Rigorous meaning of McLennan ensembles

We analyze the exact meaning of expressions for nonequilibrium stationary distributions in terms of entropy changes. They were originally introduced by McLennan for mechanical systems close to equilibrium and more recent work by Komatsu and Nakagawa has shown their intimate relation to the transient fluctuation symmetry. Here we derive these distributions for jump and diffusion Markov processes and we clarify the order of the limits that take the system both to its stationary regime and to the close-to-equilibrium regime. In particular, we prove that it is exactly the (finite) transient component of the irreversible part of the entropy flux that corrects the Boltzmann distribution to first order in the driving. We add further connections with the notion of local equilibrium, with the Green-Kubo relation and with a generalized expression for the stationary distribution in terms of a reference equilibrium process.