| Summary: | The dynamics of a probe particle is highly influenced by the nature of the bath in which it is immersed. In particular, baths composed by active (e.g., self-propelled) particles induce intriguing out-of-equilibrium effects on tracer's motion that are customarily described by integrating out the dynamics of the bath's degrees of freedom (DOFs). However, thermodynamic quantities, such as the entropy production rate, are generally severely affected by coarse-graining procedures. Here, by employing an exact integration of a subset of DOFs, we show that two classes of DOFs emerge, entropic and nonentropic. By inspecting the resulting generalized Langevin equation (GLE) for the observed variables, it turns out that active baths are associated with the presence of entropic DOFs exhibiting nonreciprocal interactions with a probe particle. Surprisingly, integrating out these DOFs inevitably results into a system-dependent increase or reduction of the entropy production rate, at variance with other coarse-graining schemes. In this case, the resulting GLE is not thermodynamically consistent. On the contrary, the entropy production rate stays invariant after integrating out nonentropic DOFs and the resulting GLE preserves the thermodynamics of the full system. Additionally, the number of nonentropic DOFs determines the dimensionality of isoentropic hypersurfaces in the parameter space. Our results shed light on the nature of active baths, revealing that the presence of a typical correlation timescale is not a sufficient condition to have nonequilibrium effects on a probe particle, and draws a path towards the understanding of thermodynamically consistent procedures to derive effective dynamics of observed DOFs.
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