IMA Postdoc Seminar (February 24, 2009)

Speaker: Stephen Wiggins (School of Mathematics, University of Bristol)

Title: Recent advances in the high dimensional Hamiltonian dynamics and geometry of reaction dynamics

Abstract:In the early development of applied dynamical systems theory it was hoped that the complexity exhibited by low dimensional nonlinear systems might somehow lead to ways of understanding the complex dynamics of high dimensional systems. Unfortunately, there has not been great progress in this area. The availability of high performance computing resources has led to many computational studies of high dimensional systems. But even under these circumstances, the problem of high dimensionality often forces one to make severe assumptions on the dynamics in order to derive physically relevant quantities from the model, e.g., ergodicity assumptions may be necessary in order to deduce a reaction rate from a computation.We approach the problem of high dimensionality from the other direction. Our interest is in the exact Hamiltonian dynamics of high dimensional systems. As applied dynamical systems theory developed and expanded throughout the 70's and 80's there was much effort in applying global, geometrical concepts and techniques to problems related to the dynamics of molecules. In the early 90's this effort began to die out in the chemistry community because the approach did not appear to apply to problems with more than two degrees-of-freedom. New concepts were required. In the past few years there has been much progress along these lines. We will discuss these recent developments and their application to the understanding of a variety of issues related to the dynamics of molecules. Theoretically, we have constructed a dynamically exact /phase space/ transition state theory, for which we can rigorously construct a "surface of (locally) no return" through which all reacting trajectories must pass. It can also be shown that the flux across the surface we construct is minimal. Central to this construction is a normally hyperbolic invariant manifold (NHIM) whose stable and unstable manifolds enclose the phase space conduits of all reacting trajectories. They enable us to determine the volume of trajectories that can escape from a potential well (the "reactive volume"), which is a central quantity in any reaction rate, and to construct a "dynamical" reaction path. Moreover, we show that the NHIM is the mathematical manifestation of the chemist's notion of the "activated complex".

The application of these ideas to concrete problems relies on the computational realisation of these structures. These can be realized locally through the Poincare-Birkhoff normal form, and then globalised. Recent advances in computational techniques enable one to carry out this procedure for systems with a large number of degrees of freedom. A similar set of techniques can be developed to deal with the corresponding quantum mechanical system. In particular a quantum normal form is used to determine quantum mechanical resonances and reaction rates with high precision. In this talk we describe the theory, applications, and computations that make this possible. We will use HCN isomerization and the Muller-Brown potential to illustrate the ideas and methods and point out a number of areas where more close collaborations between chemists and applied mathematicians could prove fruitful. For example, "rare events" from a dynamical systems point of view are homoclinic and heteroclinic trajectories. Are they related, and do they provide insight, into the "rare events" observed in reaction dynamics?