Introduction

As we know from the applications of thermodynamics, free energy is much more important than energy, since it determines phase equilibria, such as melting and boiling points and the pressure of saturated vapors, and chemical equilibria such as solubilities, binding or dissociation constants and conformational changes. Unfortunately, it is generally much more difficult to derive free energy differences from simulations than it is to derive energy differences. The reason for this is that free energy incorporates an entropic term –TS; entropy is given by an integral over phase space, while energy is an ensemble average. Only when the system is well localized in space (as a vibrating solid or a macromolecule with a well-defined average structure) is it possible to approximate the multidimensional integral for a direct determination of entropy. This case will be considered in Section 7.2.

Free energies of substates can be evaluated directly from completely equilibrated trajectories or ensembles that contain all accessible regions of configurational space. In practice it is hard to generate such complete ensembles when there are many low-lying states separated by barriers, but the ideal distribution may be approached by the replica exchange method (see Section 6.6). Once the configurational space has been subdivided into substates or conformations (possibly based on a cluster analysis of structures), the free energy of each substate is determined by the number of configurations observed in each substate.