On two possible definitions of the free energy for collective variables

The aim of this mini-review article is to clarify the relation between two distinct formulations of the thermodynamic free energy for collective variables which can be found in the molecular dynamics literature. In doing so, we discuss the different ensemble concepts underlying the two definitions and reveal their relation to strong confinement (restraints) and molecular constraints. The latter analysis is based on a variant of Federer's coarea formula which can be regarded as a generalization of Fubini's theorem for iterated integrals to curvilinear coordinates and which implies the famous "blue moon" ensemble identity for computing conditional expectations using constrained simulations. For illustration we will present a few paradigmatic examples