  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, we show that active baths are associated with
the presence of entropic DOFs exhibiting non-reciprocal interactions with a
probe particle. Surprisingly, integrating out these DOFs inevitably results
into a system-dependent increase or reduction of the entropy production rate.
On the contrary, it stays invariant after integrating out non-entropic DOFs. As
a consequence, they determine 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 time-scale is not a
sufficient condition to have non-equilibrium effects on a probe particle, and
draws a path towards the understanding of thermodynamically-consistent
procedures to derive effective dynamics of observed DOFs.
