respectively. Once in the system, the non-interacting particles move freely and collide elastically with the boundaries of the cells as well as with the discs. The particles can also leave the system through these reservoirs.

Note that in this model the discs play the role of energy tanks, allowing the redistribution of energy among the particles in a given cell and thus permitting the system to reach at large times a stationary state satisfying local thermal equilibrium (LTE). As a consequence, this model admits a well-defined notion of local temperature.

One of the main results states that, under some assumptions, the average stationary particle and energy currents through the system are governed by Fick and Fourier laws.

In this paper, we will always assume that the considered system has reached its (unique) *stationary* state and discuss only its *average* transport properties. In particular, we shall only consider average stationary currents and consequently simply call them currents.

In the present work, we shall construct a similar type of model using the Landauer-Büttiker scattering approach [@Landauer; @Landauer1970; @ButtikerFourT; @Buttiker-Scattering; @ButtikerScattering]. This will permit, in particular, to establish an effective *quantum* version of the EY-model.

Basically, in the scattering approach to charge and 