Vojkan Jaksic, Claude-Alain Pillet
Non-equilibrium steady states of finite quantum systems
coupled to thermal reservoirs
(329K, postscript)
ABSTRACT. We study the non-equilibrium statistical mechanics of a $2$-level
quantum system, ${\cal S}$, coupled to two independent free Fermi
reservoirs ${\cal R}_1$, ${\cal R}_2$, which are in thermal equilibrium
at inverse temperatures $\beta_1\not=\beta_2$. We prove that, at small
coupling, the combined quantum system ${\cal S} + {\cal R}_1 + {\cal R}_2$
has a unique non-equilibrium steady state (NESS) and that the approach
to this NESS is exponentially fast. We show that the entropy production
of the coupled system is strictly positive and relate this entropy
production to the heat fluxes through the system.
A part of our argument is general and deals with spectral theory of
NESS. In the abstract setting of algebraic quantum statistical
mechanics we introduce the new concept of $C$-Liouvillean, $L$, and
relate the NESS to zero resonance eigenfunctions of $L^\ast$.
In the specific model ${\cal S} + {\cal R}_1 + {\cal R}_2$ we study
the resonances of $L^\ast$ using the complex deformation
technique developed previously by the authors.