will be very simple and will lead to some universal transport properties. For definiteness, we assume for the rest of this paper that the following holds.

The scattering matrix $S$ does not depend on the energy.\[Assumption Gamma\]

In order to simplify some derivations, without restricting our results too much, it is convenient to assume the following.

We assume that $t_{ij} \not = 0$ for all $i,j \in \{\L,\R, 1,\dots, N\}$.\[Assumption Gamma2\]

[Recalling the properties , one sees that the Assumption A\[Assumption Gamma2\] implies that $t_{ij}
  \not = \min\{M_i,
  M_j\}$, for all $i,j \in \{\L,\R, 1,\dots, N\}$, and this permits to avoid total back-scattering ($t_{ii} \not = M_i$ for all $i$). Although not optimal, the Assumption A\[Assumption Gamma2\] also insures the existence and unicity of the solution of the self-consistently condition .]{.nodecor}

Under the assumption that the scattering matrix $S$ does not depend on the energy, one sees in the expressions for the transport coefficients – that some integrals of the following form appear (with $n = 0, 1, 2$): $$C(n) = - \int_{0}^{\infty} \left(\frac{E-\mu}{k_\B T}\right)^n \frac{\partial
  f}{\partial E}(E; T, \mu) \, dE~.$$ More explicitly, $$C^{\rm MB}(n) = \int_{x_0}^{\infty}
  x^n e^{-x} \ dx \hspace{5mm} \mbox{and} \hspace{5mm}C^{\rm \pm}(n) =  \int_{x_0}^{\infty}
  \frac{x^n e^x}{(e^x 