Volume 25, Number 5

Volume 25, Number 5, 2020
Nonlinear Dynamics in Chemical Sciences

Citation: Foreword, Regular and Chaotic Dynamics, 2020, vol. 25, no. 5, pp. 411
Hoover W. G.,  Hoover C. G.
Deterministic and time-reversible nonequilibrium molecular dynamics simulations typically generate “fractal” (fractional-dimensional) phase-space distributions. Because these distributions and their time-reversed twins have zero phase volume, stable attractors “forward in time” and unstable (unobservable) repellors when reversed, these simulations are consistent with the second law of thermodynamics. These same reversibility and stability properties can also be found in compressible baker maps, or in their equivalent random walks, motivating their careful study. We illustrate these ideas with three examples: a Cantor set map and two linear compressible baker maps, N2$(q, p)$ and N3$(q, p)$. The two baker maps’ information dimensions estimated from sequential mappings agree, while those from pointwise iteration do not, with the estimates dependent upon details of the approach to the maps’ nonequilibrium steady states.
Keywords: chaos, Lyapunov exponents, irreversibility, random walks, maps, information dimension
Citation: Hoover W. G.,  Hoover C. G., Nonequilibrium Molecular Dynamics, Fractal Phase-Space Distributions, the Cantor Set, and Puzzles Involving Information Dimensions for Two Compressible Baker Maps, Regular and Chaotic Dynamics, 2020, vol. 25, no. 5, pp. 412-423
Dhont G.,  Iwai T.,  Zhilinskií B. I.
Energy band rearrangement along a control parameter in isolated molecules is studied through axially symmetric Hamiltonians describing the coupling of two angular momenta $\mathbf{S}$ and $\mathbf{L}$ of fixed amplitude. We focus our attention on the case $S = 1$ which, albeit nongeneric, describes the global rearrangement of a system of energy bands between two welldefined limits corresponding to uncoupled and coupled momenta. The redistribution of energy levels between bands is closely related to the degeneracy of the eigenvalues of the corresponding semiquantum Hamiltonian at isolated points of the three-dimensional Cartesian product of the two-dimensional phase space and the one-dimensional control parameter space. The present paper shows that the band rearrangement for the full quantum system can be quantitatively, rather than qualitatively, reproduced with Dirac oscillator approximations. We also interpret the energy band rearrangement by comparing the evolution of the joint spectra of commuting observables (\emph{i.e.}, energy and axial angular momentum) with that of the image of the energymomentum map of the completely classical limit of the Dirac oscillator approximations.
Keywords: edge state, Chern number, Dirac oscillator, Hamiltonian monodromy
Citation: Dhont G.,  Iwai T.,  Zhilinskií B. I., A Study of Energy Band Rearrangement in Isolated Molecules by Means of the Dirac Oscillator Approximation, Regular and Chaotic Dynamics, 2020, vol. 25, no. 5, pp. 424-452
Lyu W.,  Naik S.,  Wiggins S.
In this study, we analyze how changes in the geometry of a potential energy surface in terms of depth and flatness can affect the reaction dynamics.We formulate depth and flatness in the context of one and two degree-of-freedom (DOF) Hamiltonian normal form for the saddlenode bifurcation and quantify their influence on chemical reaction dynamics [1, 2]. In a recent work, García-Garrido et al. [2] illustrated how changing the well-depth of a potential energy surface (PES) can lead to a saddle-node bifurcation. They have shown how the geometry of cylindrical manifolds associated with the rank-1 saddle changes en route to the saddle-node bifurcation. Using the formulation presented here, we show how changes in the parameters of the potential energy control the depth and flatness and show their role in the quantitative measures of a chemical reaction. We quantify this role of the depth and flatness by calculating the ratio of the bottleneck width and well width, reaction probability (also known as transition fraction or population fraction), gap time (or first passage time) distribution, and directional flux through the dividing surface (DS) for small to high values of total energy. The results obtained for these quantitative measures are in agreement with the qualitative understanding of the reaction dynamics.
Keywords: Hamiltonian dynamics, bifurcation theory, phase space methods
Citation: Lyu W.,  Naik S.,  Wiggins S., The Role of Depth and Flatness of a Potential Energy Surface in Chemical Reaction Dynamics, Regular and Chaotic Dynamics, 2020, vol. 25, no. 5, pp. 453-475
Contopoulos G.,  Tzemos A. C.
This is a short review of the theory of chaos in Bohmian quantum mechanics based on our series of works in this field. Our first result is the development of a generic theoretical mechanism responsible for the generation of chaos in an arbitrary Bohmian system (in 2 and 3 dimensions). This mechanism allows us to explore the effect of chaos on Bohmian trajectories and study in detail (both analytically and numerically) the different kinds of Bohmian trajectories where, in general, chaos and order coexist. Finally, we explore the effect of quantum entanglement on the evolution of the Bohmian trajectories and study chaos and ergodicity in qubit systems which are of great theoretical and practical interest. We find that the chaotic trajectories are also ergodic, i. e., they give the same final distribution of their points after a long time regardless of their initial conditions. In the case of strong entanglement most trajectories are chaotic and ergodic and an arbitrary initial distribution of particles will tend to Born’s rule over the course of time. On the other hand, in the case of weak entanglement the distribution of Born’s rule is dominated by ordered trajectories and consequently an arbitrary initial configuration of particles will not tend, in general, to Born’s rule unless it is initially satisfied. Our results shed light on a fundamental problem in Bohmian mechanics, namely, whether there is a dynamical approximation of Born’s rule by an arbitrary initial distribution of Bohmian particles.
Keywords: chaos, Bohmian mechanics, entanglement
Citation: Contopoulos G.,  Tzemos A. C., Chaos in Bohmian Quantum Mechanics: A Short Review, Regular and Chaotic Dynamics, 2020, vol. 25, no. 5, pp. 476-495
Kuchelmeister M.,  Reiff J.,  Main J.,  Hernandez R.
In chemical reactions, trajectories typically turn from reactants to products when crossing a dividing surface close to the normally hyperbolic invariant manifold (NHIM) given by the intersection of the stable and unstable manifolds of a rank-1 saddle. Trajectories started exactly on the NHIM in principle never leave this manifold when propagated forward or backward in time. This still holds for driven systems when the NHIM itself becomes timedependent. We investigate the dynamics on the NHIM for a periodically driven model system with two degrees of freedom by numerically stabilizing the motion. Using Poincaré surfaces of section, we demonstrate the occurrence of structural changes of the dynamics, viz., bifurcations of periodic transition state (TS) trajectories when changing the amplitude and frequency of the external driving. In particular, periodic TS trajectories with the same period as the external driving but significantly different parameters — such as mean energy — compared to the ordinary TS trajectory can be created in a saddle-node bifurcation.
Keywords: transition state theory, rank-1 saddle, normally hyperbolic invariant manifold, stroboscopic map, bifurcation
Citation: Kuchelmeister M.,  Reiff J.,  Main J.,  Hernandez R., Dynamics and Bifurcations on the Normally Hyperbolic Invariant Manifold of a Periodically Driven System with Rank-1 Saddle, Regular and Chaotic Dynamics, 2020, vol. 25, no. 5, pp. 496-507

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