Collective Variables from Local Fluctuations

We introduce a method to obtain one-dimensional collective variables for studying rarely occurring transitions between two metastable states separated by a high free energy barrier. No previous information, not even approximated, on the path followed during the transition is needed. The only requirement is to know the fluctuations of the system while in the two metastable states. With this information in hand, we build the collective variable using a modified version of Fisher's linear discriminant analysis. The usefulness of this approach is tested on the metadynamics simulation of two representative systems. The first is the freezing of silver iodide into the supertonic alpha-phase, and the second is the study of a classical Diels-Alder reaction. The collective variable works very well in these two diverse cases.