& \dots &
  L^{(k)}_{1 N} & L^{(k)}_{1 \R}\\ \vdots & \vdots &  \vdots & \vdots
  & \vdots \\ L^{(k)}_{N \L} &
  L^{(k)}_{N 1} & \dots & L^{(k)}_{N N} & L^{(k)}_{N \R} \\ L^{(k)}_{\R \L} &
  L^{(k)}_{\R 1} & \dots & L^{(k)}_{\R N} & L^{(k)}_{\R \R} \\ \end{array}\hspace{-2mm}\right)}~.$$

A quick glance at the expressions – for the transport coefficients $L_{ij}^{(k)}$, remembering that $\Gamma_{ij}(E) =
M_i \delta_{ij} - t_{ij}(E)$, shows that the Onsager relations hold, if $S(E)$ is symmetric (for each $E > 0$), then $L$ is symmetric. More generally, if the system satisfies the micro-reversibility property $S_{ij,mn}(E,B) = S_{ji,nm}(E,-B)$, where $B$ is some applied magnetic field, then $L_{ij}(B) = L_{ji}(-B)$.

In general, the scattering matrix $S$ depends on the energy. Nevertheless, in this paper, we shall restrict our attention to the energy-independent situations. The assumption that $S$ does not depend on the energy is a good approximation in some limiting cases (looking at – one sees that for electrons at low temperature one may consider $S(E) \equiv S(E_\F)$, where $E_\F$ is the Fermi energy, since $(-\partial f/\partial E)(E) \approx
\delta(E-E_\F)$) and leads to interesting consequences. In particular, we will see that the relations among the transport coefficients, $L_{ij}^{(0)}$, $L_{ij}^{(1)}$ and $L_{ij}^{(2)}$, 