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

Gregory S Ezra1, Stephen Wiggins2

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Summary
This summary is machine-generated.

This study validates a numerical method for constructing phase space dividing surfaces (DS) from normally hyperbolic invariant manifolds (NHIM). The method accurately samples points on the DS with correct microcanonical density for dynamical systems.

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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.