\sum_{j} t_{ij}(E) =
M_i~, \hspace{2mm} \forall i~.\label{Property Transmission}$$

[The sums appearing in are over the set $\{\L, \R, 1, \dots, N\}$ of *all* terminals. Unless stated, in what follows every sum will be understood over this set.]{.nodecor}

The Currents
------------

In this paper, we shall always assume that the following holds.

-   The particles do not interact with each other.

-   The system admits a unique stationary state (out of equilibrium).

\[Assumptions\]

Under these assumptions, one can derive for any of the distributions – the expressions for the (average) *electric currents* in a multi-terminal system [@ButtikerScattering; @Buttiker-Review; @Datta; @CA]. One finds $$I_i = \frac{e}{h} \sum_{j} \int_{0}^{\infty} \left[t_{ji}(E) f_i(E) - t_{ij}(E) f_j(E)\right] dE~,\label{Electrical Current 1}$$ where $e > 0$ is the charge carried by the particles and $h$ is Planck’s constant.

[To our knowledge, the expression for the average electric current in a multi-terminal system was first introduced, through formal arguments, by Büttiker in [@ButtikerScattering] and was very recently given a rigorous and general derivation (together with ) in a work by Aschbacher *et al.* [@CA] using a $C^{*}$ algebraic approach.]{.nodecor}

The expression can be interpreted as follows: $t_{ji}(E) f_i(E)$ is the average number of particles with energy $E$ that are 