Maria Cameron
Courant Institute of Mathematical Sciences, NYU
"Analysis of methods for the study of rare events and transition paths"
I will consider two methods for the study of rare events, the string method and the maxflux functional.
The string method is designed to compute minimum energy paths (MEP's) on a given energy landscape. These correspond to the most likely transition paths at temperature zero. I will show that the evolution of a curve according to the string method equation is equivalent to its evolution according to the gradient flow. The omega-limit set of a curve evolving according to the gradient flow consists of MEP's, but it is not necessarily a curve. The limit set might be multidimensional, and the evolving curve might both fill it or endlessly wander around it. This complex dynamics is linked to the presence of Morse index 2 or higher points. I will formulate a criterion of when the limit set of a curve is a curve and explain why the string method always gives an MEP in practice. However the result must be interpreted with care.
One possible cure for these problems is the maxflux functional. The maxflux functional has been around for almost 30 years but not widely used. For a given finite temperature, a path minimizing the maxflux functional is the path along which the reactive current is maximal. I will show two ways to derive it in the framework of the transition path theory and discuss its range of applicability. I will present an efficient way to minimize the maxflux functional numerically and 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.
Host: Michael Weinstein
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