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Joint Fluid Dynamics /Waves Seminar

Wednesday, Oct 21, 2009, 2:30 PM
Cullimore Lecture Hall, Room 611
New Jersey Institute of Technology

Analysis of methods for the study of rare events and transition paths

Maria Cameron

Courant Institute of Mathematical Sciences, New York University

Abstract

The overdamped Langevin equation is often used as a model in the molecular dynamics. At low temperatures, a system evolving according to such an equation spends most of the time near the potential minima and performs rare transitions between the neighborhoods of different minima. A number of methods have been developed to study the most likely transition paths. I will focus on two of them: the string method and the MaxFlux functional. The string method is designed to find so called minimum energy paths, which correspond to the most likely transition paths at temperature zero. I will consider the string method as a dynamical system in the continuous-time, continuous-space setting. If the potential contains Morse index 2 or higher critical points, there are continuum families of minimum energy paths such that the transitions along each of them have the same likelihood. I will discuss the possible evolution of the path according to the string method equation in the case of such physical ambiguity. The MaxFlux functional has been around for almost 30 years but not widely used because it is hard-to-minimize. Its minimizer provides a path of maximum likelihood at a given finite temperature. I will show two ways to derive it in the framework of the transition path theory and present an efficient way to minimize the maxflux functional numerically. I will demonstrate its the application to the problem of finding the most likely transition paths in the Lennard-Jones-38 cluster between the face-centered-cubic and the icosahedral structure.