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

Area of Science:

• Chemical Physics
• Dynamical Systems Theory
• Computational Chemistry

Background:

• Investigates the construction of phase space dividing surfaces (DS) from normally hyperbolic invariant manifolds (NHIM).
• Builds upon prior work on sampling procedures for DS construction (Wiggins et al., 2016).
• Focuses on the interplay between geometric structures and dynamics in 2 and 3 degree of freedom (DoF) systems.

Purpose of the Study:

• To demonstrate that a previously described numerical procedure for DS construction yields results consistent with sampling methods based on the explicit form of the NHIM.
• To apply and validate the general sampling procedure for DS construction in a specific case (quadratic Hamiltonian normal form near an index-one saddle).
• To confirm that the sampling procedure generates points on the analytically defined DS with correct microcanonical density.

Main Methods:

• Utilized Poincaré-Birkhoff normal form theory to obtain explicit equations for the NHIM and DS.
• Applied a general sampling procedure to generate points on the DS.
• Compared analytical averages with numerical averages of quadratic energy terms over the DS.

Main Results:

• Confirmed that the numerical procedure for DS construction from NHIM matches sampling methods when NHIM/DS equations are known.
• Demonstrated that the sampling procedure accurately places points on the analytically defined DS with correct microcanonical density for 2 and 3 DoF systems.
• Achieved excellent agreement between analytical and numerical averages of quadratic energy terms over the DS across various energies.

Conclusions:

• The validated numerical sampling procedure provides a reliable method for constructing and sampling phase space dividing surfaces from normally hyperbolic invariant manifolds.
• This approach is effective for both 2 and 3 degree of freedom systems, particularly near index-one saddles.
• The findings support the use of these methods for accurate dynamical analysis in chemical physics and related fields.