Successes and failures in the construction of NESS-states

We construct non-equilibrium states by coupling a large but finite system with quasifree evolution to two temperature baths of free fermions or bosons with different temperature. We are interested in the resulting behaviour of the temperature baths as well as in the consequences for the finite system, especially what happens in the limit when the finite system tends to infinity. As a special example we consider the Kronig-Penney model and the tight binding model. Here a heat current remains that changes the baths but does not allow a limiting behaviour when the finite system tends to infinity. For the random versions of these models, especially the Anderson model, the heat current disappears exponentially with the size of the system. We show that a limit state is attained that is independent of the time direction and does not show any kind of symmetry breaking.