  Non-equilibrium steady states (NESS) describe particularly simple and
stationary non-equilibrium situations. A possibility to obtain such states is
to consider the asymptotic evolution of two infinite heat baths brought into
thermal contact. In this work we generalise corresponding results of
Doyon~et.~al. (J.\ Phys.\ A 18 (2015) no.9) for free Klein-Gordon fields in
several directions. Our analysis is carried out directly at the level of
correlation functions and in the algebraic approach to QFT. We discuss
non-trivial chemical potentials, condensates, inhomogeneous linear models and
homogeneous interacting ones. We shall not consider a sharp contact at initial
time, but a smooth transition region. As a consequence, the states we construct
will be of Hadamard type, and thus sufficiently regular for the perturbative
treatment of interacting models. Our analysis shows that perturbatively
constructed interacting NESS display thermodynamic properties similar to the
ones of the NESS in linear models. In particular, perturbation theory appears
to be insufficient to describe full thermalisation in non-linear QFT.
Notwithstanding, we find that the NESS for linear and interacting models is
stable under small perturbations, which is one of the characteristic features
of equilibrium states.
