Theory of Non-Equilibrium Stationary States as a Theory of Resonances. Existence and Properties of NESS

We study a small quantum system (e.g. a simplified model for an atom or molecule) interacting with two bosonic or fermionic reservoirs (say, photon or phonon fields) at different temperatures T1 and T2. We show that if T1 is not equal to T2 then the combined system has a stationary, non-equilibrium state (NESS). We show that this state has nonvanishing heat fluxes and positive entropy production and that it is dynamically asymptotically stable. The latter means that the evolution with an initial condition, normal with respect to any state where the reservoirs are in equilibria at temperatures T1 and T2, converges to this NESS. Our results are valid for the temperatures satisfying the bound min(T1, T2) > g , where g is the coupling constant and 0 < α < 1 is a power related to the infra-red behaviour of the coupling functions. This restriction is introduced in order to present the setup and techniques without extra hard and lengthy technical estimates. In a subsequent work we combine the present setup and techniques with the spectral renormalization group method to obtain the results as above for all temperatures.