Chaotic dynamics in multidimensional transition states

The crossing of a transition state in a multidimensional reactive system is mediated by invariantgeometric objects in phase space: An invariant hyper-sphere that represents the transition stateitself and invariant hyper-cylinders that channel the system towards and away from the transitionstate. The existence of these structures can only be guaranteed if the invariant hyper-sphere isnormally hyperbolic, i.e., the dynamics within the transition state is not too strongly chaotic. We study thedynamics within thetransition state for the hydrogen exchangereaction in three degrees of freedom. As the energy increases, the dynamics within the transition statebecomes increasingly chaotic. We find that the transition state first looses and then, surprisingly,regains its normal hyperbolicity. The important phase space structures of transition state theory will, therefore,exist at most energies above the threshold

Characterising the teaching of university mathematics: a case of linear algebra

This paper focuses on university mathematics teaching where the topic is linear algebra. The research team includes two mathematics educators and a mathematician who collaborate to study the teaching approach and the issues it raises for teaching-learning at university level. We see university mathematics to constitute a community of practice in which the practitioners are those who do mathematics. Such a perspective draws sociohistorically on established practices in doing, learning and teaching mathematics within a university. The paper offers an interpretation of these theoretical perspectives in relation to a first year course on Linear Algebra. We look at how teaching is constructed within the particular setting, with a critical eye on the learners, on learning outcomes and on the tensions experienced by the lecturer in satisfying student needs and mathematical values

Communication: transition state trajectory stability determines barrier crossing rates in chemical reactions induced by time-dependent oscillating fields

When a chemical reaction is driven by an external field, the transition state that the system must pass through as it changes from reactant to product—for example, an energy barrier—becomes timedependent. We show that for periodic forcing the rate of barrier crossing can be determined through stability analysis of the non-autonomous transition state. Specifically, strong agreement is observed between the difference in the Floquet exponents describing stability of the transition state trajectory, which defines a recrossing-free dividing surface [G. T. Craven, T. Bartsch, and R. Hernandez,“Persistence of transition state structure in chemical reactions driven by fields oscillating in time,”Phys. Rev. E 89, 040801(R) (2014)], and the rates calculated by simulation of ensembles of trajectories. This result opens the possibility to extract rates directly from the intrinsic stability of the transition state, even when it is time-dependent, without requiring a numerically expensive simulation of the long-time dynamics of a large ensemble of trajectorie

Persistence of transition-state structure in chemical reactions driven by fields oscillating in time

Chemical reactions subjected to time-varying external forces cannot generally be described through a fixed bottleneck near the transition-state barrier or dividing surface. A naive dividing surface attached to the instantaneous, but moving, barrier top also fails to be recrossing-free. We construct a moving dividing surface in phase space over a transition-state trajectory. This surface is recrossing-free for both Hamiltonian and dissipative dynamics. This is confirmed even for strongly anharmonic barriers using simulation. The power of transition-state theory is thereby applicable to chemical reactions and other activated processes even when the bottlenecks are time dependent and move across space

The geometry of transition states: How invariant manifolds determine reaction rates

Over the last years, a new geometrical perspective on transition state theory has been developed that provides a deeper insight on the reaction mechanisms of chemical systems. This new methodology is based on the identification of the invariant structures that organize the dynamics at the top of the energetic barrier that separates reactants and products. Moreover, it has allowed to solve, or at least circumvent, the recrossing-free problem in rate calculations. In this paper, we will discuss which kind of objects determine the reaction dynamics in the presence of dilute, dense and condensed phase baths

Gluing torus families across a singularity: The lens space for the hydrogen atom in crossed fields

We demonstrate that topological information that can be extracted from periodic orbits in a near-integrable system can lead to a complete topological characterization of families of invariant 2-tori in terms of lens spaces. This approach ties in with the techniques we developed for classifying the tori in systems with more than two degrees of freedom. It therefore offers a general way to investigate families of invariant 2-tori in higher-dimensional Hamiltonian systems

Reaction rate calculation with time-dependent invariant manifolds

The identification of trajectories that contribute to the reaction rate is the crucial dynamical ingredient in any classical chemical reactivity calculation. This problem often requires a full scale numerical simulation of the dynamics, in particular if the reactive system is exposed to the influence of a heat bath. As an efficient alternative, we propose here to compute invariant surfaces in the phase space of the reactive system that separate reactive from nonreactive trajectories. The location of these invariant manifolds depends both on time and on the realization of the driving force exerted by the bath. These manifolds allow the identification of reactive trajectories simply from their initial conditions, without the need of any further simulation. In this paper, we show how these invariant manifolds can be calculated, and used in a formally exact reaction rate calculation based on perturbation theory for any multidimensional potential coupled to a noisy environment

Finite-barrier corrections for multidimensional barriers in colored noise

The usual identification of reactive trajectories for the calculation of reaction rates requires very timeconsuming simulations, particularly if the environment presents memory effects. In this paper, we develop a method that permits the identification of reactive trajectories in a system under the action of a stochastic colored driving. This method is based on the perturbative computation of the invariant structures that act as separatrices for reactivity. Furthermore, using this perturbative scheme, we have obtained a formally exact expression for the reaction rate in multidimensional systems coupled to colored noisy environments

Transition state theory for activated systems with driven anharmonic barriers

Classical transition state theory has been extended to address chemical reactions across barriers that are driven and anharmonic. This resolves a challenge to the naive theory that necessarily leads to recrossings and approximate rates because it relies on a fixed dividing surface. We develop both perturbative and numerical methods for the computation of a time-dependent recrossing-free dividing surface for a model anharmonic system in a solvated environment that interacts strongly with an oscillatory external field. We extend our previous work, which relied either on a harmonic approximation or on periodic force driving.We demonstrate that the reaction rate, expressed as the long-time flux of reactive trajectories, can be extracted directly from the stability exponents, namely, Lyapunov exponents, of the moving dividing surface. Comparison to numerical results demonstrates the accuracy and robustness of this approach for the computation of optimal (recrossing-free) dividing surfaces and reaction rates in systems with Markovian solvation forces. The resulting reaction rates are in strong agreement with those determined from the long-time flux of reactive trajectories

Transition state theory for laser-driven reactions

Recent developments in Transition State Theory brought about by dynamical systems theory are extended to time-dependent systems such as laser-driven reactions. Using time-dependent normal form theory, we construct a reaction coordinate with regular dynamics inside the transition region. The conservation of the associated action enables one to extract time-dependent invariant manifolds that act as separatrices between reactive and non-reactive trajectories and thus make it possible to predict the ultimate fate of a trajectory. We illustrate the power of our approach on a driven H´enon-Heiles system, which serves as a simple example of a reactive system with several open channels. The present generalization of Transition State Theory to driven systems will allow one to study processes such as the control of chemical reactions through laser pulses