Sampling Phase Space Dividing Surfaces Constructed from Normally Hyperbolic Invariant Manifolds (NHIMs)

G. S. Ezra and S. Wiggins, Sampling Phase Space Dividing Surfaces Constructed from Normally Hyperbolic Invariant Manifolds (NHIMs), J. Phys. Chem. A., 122(42), 8354–8362 (2018).

https://pubs.acs.org/doi/10.1021/acs.jpca.8b07205

In this paper, we further investigate the construction of a phase space dividing surface (DS) from a normally hyperbolic invariant manifold (NHIM) and the sampling procedure for the resulting dividing surface described in earlier work (Wiggins, S.J. Chem. Phys. 2016, 144, 054107). Our discussion centers on the relationship between geometrical structures and dynamics for 2 and 3 degree of freedom (DoF) systems, specifically, the construction of a DS from a NHIM. We show that if the equation for the NHIM and associated DS is known (e.g., as obtained from Poincaré–Birkhoff normal form theory), then the numerical procedure described in Wiggins et al. ( J. Chem. Phys. 2016, 144, 054107) gives the same result as a sampling method based upon the explicit form of the NHIM. After describing the sampling procedure in a general context, it is applied to a quadratic Hamiltonian normal form near an index-one saddle where explicit formulas exist for both the NHIM and the DS. It is shown for both 2 and 3 DoF systems that a version of the general sampling procedure provides points on the analytically defined DS with the correct microcanonical density on the constant-energy DS. Excellent agreement is obtained between analytical and numerical averages of quadratic energy terms over the DS for a range of energies.

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