Computing transition rates for rare events: When Kramers theory meets the free-energy landscape

Computing reactive trajectories and free-energy landscapes associated with rare-event kinetics is key to understanding the dynamics of complex systems. The analysis of the free-energy landscape on which the underlying dynamics takes place has become central to compute transition rates. In the overdamped limit, most often encountered in biophysics and soft condensed matter, the Kramers theory and its multidimensional extension derived by Langer have proved to be quite successful in recovering correct kinetics. However, the additional calculation to obtain rate constants in complex systems where configurational entropy is competing with energy is still challenging conceptually and computationally. Building on the Kramers theory and the metadynamics framework, we propose an expression for the rate in terms of the height of the free-energy barrier measured along the minimum free-energy path and an auxiliary measure of the configurational entropy in terms of the joint probability distribution of the reactive and nonreactive coordinates representing the slow modes of the system. We apply the formalism to three different problems where our approach shows good agreement with simulations and experiments and can present significant improvement over the Kramers-Langer's framework when slow entropic contribution, such as configurational entropy, predominates over enthalpic contribution.