Analysis of the effective long time-behaviour of molecular 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.

The immediate beneficiaries of the research will be the community of mathematicians working on dynamical systems with several time scales. This includes researchers in applied stochastic processes and applied analysis, and in particular people studying the effective numerical simulation of molecular systems. This represents a substantial international community, with significant groups and keyplayers in the UK.

One aim of the research is to foster the growing community working on the mathematical analysis of problems arising in computational chemistry (thermostatting, free energy calculations, rare events). The research brings together applied analysis and applied stochastic processes in a novel way.

The core challenge of the proposed research is to derive effective descriptions of high-dimensional complex systems. If the objectives of this project are achieved, it will provide an in-depth understanding of model problems which are amenable to a rigorous analysis. The understanding gained there can then be extrapolated to realistic models appearing in a range of application areas. The benefit to the wide community working on problems in this kind, both in mathematics and the sciences, is thus an improved understanding of how to derive effective descriptions of complex systems. This understanding can the inform the design of algorithms, giving scientists a handle how to study systems which are too large and have to disparate time-scales to be simulated directly. We anticipate a number of interdisciplinary follow-on projects, with the overarching tool of developing and applying new tools for understanding, predicting and sometimes controlling the evolution of complex systems.