  Starting from the stochastic thermodynamics description of two coupled
underdamped Brownian particles, we showcase and compare three different
coarse-graining schemes leading to an effective thermodynamic description for
the first of the two particles: marginalization over one particle, bipartite
structure with information flows and the Hamiltonian of mean force formalism.
In the limit of time-scale separation where the second particle with a fast
relaxation time scale locally equilibrates with respect to the coordinates of
the first slowly relaxing particle, the effective thermodynamics resulting from
the first and third approach are shown to capture the full thermodynamics and
to coincide with each other. In the bipartite approach, the slow part does not,
in general, allow for an exact thermodynamic description as the entropic
exchange between the particles is ignored. Physically, the second particle
effectively becomes part of the heat reservoir. In the limit where the second
particle becomes heavy and thus deterministic, the effective thermodynamics of
the first two coarse-graining methods coincides with the full one. The
Hamiltonian of mean force formalism however is shown to be incompatible with
that limit. Physically, the second particle becomes a work source. These
theoretical results are illustrated using an exactly solvable harmonic model.
