boxes containing the discs have finite energy, while the reservoirs have infinite energy, we believe that the transport properties of the system *in the stationary state* (assuming there is one) will be similar in both cases.

    Usually, the condition of zero net (average) particle and heat currents, between the system and a reservoir, is referred to as the self-consistency condition [@BolRV; @RichViss; @Visscher; @Davies; @BLebL; @roy]. If one is only interested in the electric current, the self-consistent reservoirs correspond to the *voltage probes* used in mesoscopic physics, originally introduced to model the inelastic scattering occurring in a conductor [@Buttiker-Reservoir; @ButtikerQuantum; @ButtikerBarriers; @Amato; @ButtikerQuantum2; @Buttiker-Review; @Ando; @Pilgram; @Heidi].

    A *crucial* distinction appears here: In voltage probes *only* the electric currents are set to zero while in this paper the self-consistent reservoirs require to set *both* the electric and heat currents to zero. To our knowledge, this fundamental difference has never been discussed in the formalism of Landauer-Büttiker and consequently makes one of the novel aspects of the present work. In particular, our results generalise works by, for example, Büttiker [@Buttiker-Reservoir; @ButtikerQuantum; @ButtikerBarriers], D’Amato and Pastawski [@Amato], Ando [@Ando] and very recently Roy and Dhar [@roy].

    If the couplings between the system 