will be crucial that the ratio $\mathcal{R}$ satisfies $0 < \mathcal{R} < 1$. Assume first that $C_{ij} \leq 0$, then ($
\mathcal{R} > 0$) $$\label{In1a}
I_{ij} \geq (X_i - X_j)^2 + (Z_i - Z_j)^2 \geq 0~.$$ Assume next that $C_{ij} > 0$, then ($\mathcal{R} < 1$) $$I_{ij} >  (X_i - X_j)^2 + (Z_i - Z_j)^2  - 2 C_{ij} = (X_i - X_j + Z_i - Z_j)^2 \geq 0~.$$ This ends the proof that $\sigma_s \geq 0$.

[The term $(X_i - X_j)^2$ accounts for the entropy production due to the electric current, and it is well known in mesoscopic physics (see [@Buttiker-Symmetry]). The term $(Z_i - Z_j)^2$ is related to the heat current, and one sees that there is also a thermoelectric term, $-2 \mathcal{R} C_{ij}$, which might take positive and negative values.]{.nodecor}

Equilibrium and Non-Equilibrium States {#Equilibrium
  and non-equilibrium states}
--------------------------------------

Here we present some equivalent characterizations of the equilibrium and non-equilibrium states of the multi-terminal system. We start with the following.

We say that the system is at *equilibrium* if $$T_\L = T_1 = \dots = T_N = T_\R  \hspace{5mm}\mbox{ and } \hspace{5mm}
\mu_\L = \mu_1 = \dots = \mu_N = \mu_\R~.$$ Otherwise, we say that the system is 