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Details of Grant 

EPSRC Reference: EP/P031587/1
Title: Nonlocal Partial Differential Equations: entropies, gradient flows, phase transitions and applications
Principal Investigator: Pavliotis, Professor G
Other Investigators:
Carrillo, Professor J
Researcher Co-Investigators:
Project Partners:
Department: Mathematics
Organisation: Imperial College London
Scheme: Standard Research
Starts: 01 October 2017 Ends: 31 March 2021 Value (£): 449,210
EPSRC Research Topic Classifications:
Mathematical Analysis Non-linear Systems Mathematics
Numerical Analysis
EPSRC Industrial Sector Classifications:
No relevance to Underpinning Sectors
Related Grants:
Panel History:
Panel DatePanel NameOutcome
28 Feb 2017 EPSRC Mathematical Sciences Prioritisation Panel March 2017 Announced
Summary on Grant Application Form
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.
Key Findings
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Further Information:  
Organisation Website: http://www.imperial.ac.uk