  We analyze the stability properties shown by KMS states for interacting
massive scalar fields propagating over Minkowski spacetime, recently
constructed in the framework of perturbative algebraic quantum field theories
by Fredenhagen and Lindner \cite{FredenhagenLindner}. In particular, we prove
the validity of the return to equilibrium property when the interaction
Lagrangian has compact spatial support. Surprisingly, this does not hold
anymore, if the adiabatic limit is considered, namely when the interaction
Lagrangian is invariant under spatial translations. Consequently, an
equilibrium state under the adiabatic limit for a perturbative interacting
theory evolved with the free dynamics does not converge anymore to the free
equilibrium state. Actually, we show that its ergodic mean converges to a non
equilibrium steady state for the free theory.
