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تسجيل مستخدم جديدFinding NHIM: Identifying High Dimensional Phase Space Structures in Reaction Dynamics using Lagrangian Descriptors
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Phase space structures such as dividing surfaces, normally hyperbolic invariant manifolds, their stable and unstable manifolds have been an integral part of computing quantitative results such as transition fraction, stability erosion in multi-stable mechanical systems, and reaction rates in chemical reaction dynamics. Thus, methods that can reveal their geometry in high dimensional phase space (4 or more dimensions) need to be benchmarked by comparing with known results. In this study, we assess the capability of one such method called Lagrangian descriptor for revealing the types of high dimensional phase space structures associated with index-1 saddle in Hamiltonian systems. The Lagrangian descriptor based approach is applied to two and three degree-of-freedom quadratic Hamiltonian systems where the high dimensional phase space structures are known, that is as closed-form analytical expressions. This leads to a direct comparison of features in the Lagrangian descriptor plots and the phase space structures intersection with an isoenergetic two-dimensional surface and hence provides a validation of the approach.
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Complementary to existing applications of Lagrangian descriptors as an exploratory method, we use Lagrangian descriptors to find invariant manifolds in a system where some invariant structures have already been identified. In this case we use the par
ametrisation of a periodic orbit to construct a Lagrangian descriptor that will be locally minimised on its invariant manifolds. The procedure is applicable (but not limited) to systems with highly unstable periodic orbits, such as the isokinetic Chesnavich CH4+ model subject to a Hamiltonian isokinetic theromostat. Aside from its low computational requirements, the method enables us to study the invariant structures responsible for roaming in the isokinetic Chesnavich CH4+ model.
Recent studies have found an unusual way of dissociation in formaldehyde. It can be characterized by a hydrogen atom that separates from the molecule, but instead of dissociating immediately it roams around the molecule for a considerable amount of t
ime and extracts another hydrogen atom from the molecule prior to dissociation. This phenomenon has been coined roaming and has since been reported in the dissociation of a number of other molecules. In this paper we investigate roaming in Chesnavichs CH$_4^+$ model. During dissociation the free hydrogen must pass through three phase space bottleneck for the classical motion, that can be shown to exist due to unstable periodic orbits. None of these orbits is associated with saddle points of the potential energy surface and hence related to transition states in the usual sense. We explain how the intricate phase space geometry influences the shape and intersections of invariant manifolds that form separatrices, and establish the impact of these phase space structures on residence times and rotation numbers. Ultimately we use this knowledge to attribute the roaming phenomenon to particular heteroclinic intersections.
We present the capability of Lagrangian descriptors for revealing the high dimensional phase space structures that are of interest in nonlinear Hamiltonian systems with index-1 saddle. These phase space structures include normally hyperbolic invarian
t manifolds and their stable and unstable manifolds, and act as codimenision-1 barriers to phase space transport. The method is applied to classical two and three degrees-of-freedom Hamiltonian systems which have implications for myriad applications in physics and chemistry.
Hamiltonian dynamical systems possessing equilibria of ${saddle} times {centre} times...times {centre}$ stability type display emph{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 emph{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 emph{Normally Hyperbolic Invariant Manifold (NHIM)}, whose emph{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) emph{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 emph{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 $R^3$, 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.
This Response is concerned with the recent Comment of Ruiz-Herrera, Limitations of the Method of Lagrangian Descriptors [arXiv:1510.04838], criticising the method of Lagrangian Descriptors. In spite of the significant body of literature asserting the
contrary, Ruiz-Herrera claims that the method fails to reveal the presence of stable and unstable manifolds of hyperbolic trajectories in incompressible systems and in almost all linear systems. He supports this claim by considering the method of Lagrangian descriptors applied to three specific examples. However in this response we show that Ruiz-Herrera does not understand the proper application and interpretation of the method and, when correctly applied, the method beautifully and unambiguously detects the stable and unstable manifolds of the hyperbolic trajectories in his examples.
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