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Starting from the stochastic thermodynamics description of two coupledunderdamped Brownian particles, we showcase and compare three differentcoarse-graining schemes leading to an effective thermodynamic description forthe first of the two particles: marginalization over one particle, bipartitestructure with information flows and the Hamiltonian of mean force formalism.In the limit of time-scale separation where the second particle with a fastrelaxation time scale locally equilibrates with respect to the coordinates ofthe first slowly relaxing particle, the effective thermodynamics resulting fromthe first and third approach are shown to capture the full thermodynamics andto coincide with each other. In the bipartite approach, the slow part does not,in general, allow for an exact thermodynamic description as the entropicexchange between the particles is ignored. Physically, the second particleeffectively becomes part of the heat reservoir. In the limit where the secondparticle becomes heavy and thus deterministic, the effective thermodynamics ofthe first two coarse-graining methods coincides with the full one. TheHamiltonian of mean force formalism however is shown to be incompatible withthat limit. Physically, the second particle becomes a work source. Thesetheoretical results are illustrated using an exactly solvable harmonic model.

Effective thermodynamics of two interacting underdamped Brownian particles