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A Hamiltonian-based model of many harmonically interacting massive particlesthat are subject to linear friction and coupled to heat baths at differenttemperatures is used to study the dynamic approach to equilibrium andnon-equilibrium stationary states. Based on the exactly calculated dynamicapproach to the stationary distribution, the functional that governs thisapproach, which is called the free entropy, is constructed. For the stationarydistribution the free entropy becomes maximal and its time derivative isminimal and vanishes. Thus, the free entropy characterizes equilibrium as wellas non-equilibrium stationary distributions by their extremal and stabilityproperties. For an equilibrium system, i.e. if all heat baths have the sametemperature, the free entropy equals the negative free energy divided bytemperature. Using a systematic perturbative scheme for calculating velocityand position correlations in the overdamped massless limit, explicit resultsfor few particles are presented: For two particles localization in position andmomentum space is demonstrated in the non-equilibrium stationary state,indicative of a tendency to phase separate. For three elastically interactingparticles heat flows from a particle coupled to a cold reservoir to a particlecoupled to a warm reservoir if the third reservoir is sufficiently hot. Activeparticle models can be described in the same general framework, which therebyallows to characterize their entropy production not only in the stationarystate but also in the approach to the stationary non-equilibrium state.Finally, the connection to non-equilibrium thermodynamics formulations thatinclude the reservoir entropy production is discussed.

Approach to equilibrium and non-equilibrium stationary distributions of interacting many-particle systems that are coupled to different heat baths