\{\L, \R, 1,\dots,N\}$): $$\begin{aligned}
I_i &=& \frac{e}{h} \sum_{j} \int_{0}^{\infty} f_j(E)
\Gamma_{ij}(E) \, dE~,\label{Electric Current Def}\\
J_i &=& \frac{1}{h} \sum_{j} \int_{0}^{\infty} f_j(E)
\Gamma_{ij}(E) (E - \mu_i) \, dE~.\label{Heat Current Def}\end{aligned}$$

From one immediately obtains the following properties: $$\sum_{i} \Gamma_{ij}(E) = 0~, \hspace{2mm} \forall j \hspace{7mm} \mbox{and} \hspace{7mm} \sum_{j} \Gamma_{ij}(E) =
0~, \hspace{2mm} \forall i~.\label{Property Conductance}$$

[Note that $\Gamma_{ij}(E) \not = \Gamma_{ji}(E)$ in general.]{.nodecor}

Using the relations one immediately sees that the expressions – for the currents satisfy the conservation of charge and energy, respectively, $$\label{conservation currents}
\sum_{i} I_{i} = 0 \hspace{7mm} \mbox{and} \hspace{7mm} \sum_{i} \left(J_{i}
+ \frac{\mu_i}{e} I_{i}\right) = 0~.$$

[The second relation in emphasises the fact that, in general, energy is conserved but not heat, in the sense that $\sum_{i} J_{i} \not = 0$ in general. However, as we shall see in the next section (see  \[remark heat\]), heat is conserved in the linear response approximation.]{.nodecor}

The Strategy {#The strategy}
------------

In  \[The null currents condition\], we will fix the values $(T_\L,\mu_\L)$ and $(T_\R,\mu_\R)$, and determine the parameters $(T_i,\mu_i)$, for $i = 1\dots,N$, for which the *self-consistency condition* is satisfied: $$I_i = 0 \quad \mbox{and} \quad J_i = 0 \quad \mbox{for} \quad i =
1,\dots,N~.\label{Condition Zero Currents}$$ Actually, this condition 