C(0) \Gamma_{ij}~,
\hspace{5mm}  L_{ij}^{(1)} = Q_1 L_{ij}^{(0)}  \hspace{5mm}
\mbox{and} \hspace{5mm}
L_{ij}^{(2)} = Q_2 L_{ij}^{(0)}~,$$ where $$Q_1 =\frac{k_\B T}{e} \frac{C(1)}{C(0)} > 0 \hspace{5mm}
\mbox{and} \hspace{5mm} Q_2 = \frac{k_\B^2 T^2}{e^2} \frac{C(2)}{C(0)} > 0~.\label{L4L1}$$

\[Autre remarks FD2\] [The quantity $Q_1$ is related to the Seebeck coefficient (see  \[Rem Seebeck\]). In the Fermi-Dirac case, $\lim_{x_0 \rightarrow  -\infty} Q_2/T^2 = (\pi^2/3) (k_\B/e)^2$ is called the Lorentz number.]{.nodecor}

Recalling that $\Gamma_{ij} =
M_i \delta_{ij} - t_{ij}$, one immediately obtains the following properties ($k = 0,1,2$): $$\label{Property Lij}
L^{(k)}_{ij} \ \left\{\begin{array}{ll} > 0 &
\mbox{if } \ i
= j\\
< 0 &
\mbox{if } \ i
\not = j
\end{array}\right.~,$$ and $$\label{Property Lij2}
\sum_{i} L^{(k)}_{ij} = 0~, \hspace{2mm} \forall j \hspace{7mm} \mbox{and} \hspace{7mm} \sum_{j} L^{(k)}_{ij} =
0~, \hspace{2mm} \forall i~.$$

As we shall see later on, the following ratio, $\mathcal{R}$, will play an important role. Using standard techniques, we show in Appendix B that (for all $x_0$):

$$\label{Ratio R}
\mathcal{R} \equiv \frac{Q_1}{\sqrt{Q_2}} =
\frac{C(1)}{\sqrt{C(0) \cdot C(2)}} \in (0,1)~.$$

[In terms of the matrix $$C = {\left(\hspace{-2mm}\begin{array}{cccccccccccccccccc}C(0) & C(1)\\C(1) & C(2)\\ \end{array}\hspace{-2mm}\right)},$$ the inequality $0 < \mathcal{R} < 1$ is equivalent to $\det(C) = C(0)
C(2) - C(1)^2 > 0$.]{.nodecor}

The Entropy Production {#The entropy production}
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