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

Quantum Transport through Partial Barriers in Higher-Dimensional Systems

Partial transport barriers in the chaotic sea of Hamiltonian systems influence classical transport, as they allow for a small flux between chaotic phase-space regions only. We find for higher-dimensional systems that quantum transport through such a partial barrier is more restrictive than expected from two-dimensional maps. We establish a universal transition from quantum suppression to mimicking classical transport. The scaling parameter involves the flux, the size of a Planck cell, and the localization length due to dynamical localization along a resonance channel. This is numerically demonstrated for coupled kicked rotors with a partial barrier that generalizes a cantorus to higher dimensions.

Instantons and the quantum bound to chaos

The rate at which information scrambles in a quantum system can be quantified using out-of-time-ordered correlators. A remarkable prediction is that the associated Lyapunov exponent [Formula: see text] that quantifies the scrambling rate in chaotic systems obeys a universal bound [Formula: see text]. Previous numerical and analytical studies have indicated that this bound has a quantum-statistical origin. Here, we use path-integral techniques to show that a minimal theory to reproduce this bound involves adding contributions from quantum thermal fluctuations (describing quantum tunneling and zero-point energy) to classical dynamics. By propagating a model quantum-Boltzmann-conserving classical dynamics for a system with a barrier, we show that the bound is controlled by the stability of thermal fluctuations around the barrier instanton (a delocalized structure which dominates the tunneling statistics). This stability requirement appears to be general, implying that there is a close relation between the formation of instantons, or related delocalized structures, and the imposition of the quantum-chaos bound.

Regular reaction dynamics in analytical form in the vicinity of symmetrical transition states. Central barrier crossings in SN2 reactions

When an activated complex, as defined in transition state theory (TST), has a polyhedral shape, its kinetic energy is found to be diagonal in a system of spherical polar coordinates. If, in addition, the polyhedron is characterized by a high symmetry, then its dynamics considerably simplifies. An application of this approach to the most symmetrical TS known to date, i.e., that which controls the Cl- + CH3Cl → ClCH3 + Cl- SN2 nucleophilic substitution, is presented and an analytical expression of its potential energy surface is provided. In a substantial range around the saddle point, approximate equations of motion for the two components of the reaction coordinate, i.e., the antisymmetrical stretching motion of the ClCCl core and the wagging motion of the hydrogen triad, can be derived in an analytical form. During an extensive period of time, the main component of the reaction coordinate is governed by an unexpectedly simple equation of motion that depends on a single initial condition, irrespective of the other ones and of the internal energy. Reactive trajectories are observed to form a perfectly collimated bundle characterized by undetectable dispersion, thereby giving a spectacular example of regular dynamics in an anharmonic potential. Regularity and collimation are brought about by local symmetry, which is a widespread feature of potential energy surfaces. Anharmonicity is observed to influence the dynamics only at a late stage. As energy increases, trajectories tend to fan out and to deviate from the analytical equation. For the wagging motion, chaos sets in at much lower energies.

Navigating phase space transport with the origin-fate map

We introduce and demonstrate the usage of the origin-fate map (OFM) as a tool for the detailed investigation of phase space transport in reactant-product-type systems. For these systems, which exhibit clearly defined start and end states, it is possible to build a comprehensive picture of the lobe dynamics by considering backward and forward integration of sets of initial conditions to index their origin and fate. We illustrate the method and its utility in the study of a two degrees of freedom caldera potential with four exits, demonstrating that the OFM not only recapitulates results from classical manifold theory but even provides more detailed information about complex lobe structures. The OFM allows the detection of dynamically significant transitions caused by the creation of new lobes and is also able to guide the prediction of the position of unstable periodic orbits (UPOs). Further, we compute the OFM on the periodic orbit dividing surface (PODS) associated with the transition state of a caldera entrance, which allows for a powerful analysis of reactive trajectories. The intersection of the manifolds corresponding to this UPO with other manifolds in the phase space results in the appearance of lobes on the PODS, which are directly classified by the OFM. This allows computations of branching ratios and the exploration of a fractal cascade of lobes as the caldera is stretched, which results in fluctuations in the branching ratio and chaotic selectivity. The OFM is found to be a simple and very useful tool with a vast range of descriptive and quantitative applications.

Partial barriers to chaotic transport in 4D symplectic maps

Chaotic transport in Hamiltonian systems is often restricted due to the presence of partial barriers, leading to a limited flux between different regions in phase space. Typically, the most restrictive partial barrier in a 2D symplectic map is based on a cantorus, the Cantor set remnants of a broken 1D torus. For a 4D symplectic map, we establish a partial barrier based on what we call a cantorus-NHIM-a normally hyperbolic invariant manifold with the structure of a cantorus. Using a flux formula, we determine the global 4D flux across a partial barrier based on a cantorus-NHIM by approximating it with high-order periodic NHIMs. In addition, we introduce a local 3D flux depending on the position along a resonance channel, which is relevant in the presence of slow Arnold diffusion. Moreover, for a partial barrier composed of stable and unstable manifolds of a NHIM, we utilize periodic NHIMs to quantify the corresponding flux.

Finding normally hyperbolic invariant manifolds in two and three degrees of freedom with Hénon-Heiles-type potential

We present a method based on a Lagrangian descriptor 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 a normally hyperbolic invariant manifold and its stable and unstable manifolds, which act as codimension-1 barriers to phase space transport. In this article, finding the invariant manifolds in high-dimensional phase space will constitute identifying coordinates on these invariant manifolds. The method of Lagrangian descriptor is demonstrated by applying to classical two and three degrees of freedom Hamiltonian systems which have implications for myriad applications in chemistry, engineering, and physics.