Numerical Methods For The Study Of Transition Paths In Rare Events For Complex Systems

The dynamics of complex systems is often driven by rare but important events. Well-known examples include chemical reactions, diffusion processes and conformational changes in macromolecules. The disparity between the effective thermal energy and the typical energy barrier of the system results in a wide separation of time scales. The separation of time scales presents serious computational challenges in the study of rare events. In the dissertation, using an adaptive zero-temperature string method, which was presented based on the zero-temperature string method, carries out the study of transition paths in rare events for complex systems with smooth energy landscape. As zero-temperature string method, the adaptive zero-temperature string method only requires the computation of the gradient of the potential, and makes any initial string converge to the minimal energy path and the error be or smaller. The critical points on the minimal energy path, which often denote stable states or transition states, are concerned. By choosing an appropriate monitor function, the density of mesh points near the critical points can be bigger than that near other points on the minimal energy path. So the method can accurately locate the positions of the critical points and calculate the energy values of the critical points. The adaptive zero-temperature string method is applied to the double well system, the Mueller potential problem and a seven-atom Lennard-Jones cluster problem, and it appears satisfactory.The dissertation also presents an adaptive finite-temperature string method to study transition paths in rare events for complex systems with rough energy landscape. Like the adaptive zero-temperature string method, by choosing an appropriate monitor function, the density of mesh points near the critical points can be bigger than that near other points on the effective minimal energy path. So the method can accurately locate the positions of the critical points and calculate the averaged energy values of the critical points.The proposed adaptive string method enriches numerical methods for the study of transition paths in rare events.