  In recent years, there has been tremendous focus on identifying whether
effective descriptions of open quantum systems such as master equations,
provide the correct steady state in the long time limit. The correct steady
state is usually not known, however it can be approximated by means of the Mean
Force Hamiltonian up to some fixed order, the reaction coordinate mapping or
other pseudo-mode like approaches. A few years ago a controversy arose
concerning the thermalization of master equations, namely, whether steady state
coherences actually arise in systems with composite interactions or if they are
some artifact caused by nonpositivity of the effective description, in this
manuscript we confirm the existence of such steady state coherences in
numerically exact dynamics, and see reminiscences of it in a CPTP map, while
the steady state coherence seems to be real its oscillatory characyer seems to
be an artifact of second order approximations, indicating that rather than
being steady state coherences as one may think, the equilibrium state of the
system is not diagonal in the basis of the system Hamiltonian. This paper also
shows evidence than comparing Hamiltonian corrections calculated from the mean
force approach and the quasi-steady states of dynamical equations may be
deceiving, as dynamics may not tend to the state with the correction as shown
in this paper.
