The string method as a dynamical system
The string method is a numerical scheme for exploring complex potential
landscapes. It is typically used to find mountain pass separating local
minima, thereby determining (under suitable hypotheses) the pathways
and rates of thermally-activated transitions between metastable states.
Before discretization, the string method can be viewed as an evolution
law for curves. We examine its large-time behavior, asking in
particular whether it converges to a steady state (a ``minimum energy
path or MEP''). The answer is negative in general; we give a collection
of examples accompanied with movies demonstrating different types of
complex large-time behavior of the evolving path. The answer is
affirmative, however, under suitable hypotheses on the structure of the
energy landscape or the choice of the initial curve.
This is a joint work with Bob kohn and Eric Vanden-Eijnden.