Volume 25, Number 5
Volume 25, Number 5, 2020
Nonlinear Dynamics in Chemical Sciences
Hoover W. G., Hoover C. G.
Nonequilibrium Molecular Dynamics, Fractal PhaseSpace Distributions, the Cantor Set, and Puzzles Involving Information Dimensions for Two Compressible Baker Maps
Abstract
Deterministic and timereversible nonequilibrium molecular dynamics simulations
typically generate “fractal” (fractionaldimensional) phasespace distributions. Because these
distributions and their timereversed 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.

Dhont G., Iwai T., Zhilinskií B. I.
A Study of Energy Band Rearrangement in Isolated Molecules by Means of the Dirac Oscillator Approximation
Abstract
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 threedimensional Cartesian product of the
twodimensional phase space and the onedimensional 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.

Lyu W., Naik S., Wiggins S.
The Role of Depth and Flatness of a Potential Energy Surface in Chemical Reaction Dynamics
Abstract
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 degreeoffreedom (DOF) Hamiltonian normal form for the saddlenode
bifurcation and quantify their influence on chemical reaction dynamics [1, 2]. In a recent
work, GarcíaGarrido et al. [2] illustrated how changing the welldepth of a potential energy
surface (PES) can lead to a saddlenode bifurcation. They have shown how the geometry of
cylindrical manifolds associated with the rank1 saddle changes en route to the saddlenode
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.

Contopoulos G., Tzemos A. C.
Chaos in Bohmian Quantum Mechanics: A Short Review
Abstract
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.

Kuchelmeister M., Reiff J., Main J., Hernandez R.
Dynamics and Bifurcations on the Normally Hyperbolic Invariant Manifold of a Periodically Driven System with Rank1 Saddle
Abstract
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 rank1 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 saddlenode bifurcation.
