  A Hamiltonian-based model of many harmonically interacting massive particles
that are subject to linear friction and coupled to heat baths at different
temperatures is used to study the dynamic approach to equilibrium and
non-equilibrium stationary states. Based on the exactly calculated dynamic
approach to the stationary distribution, the functional that governs this
approach, which is called the free entropy, is constructed. For the stationary
distribution the free entropy becomes maximal and its time derivative is
minimal and vanishes. Thus, the free entropy characterizes equilibrium as well
as non-equilibrium stationary distributions by their extremal and stability
properties. For an equilibrium system, i.e. if all heat baths have the same
temperature, the free entropy equals the negative free energy divided by
temperature. Using a systematic perturbative scheme for calculating velocity
and position correlations in the overdamped massless limit, explicit results
for few particles are presented: For two particles localization in position and
momentum space is demonstrated in the non-equilibrium stationary state,
indicative of a tendency to phase separate. For three elastically interacting
particles heat flows from a particle coupled to a cold reservoir to a particle
coupled to a warm reservoir if the third reservoir is sufficiently hot. Active
particle models can be described in the same general framework, which thereby
allows to characterize their entropy production not only in the stationary
state but also in the approach to the stationary non-equilibrium state.
Finally, the connection to non-equilibrium thermodynamics formulations that
include the reservoir entropy production is discussed.
