are independent and inject particles into the system according to some distribution function. We assume that they can also absorb particles without changing their state. In this paper, we shall consider the following three cases ($i \in \{\L, \R, 1, \dots, N\}$): $$\label{Maxwell-Boltzmann}
f^{\rm MB}_i(E) \equiv f^{\rm MB}(E; T_i, \mu_i) = \exp\left(-\frac{E-\mu_i}{k_\B
    T_i}\right)~,$$ $$\label{Fermi-Dirac}
f^{\rm FD}_i(E) \equiv f^{\rm FD}(E; T_i, \mu_i) = \left[\exp\left(\frac{E-\mu_i}{k_\B
    T_i}\right) + 1\right]^{-1}~,$$

$$\label{Bose-Einstein}
f^{\rm BE}_i(E) \equiv f^{\rm BE}(E; T_i, \mu_i) = \left[\exp\left(\frac{E-\mu_i}{k_\B
    T_i}\right) - 1\right]^{-1}~.$$

Here, $E \in [0,\infty)$ is the energy, $k_\B$ is Boltzmann’s constant and $T_i > 0$ is the temperature of the $i$-th terminal. For the chemical potentials there is one subtlety: for the Maxwell-Boltzmann and Fermi-Dirac functions, one has $\mu_i \in (-\infty, \infty)$, while for the Bose-Einstein function, one has $\mu_i \in (-\infty, 0)$. Let us emphasise that, for each $i$, the parameters $T_i$ and $\mu_i$ are *independent* of each other.

We consider that the transport properties of the $k$-th quantum dot are described by a scattering matrix $S^{(k)}(E)$ at energy $E > 0$: $$S^{(k)}(E) = \left(S^{(k)}_{ij;mn}(E)\right)~, \hspace{5mm} k \in \{1, \dots, N\}~,$$ where $i,j$ denote the three possible entrances of the $k$-th dot and the indices $m,n$ denote their corresponding channels. 