Analysis of the Effective Long Time-Behaviour of Molecule Systems

Playing golf means: moving the golf ball from the given initial position on the tee through a landscape of hills and valleys to the hole as the final location. Somewhat simplified, the aim of this project is to understand how to play golf not in the three-dimensional space we are accustomed to, but in a complicated landscape with hundreds of dimensions. While this sounds like a mathematical folly, this is what in fact happens in many complex systems, such as molecules or DNA. For example, molecules can exist in several distinctly different states, sometimes called conformations. These states can be pictured as wells of a complex energy landscape. Let us say that the molecule consists of N atoms, each described by 3 space coordinates and 3 momentum coordinates. Then the wells are in a 6N dimensional space. Starting with one conformation of the molecule (one of the wells, as the chemical equivalent of the tee), the atoms will then spend most of the time jostling around in that well before a rare spontaneous fluctuation occurs that lifts the atoms of the reactant over the barrier into the next valley, the well of the other conformation (corresponding to the hole in the golf analogy). This is an example of a so-called rare event. And while these events are indeed rare, they normally carry crucial information on the system in question. So one would like to understand and predict these transitions and rare events. However, a direct molecular simulation would need to resolve the atomistic timescale, while rare events take place on timescales which can be larger by many orders of magnitude; this renders direct simulations unfeasible even on the largest supercomputers. Instead, the aim of this project is to derive rigorously reduced models that capture the effective long-time behaviour of high-dimensional complex systems. A particular focus will be on rare events and transition states. A variety of model problems will be investigated, chosen to capture key challenges present in a number of more complicated problems in various application areas.