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

Abstract

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.