Many processes in nature are rare events. These include chemical reactions, nucleation events, and conformational changes of bio-molecules. These events are rare but they have a profound impact on the system. As a result of such events, the system transforms from one metastable state to another. The question of interest is the mechanism of such transitions and the transition rates. Classical transition-state theory works very well for systems with relatively simple energy landscapes. In this case, the most probable transition path is the so-called minimal energy path (MEP). Once this path is found, the transition rate can be calculated by various linearization procedures. However, often the systems of interest in applications have very complex energy landscapes, and the notion of MEP becomes irrelevant.
In the last several years, Weinan E, Eric Vanden-Eijnden, and I have developed a theoretical framework (the transition-path theory) as well as efficient numerical tools (finite-temperature string method) for dealing with such rare events in systems with complex energy landscapes. The theoretical framework can be viewed as a statistical-mechanics framework for paths. We think of the transition paths in configuration space. This allows us to make use of the Fokker-Planck formalism and the associated variational principles, which in turn can be used to reduce the complex many-body problem in a Fokker-Planck formalism to a system of coupled one-body problems, much in the same way as in density functional theory, which reduces the quantum many-body problem to coupled one-body problems.
The numerical methods have been sucessfully applied to a variety of problems and proven to be efficient. However, a systematic study of the numerical issues in the methods is still lacking. In the proposed work, I will study the convergence and accuracy of the finite-temperature string method, as well as its dependence on the input parameters. Our aim is to make the method robust and more efficient. In the meanwhile, I will also apply the method to study problems arising from solids and chemical reactions which are of practical interest.
