Accession Number:

ADA513258

Title:

Geometrical Models of the Phase Space Structures Governing Reaction Dynamics

Descriptive Note:

Preprint

Corporate Author:

MINNESOTA UNIV MINNEAPOLIS INST FOR MATHEMATICS AND ITS APPLICATIONS

Personal Author(s):

Report Date:

2009-08-01

Pagination or Media Count:

46.0

Abstract:

Hamiltonian dynamical systems possessing equilibria of saddle x centre x llll x centre 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 behaviour. 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 threedimensional space R3, 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.

Subject Categories:

  • Numerical Mathematics
  • Computer Systems

Distribution Statement:

APPROVED FOR PUBLIC RELEASE