Geometrical models of the phase space structures governing reaction dynamics
- 171 Downloads
Hamiltonian dynamical systems possessing equilibria of saddle × center × ... × center stability type display reaction-type dynamics for energies close to the energy of such equilibria; entrance and exit from certain regions of the phase space is only possible via narrow bottlenecks created by the influence of the equilibrium points. In this paper we provide a thorough pedagogical description of the phase space structures that are responsible for controlling transport in these problems. Of central importance is the existence of a Normally Hyperbolic Invariant Manifold (NHIM), whose stable and unstable manifolds have sufficient dimensionality to act as separatrices, partitioning energy surfaces into regions of qualitatively distinct behavior. This NHIM forms the natural (dynamical) equator of a (spherical) dividing surface which locally divides an energy surface into two components (“reactants” and “products”), one on either side of the bottleneck. This dividing surface has all the desired properties sought for in transition state theory where reaction rates are computed from the flux through a dividing surface. In fact, the dividing surface that we construct is crossed exactly once by reactive trajectories, and not crossed by nonreactive trajectories, and related to these properties, minimizes the flux upon variation of the dividing surface.
We discuss three presentations of the energy surface and the phase space structures contained in it for 2-degree-of-freedom (DoF) systems in the three-dimensional space ℙ3, and two schematic models which capture many of the essential features of the dynamics for n-DoF systems. In addition, we elucidate the structure of the NHIM.
Key wordshigh dimensional Hamiltonian dynamics phase space structure and geometry normally hyperbolic invariant manifold Poincaré-Birkhoff normal form theory chemical reaction dynamics transition state theory
MSC2000 numbers37J05 37N99 70Hxx 92E20
Unable to display preview. Download preview PDF.
- 2.Arnol’d, V. I., Kozlov, V. V., and Neishtadt, A. I., Mathematical Aspects of Classical and Celestial Mechanics, in Dynamical Systems III, Encyclopaedia of Mathematical Sciences, vol. 3, Berlin: Springer, 1988.Google Scholar
- 4.Chisholm, M., The Sphere in Three Dimensions and Higher: Generalizations and Special Cases, 2000, http://www.theory.org/geotopo/.
- 5.de Oliveira, H. P., Ozorio de Almeida, A. M., Damĩao Soares, I., and Tonini, E. V., Homoclinic Chaos in the Dynamics of a General Bianchi Type-IX Model, Phys. Rev. D. (3), 2002, vol. 65, no. 8, 083511, 9 pp.Google Scholar
- 10.Guillemin, V., Moment Maps and Combinatorial Invariants of Hamiltonian T n-spaces, Boston: Birkhäser, 1994.Google Scholar
- 19.McGehee, R. P., Some Homoclinic Orbits for the Restricted Three-Body Problem, Ph.D. thesis, University of Wisconsin, (1969.Google Scholar
- 21.Meyer, K., A Lie Transform Tutorial II, in Computer Aided Proofs in Analysis (Cincinnati, OH, 1989), The IMA Volumes in Mathematics and its Applications, vol. 28, Berlin: Springer, 1991, pp. 190–210.Google Scholar
- 33.Pollak, E. and Talkner, P., Reaction Rate Theory: What Is Was, Where It Is Today, and Where Is It Going? Chaos, 2005, vol. 15, 026116, 11 pp.Google Scholar
- 46.Wiggins, S., Normally Hyperbolic Invariant Manifolds in Dynamical Systems, Springer-Verlag, 1994.Google Scholar