Large deviation techniques for model coarse graining

Physical systems of interest to society are extremely complex. For example, the atmosphere and ocean of Earth, relevant for questions of global warming, long time climate estimates or prediction of extreme whether events, consists of a tremendeous number of interacting systems, each with many degrees of freedom. It is impossible to consider the system in its full complexity when trying to make predictions, in particular if the questions concern long-time prediction or rare events.

Indeed, multiscale systems are ubiquitous in nature: the underlying processes relevant to many physical phenomena often happen on vastly different length- or time-scales. This opens up the possibility of coarse-graining and averaging, where fast, fluctuating degrees of freedom can be considered as effective noise on slow degrees of freedom. In particular, this helps to reduce, or coarse-grain, complex physical models into much simpler models that are tractable analytically or numerically to enable prediction and deeper understanding of the involved processes.

Rare events, for example conformational changes of the relevant unknowns, are particularly interesting and rich in such a setup. In stochastic systems, unlikely fluctuations can push the system from its typical state into other, meta-stable configurations with often vastly different properties. Examples are chemical reactions, phase transitions, weather patterns, protein folding, or persistent structures in fluid flow. In such situations, large deviation theory gives precise and rigorous estimates of the probabilities and mechanisms of these conformational changes, by generalising the notion of free energy and entropy to arbitrary stochastic systems.

Obtaining explicit large deviation principles in this multiscale setup is a big challenge, since the associated fluctuations stem from averaging of complex physical processes, and therefore are generally non-linear, non-Gaussian, or even non-Markovian. The computation of large deviation principles in such a setup is of high importance, as it would allow us to estimate transition probabilities on the effective, coarse-grained model, without the need to consider all (fast, unimportant) degrees of freedom, thus making computation feasible.

The proposal concerns itself with the development of theory and numerical algorithms in the above situation, and to make available the developed techniques to applied sciences. The PI will apply these large deviation methods for multiscale systems and coarse-grained models to three concrete problems: (i) Metastability in atmospheric jets, where turbulent fluctuations facilitate the disappearance of planetary jets in atmospheric flow, (ii) magnetically confined fusion experiments, where conformational changes in the boundary layer in plasma reactors prevent efficient confinement, and (iii) fibre-optics communications, where random fluctuations in optical fibres lead to bit-flips in photonic communication.

All theoretical research efforts will result in the development of algorithms or software implementations permitting the re-use by researchers in other fields that are concerned with rare events in multiscale systems.

Multiscale systems are ubiquitous in nature. In those, certain phenomena cannot be adequately modeled without the technology I am developing: Whenever the system is too complex for analytical treatment, and numerical experiments are too costly, rare events can no longer be treated with traditional methods. In this research project, I will develop methods to effectively compute rare events in stochastic systems with scale separation using algorithms based on large deviation theory. As such, it provides valuable tools to parties interested in quantifying the risk of such events, in particular in situations where direct experiments are hard to perform. An increased confidence in rare event probability predictions has the potential of acting as guideline for policy makers in devising strategies to counter the associated challenges.

Specifically, the PI has taken care to identify and address concrete applications in theoretical physics and engineering that are directly impacted by the proposed research and can benefit from the research output. In particular, the PI already established contact to three possible applications in the fields of atmosphere dynamics, plasma physics, and photonics, which themselves are fields of high societal and economic impact with respect to the possibilities of their prospective breakthroughs in climate science, power generation, and communication technologies.

Concretely, as immediate impact, the research output and theoretical results of the proposed programme will be framed in the form of either implementable algorithms or software packages with reference implementations, and will be made available to interested researchers as well as the public sector in the form of open source software. This amplifies the potential impact of the scientific output by providing third parties with immediate steps to apply the theoretical results to their own field.

In the medium term, in particular towards the end of the project, and possibly further, spin-off results for fields of application distinct from the above are envisioned. In particular, the research output will be of use for investigating population dynamics, fluid turbulence, and ocean surface waves. The PI intends to disseminate the research output to those fields by strengthening his existing network and establishing further collaborations.

In the long term, broader impact of the proposed research arises from its fundamental contributions to the computation of rare events in spatially extended stochastic systems, which form the basis of almost every field in physics, biology, and engineering. As such, the scientific output of the programme allows researchers in a broad variety of fields to obtain more reliable predictions of rare event probabilities at higher computational efficiency, and in particular for systems for which such quantitative estimates are infeasible with other methods.