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Report

Teaser, summary, work performed and final results

Periodic Reporting for period 1 - TraX (Stability and Transitions in Physical Processes)

Teaser

Summary

In many dynamical processes in natural sciences, understanding qualitative changes of a system, i.e., how, where and when reorganization of the dynamics takes place, provides the key to the understanding of the mechanisms at play and to the global understanding of the dynamics. This applies to processes as diverse as chemical reactions, the rearrangement of clusters, the ionization of atoms, the capture of asteroids by large celestial bodies, phase transitions in cosmology, and other systems. The most important problems in the study of reorganization processes are to predict and, where possible, to control whether the reorganization will happen.

Many reorganization processes share a common formal structure that can be exploited in their study: The qualitative structure of the dynamics is determined by invariant geometric objects in phase space, called invariant manifolds. These manifolds act as barriers that channel the dynamics of typical trajectories. Once the invariant manifolds are known, it can be predicted which initial conditions/states of the system will or will not lead to a qualitative reorganization. This structural description also yields quantitative information like, for example, chemical reaction rates, ionization yields and other transport properties.

In the figure, the red circle symbolizes an unstable periodic orbit that lies at the boundary of different regions of phase space. It might, for example, separate the initial from the final configurations of the molecules in a chemical reaction. The green and blue cylinders are the invariant manifolds, often called â€œreaction tubesâ€. If the molecules start with a configuration inside the green cylinder, they will be channeled through the red loop and away inside one of the blue cylindersâ€”a reaction will take place. If the molecules start outside the green cylinders, they will fail to make the reaction. Thus, if we know the exact shape of the cylinder, we can predict whether a reaction will happen or not.

Unfortunately, in a realistic situation the invariant manifolds are often quite difficult to compute, in particular in a high-dimensional system. It is also not always as easy to see how to use the manifolds once they are known. In TraX, we tackle both these problems: We develop computational methods of wide applicability along with the mathematical theory that underlies them, and to demonstrate the applicability of these methods in various fields of science. The research of the network includes applications to celestial mechanics, chemistry and atomic physics.

Work performed

TraX has developed a variety of numerical schemes for the computation of invariant manifolds, in particular in systems that are subject to time-dependent driving. These methods were crucial for advances in the various applied fields:

1. In celestial mechanics we investigate the dynamics of small bodies in the Earth-Moon system. These can be Near Earth Asteroids, dust particles, or spacecraft. We have numerically investigated trajectories in the Earth-Moon system that are captured into a zone of stable equilibrium. We are developing a simplified model to understand the dynamical landmarks that mediate the captures.

2. In chemistry we study the dynamics of chemical reactions in complex environments. This moniker encompasses a variety of important reactions, including reactions of biomolecules in living systems. We have achieved a theory of decay rates based on invariant manifolds that can be used, with an unprecedented level of detail, to describe not only time-averaged decay rates in a driven system, but also instantaneous rates, and it applies to multidimensional as well as to one-dimensional systems. This advance provides a detailed view of the reaction dynamics and also a computational algorithm for rate calculation that is much more efficient than earlier methods.

3. In atomic physics we focus on the question: How and when are ionized electrons driven back to the ionic core by an ultrastrong, ultrashort laser pulse? The answer to this question holds the key to future breakthroughs like the real-time imaging of biomolecules with bright short-wave light sources. In a strong field most ionized electrons drift far from the core and cannot contribute to energy transfer processes. However, a few â€“very fewâ€“ find their way back to the core and bring with them the energy they absorbed from the laser, thereby becoming the drivers of the all-important phenomenon of High Harmonic Generation. We have developed a hierarchy of models for this process that allows a detailed interpretation of the resulting spectrum.

Final results

For the remaining duration of the project we expect further advances in all areas of application.

1. In celestial mechanics we will investigate the possibility to attach a propulsion system to a Near Earth Asteroid and capture it into a suitable orbit of the Earth-Moon system by means of a small manoeuvre, as planned, for example, in NASA\'s Asteroid Redirect Mission. Such a capture would allow the exploitation of raw materials from asteroids, e.g., gold, iridium and platinum.

2. In chemistry, the characterization of rates and pathways for chemical reactions in complex solvents remains a primary concern. They determine which design pathways can realistically be used to accelerate or suppress particular reactions. We will spell out the details of this process, and the possibilities for its control, in realistic molecules.

3. In atomic physics we will propose a new recollision scenario for an atom subjected to an elliptically polarized laser pulse. This scenario, based on the construction of invariant manifolds of certain periodic orbits, opens the possibility for controlling the amount of energy brought back by the pre-ionized electron, and hence will have an impact on the spectrum of high harmonic radiation, a critical step towards the production of laser sources with short wavelength.