Weighted Detailed Balance and Local KMS Condition for Non-Equilibrium Stationary States

We give a general denition 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 denition 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 dierent temperatures and study its cycle dynamics and entropy production.