Comment on two distinct notions of free energy

In classical molecular dynamics free energy is one of the most important quantities in analyzing a molecular system. In addition to the standard free energy there is a related free energy concept that is prevalent in transition state theory, but which relies on a different ensemble concept. We show that problems that rely on either definition can be treated in a uniform way using constrained molecular dynamics with no need for unbiasing the respective probability ensembles. Not only does this prove useful in designing algorithms that sample the free energy landscape, but also it clarifies the relation between various results that are available in the literature. In particular we explain that the famous Blue Moon formula is an instance of Federer’s co-area formula, and can easily be generalized to phase space observables. We moreover argue that Blue Moon reweighting also becomes an issue for first-order dynamical systems (e.g., Brownian motion).