Annales Henri Poincaré

, Volume 8, Issue 8, pp 1539-1593

First online:

Open Access This content is freely available online to anyone, anywhere at any time.

Theory of Non-Equilibrium Stationary States as a Theory of Resonances

  • Marco MerkliAffiliated withDepartment of Mathematics and Statistics, Memorial University of Newfoundland Email author 
  • , Matthias MückAffiliated withFachbereich Mathematik, Johannes Gutenberg-Universität
  • , Israel Michael SigalAffiliated withDepartment of Mathematics, University of Toronto


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). We show that the combined system has a family of stationary states parametrized by two numbers, T 1 and T 2 (‘reservoir temperatures’). If T 1T 2, then these states are non-equilibrium stationary states (NESS). In the latter case we show that they have nonvanishing heat fluxes and positive entropy production and are 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 T 1 and T 2, converges to the corresponding NESS. Our results are valid for the temperatures satisfying the bound min (T 1,T 2) > g 2 + α, where g is the coupling constant and 0 < α < 1 is a power related to the infra-red behaviour of the coupling functions.