[Sem-coll] AM Colloquium August 25
Joe Millham
jmillhamiit at gmail.com
Wed Aug 20 17:23:00 CDT 2008
Greetings all,
Please join the Applied Math department for the following colloquium
next week. All are encouraged to come, and students registered for
Math 593 are reminded this is a qualifying colloquium. Refreshments
will be served.
Who: Stephen Wiggins, University of Bristol, UK
What: Recent Advances in the High Dimensional Hamiltonian Dynamics and
Geometry of Reaction Dynamics
When: Monday, August 25, 4:40 pm
Where: E1 106
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?
See you there!
--
Joe Millham
Administrative Assistant
Department of Applied Mathematics
Illinois Institute of Technology
10 W. 32rd St.
Chicago IL 60616
312.567.8984 (Phone)
312.567.3135 (Fax)
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