  We consider translation-invariant interacting particle systems on the lattice
with finite local state space admitting at least one Gibbs measure as a
time-stationary measure. The dynamics can be irreversible but should satisfy
some mild non-degeneracy conditions. We prove that weak limit points of any
trajectory of translation-invariant measures, satisfying a non-nullness
condition, are Gibbs states for the same specification as the time-stationary
measure. This is done under the additional assumption that zero entropy loss of
the limiting measure w.r.t. the time-stationary measure implies that they are
Gibbs measures for the same specification. We show how to prove the
non-nullness for a large number of cases, and also give an alternate version of
the last condition such that the non-nullness requirement can be dropped. As an
application we obtain the attractor property if there is a reversible Gibbs
measure. Our method generalizes convergence results using relative entropy
techniques to a large class of dynamics including irreversible and non-ergodic
ones.
