  We continue the investigation of kinetic models of a system in contact via
stochastic interactions with several spatially homogeneous thermal reservoirs
at different temperatures. Considering models different from those investigated
in earlier work, we explicitly compute the unique spatially uniform
non-equilibrium steady state (NESS) and prove that it is approached
exponentially fast from any uniform initial state. This leaves open the
question of whether there exist NESS that are not spatially uniform. Making a
further simplification of our models, we then prove non-existence of such NESS
and exponential approach to the unique spatially uniform NESS (with a
computably boundable rate). The method of proof relies on refined Doeblin
estimates and other probabilisitic techniques, and is quite different form the
analysis in earlier work that was based on contraction mapping methods.
