Weighted detailed balance and local KMS condition for non-equilibrium stationary states

We give a general definition of the local KMS condition and we prove its equivalence with a nonlinear Gibbs prescription. We discuss the irreversible $(H, \beta)$-KMS condition, its connections with the local KMS condition and we study the irreversible $(H, \beta)$-KMS condition for Markov generators of stochastic limit type. We introduce a definition of weighted detailed balance based on the notion of current decomposition and discuss invariant states with constant micro--currents. As an example, we construct a non-equilibrium steady state for a quantum spin chain coupled to two reservoirs at different temperatures and study its cycle dynamics and entropy production.