Project overview

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 above, 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 “reaction tubes”. 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 cylinders—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.

The acronym TraX refers to the TRAnsitions we intend to study and the X has a double meaning: It refers to a hyperbolic point (or more generally to a hyperbolic object) responsible for the transitions, acting as a common denominator for all the physical processes we plan to study, and it also refers to the staff Xchange programme we propose to implement to reach the scientific objectives.