NUMERICAL METHODS FOR COMPUTING TRANSITION PATHWAYS IN BARRIER-CROSSING EVENTS

Abstract

The dynamics of complex systems are often driven by rare but important events. Well-known examples include nucleation events during phase transitions, conformational changes in macromolecules, and chemical reactions. The long time scale associated with these rare events is a consequence of the disparity between the effective thermal energy and typical energy barrier of the systems. The dynamics proceeds by long waiting periods around metastable states followed by sudden jumps from one state to another. These jumps happen on a time scale that is much larger than the system's intrinsic time scales, so they are called rare events.

The important object in the study of rare events is to understand the transition mechanism, i.e. the transition pathways. In this thesis, we present two approaches to compute transition pathways between two metastable states. The first is called the string method. The method evolves a curve in the path space by gradient flow. The curve is parameterized by its intrinsic arc length. The string method aims to compute the minimum energy path, which is the maximum likelihood path in zero-temperature limit. The second one is importance sampling using the cross-entropy method. The basic idea is to use an alternative sampling distribution to increase numerical efficiency. We use it to create a higher probability for the successive reactions. This method has the advantage of being able to sample transition paths at finite temperatures. In the project we apply the cross-entropy method iteratively, starting from larger noise, gradually decreasing to smaller one. The numerical results from the numerical experiments show a considerable improvement of the performance of the importance sampling scheme.