Quantum equilibration under constraints and transport balance.

For open quantum systems coupled to a thermal bath at inverse temperature β, it is well known that under the Born, Markov, and secular approximations, the system density matrix will approach the thermal Gibbs state with the bath inverse temperature β. We generalize this to systems where there exists a conserved quantity (e.g., the total particle number), where for a bath characterized by inverse temperature β and chemical potential μ, we find equilibration of both temperature and chemical potential. For couplings to multiple baths held at different temperatures and different chemical potentials, we identify a class of systems that equilibrates according to a single hypothetical average but in general nonthermal bath, which may be exploited to generate desired nonthermal states. Under special circumstances, the stationary state may again be described by a unique Boltzmann factor. These results are illustrated by several examples.