  We reconsider the structure-based route to coarse graining in which the
coarse-grained model is defined in such a way to reproduce some distributions
functions of the original system as accurately as possible. We consider
standard expressions for pressure and chemical potential applied to this family
of coarse-grained models with density-dependent interactions and show that they
only provide approximations to the pressure and chemical potential of the
underlying original system. These approximations are then carefully compared in
two cases: we consider a generic microscopic system in the low-density regime
and polymer solutions under good-solvent conditions. Moreover, we show that the
state-dependent potentials depend on the ensemble in which they have been
derived. Therefore, care must be used in applying canonical state-dependent
potentials to predict phase lines, which is typically performed in other
ensembles.
