array}{ll}\delta \mu_{i-1}/e & \mbox{ if } \; \, i = 1,
  \dots, N+2\\ \delta T_{i-(N+3)}/T &  \mbox{ if } \; \; i = N+3, \dots, 2N+4\end{array}\right. ~.$$ Here, we have set $\L = 0$ and $\R = N+1$. Let $X_i = \delta \mu_i/e$ and $Z_i = \sqrt{Q_2} \ \delta
T_i/T$, for $i \in \{0, 1, \dots, N, N+1\}$. Then, in Appendix C, we show that one can rewrite the entropy production rate as follows: $$\label{sigmas expression}
\sigma_s = \sum_{\scriptsize\begin{array}{c}
    i,j = 0\\ i < j \end{array}}^{N+1} (-L^{(0)}_{ij}) \ I_{ij}~,$$ where $$I_{ij} = (X_i - X_j)^2
  + (Z_i - Z_j)^2 -  2 \mathcal{R} C_{ij}~, \hspace{5mm} C_{ij} = X_i Z_j + X_j Z_i - X_i Z_i - X_j Z_j~.$$ Observe now that $(-L^{(0)}_{ij}) > 0$ for all $i \not = j$. Therefore, to show that $\sigma_s \geq 0$ it is sufficient to show that $I_{ij} \geq 0$ for all $i \not = j$. As we shall see, it 