\, \frac{\delta T_j}{T}~,\label{The heat
  current}\end{aligned}$$ where $$\begin{aligned}
L_{ij}^{(0)} &=& -\frac{e^2}{h} \int_{0}^{\infty} \frac{\partial
  f}{\partial E}(E; T, \mu) \Gamma_{ij}(E) \, dE~,\label{transp 1}\\
L_{ij}^{(1)} &=& -\frac{e}{h} k_\B T \int_{0}^{\infty} \left(\frac{E-\mu}{k_\B T}\right) \frac{\partial
  f}{\partial E}(E; T, \mu) \Gamma_{ij}(E) \, dE~,\label{transp 2}\\
L_{ij}^{(2)} &=& -\frac{(k_\B T)^2}{h} \int_{0}^{\infty} \left(\frac{E-\mu}{k_\B T}\right)^2 \frac{\partial
  f}{\partial E}(E; T, \mu) \Gamma_{ij}(E)  \, dE~.\label{transp 3}\end{aligned}$$

\[remark heat\] [Using the relations , one sees that the first order approximations for the heat currents satisfy $$\sum_{i} J_{i} = 0~.\label{cons heat}$$ In  \[The null currents condition\], we will impose the self-consistency condition, $I_i = J_i = 0$ for $i =
1,\dots,N$, and the conservation laws and will permit to interpret $I_\L = -I_\R$ and $J_\L = -J_\R$ as the electric and heat currents flowing through the system (from left to right).]{.nodecor}

The Transport Matrix {#The transport matrix}
--------------------

Here we establish some basic properties of the transport matrix. From the expressions – for the currents we are naturally led to the following definition.

\[def transport matrix\] We denote by $L$ the transport matrix: $$L = {\left(\hspace{-2mm}\begin{array}{cccccccccccccccccc}L^{(0)} & L^{(1)}\\ L^{(1)} & L^{(2)}\\ \end{array}\hspace{-2mm}\right)}~, \hspace{3mm} \mbox{where}  \hspace{3mm}  L^{(k)} =
  {\left(\hspace{-2mm}\begin{array}{cccccccccccccccccc}L^{(k)}_{\L \L} & \L^{(k)}_{\L 1} & \dots & L^{(k)}_{\ LN} &
  \L^{(k)}_{\L \R}\\ L^{(k)}_{1 \L} & \L^{(k)}_{1 1} 