Phase space structure and escape time dynamics in a Van der Waals model for exothermic reactions

Abstract

We study the phase space structures that control the transport in a classical Hamiltonian model for a chemical reaction. This model has been proposed to study the yield of products in an ultracold exothermic reaction [1]. In the considered model, two elements determine the evolution of the system: a Van der Waals force and short-range force associated with the many-body interactions. In the previous work has been used small random periodic changes in the direction of the momentum to simulate the short-range many-body interactions. In the present work, random Gaussian bumps have been added to the Van der Waals potential energy simulate the short-range effects between the particles in the system. We compare both variants of the model and explain their differences similarities and differences from a phase space perspective. In order to visualize the structures that direct the dynamics in the phase space, we construct a natural Lagrangian descriptor for Hamiltonian systems based on the Maupertuis action S0 = q f q i p • dq.

Transport and roaming on the double van der Waals potential energy surface

“…The fate maps and Lagrangian descriptors for the lowest value of energy E = 0.01, close to the threshold energy are very similar to the results for the double Morse potential well, see Fig 11 in the reference [37]. This similarity is related to the fact that the dynamics associated with the transport between the three regions is driven only by finite segments of the stable and unstable manifolds of the hyperbolic periodic orbits.…”

“…The general idea behind this diagnostic tool is to integrate a positive scalar quantity along the trajectories of the dynamical system. For example, other definitions of LDs include the use of the p-norm of the components of the vector field that defines the flow of the system [36], or the computation of the action integral [37], which is the approach we follow in this paper. For an extended review on the method of Lagrangian descriptors that includes an introductory tutorial on its application to classical Hamiltonian systems with a Python software package, refer to [38].…”

“…In order to study the transport between the wells and the roaming trajectories, we consider the dividing surfaces associated with each of the three periodic orbits Γ 1 , Γ 2 , and Γ 3 . These surfaces have been previously used to study the double Morse PES in [26,37]. The dividing surface of a periodic orbit has three important properties for the analysis of the transport:…”

Lagrangian Descriptors and the Action Integral of Classical Mechanics

“…Hence, we believe that a connection between these two trajectory diagnostics has the potential to yield greater insights into global dynamics and phase space structure. We note that use of the action as a Lagrangian descriptor was described in [21], and has also been used in [16,26,27]. See also [28].…”

Visualizing the Phase Space of the HeI_2 van der Waals Complex using Lagrangian Descriptors

“…The rare gas-diatomic halogen van der Waals (vdW) complexes, such as HeI 2 in Fig. 1 have been proven to be ideal model systems for fundamental investigations of molecular dynamics [42,45,48], because of their relative simplicity. The authors in [47] produced a model that is a three DoF system, and then, by imposing a periodic kick they converted it into a Poincaré map, which is the 4D symplectic map that we will analyze in this paper.…”