Nonlocal Partial Differential Equations: entropies, gradient flows, phase transitions and applications

Understanding the qualitative properties of large systems of interacting particles is of crucial importance in many applications in physics and biology such as molecular dynamic simulations, particles immersed in a fluid, organogenesis modelling, swarming methods for optimization or herding in the social sciences and models for opinion formation, to name a few. This project will be devoted to the further advancement in our understanding of the connection between particle descriptions and continuum models via the passage to the thermodynamic limit. One of our main goals will be to study the thermodynamic (mean field) limit for systems of interacting particles in rugged, multiscale energy landscapes, of the type that one frequently encounters in applications such as biophysics and chemical physics. The dynamics in such a potential exhibit metastable behavior at all scales. In particular, we want to understand the effect of the multiscale structure on the existence and stability of stationary states of the mean field dynamics. Phase transitions, i.e., abrupt changes of behavior, driven by noise will be analyzed for rugged energy landscapes. We will then employ tools from control theory in order to develop algorithms for stabilizing unstable steady states. In addition, we will develop efficient numerical techniques for solving nonlocal, nonlinear mean field equations and we will apply them to diverse problems such as dynamical density functional theory and mathematical models from the social sciences, including models for opinion formation. Furthermore, we will use appropriate systems of interacting particles and their corresponding mean field limit in order to develop consensus-based global optimization algorithms that can be applied to potentials with a multiscale structure characterized by (infinitely) many local minima.

It is anticipated that the successful completion of the proposed research will have substantial impact to the academic community, in particular to researchers working in several areas of applied mathematics and mathematical physics, including the theory of partial differential equations, statistical mechanics and optimization and control. Furthermore, the computational algorithms that we will develop will be of use to practitioners working in computational physics and in the mathematical modeling in the social sciences.

Methods for Communication.- The team will have a common strategy for disseminating and communicating the research results that we detail next:

Publications.- The PI and Co-PI have an outstanding publication record with articles in the leading international research journals in applied mathematics and mathematical physics. It is anticipated that the work outlined in this proposal with be submitted to top journals in mathematics, applied mathematics, mathematical physics, and mathematical biology.

Workshops and Scientific Events.- An important route to obtain feedback and to engage the closest scientific community in this topic will be a dedicated workshop that we will organise and host to be held provisionally in Spring 2019. This workshop will count with the top-world researchers in the field. We will organize annual small (2-3 days, 20 participants) focused workshops in which we plan to invite researchers mostly from the UK and European countries to increase the number of collaborations at Imperial with top groups in near geographical areas. These small workshops will be funded from visitors expenses already budgeted.

The PI and Co-PI have considerable experience in organising successful workshops and summer schools as detailed in the track of record. Furthermore, 2018 will be the year of mathematical biology in Europe organized by the EMS and the ESMTB in which JAC is one of the main organisers as chairman of the Applied Mathematics Committee of the European Mathematical Society. During that year there will be several events organized across Europe in which this team will be involved trying to understand how swarming models can be used in organogenesis and in other instances in mathematical biology through experimental real data.

Collaborations and Visitors.- Additionally the PI and Co-PI have an active and extensive network of international collaborations with visitors planned to come to Imperial, and the investigators of the team will be visiting regularly research groups in the UK and overseas. This will aid in the dissemination of the research and increase its impact through the academic community. This is ensured by the high visibility of the research done as measured by the number of researchers citing publications of the PI and Co-PI.

Public engagement: communicating research and scientific values.- Imperial College has an efficient press office promoting innovative research conducted at the College. With their help, we plan to promote the numerous aspects of the research that have significant impact and that would be of interest to the public. This includes initiating contact with interested media sources, and possibly traveling to schools around London area to promote mathematics, and the research proposed here in particular. Furthermore, we plan to write popular science articles and to give popular science talks on the applications of applied mathematics and of statistical physics to the social sciences.