  We analyze the relative entropy of certain KMS states for scalar
self-interacting quantum field theories over Minkowski backgrounds that have
been recently constructed by Fredenhagen and Lindner in [FL14] in the framework
of perturbative algebraic quantum field theory. The definition we are using is
a generalization of the Araki relative entropy to the case of field theories.
In particular, we shall see that the analyzed relative entropy is positive in
the sense of perturbation theory, hence, even if the relative modular operator
is not at disposal in this context, the proposed extension is compatible with
perturbation theory. In the second part of the paper we analyze the adiabatic
limits of these states showing that also the density of relative entropy
obtained dividing the relative entropy by the spatial volume of the region
where interaction takes place is positive and finite. In the last part of the
paper we discuss the entropy production for states obtained by an ergodic mean
(time average) of perturbed KMS states evolved with the free evolution recently
constructed by the authors of the present paper. We show that their entropy
production vanishes even if return to equilibrium [Ro73, HKT74] does not hold.
This means that states constructed in this way are thermodynamically simple,
namely they are not so far from equilibrium states.
