Hydrodynamic limits of systems of geometrically enriched particles

Context of the research including potential impact : Classical particle systems in physics such as gases consist of point particles. However, there are many examples in nature where particles have more complex geometrical structure. For instance, granular media consist of spherical or polyedral particles. Liquid crystals, bacterial colonies or some organs like muscles are made of rod-like particles. The three dimensional attitude of fish, birds or drones is encoded into rotations. Particle dynamics (such as particle swarm optimization) are used to solve data science problems where one needs to preserve the underlying geometrical structure of the data. Particles which carry an underlying geometrical structure are referred to as geometrically enriched particles.

Systems of geometrically enriched particles produce patterns that are significantly different from those observed in gases. Patterning and self-organization underpin many phenomena in the living world including organ development and cancer. Their investigation will deepen our knowledge of these and help progress towards the resolution of tremendously important questions in health and well-being. Data science deals with high dimensional problems and the design of particle-based processing methods preserving an underlying geometrical structure in large dimensions present specific challenges which have a wide array of applications.

Aims and objectives : This project aims at investigating the different types of emerging patterns in systems of geometrically enriched particles and will contribute to generating new algorithms to process large sets of geometrically structured data in high dimensions.

Novelty of the research methodology : the dynamics of particle systems is practically intractable, due to the large number of its constituents. Continuum models providing averaged information such as the particle density need to be derived. The coarse-graining of particle systems into continuum models requires several mathematical steps that lead to singular perturbation problems whose limits are the targetted continuum models. This procedure is well established for gases and is called the hydrodynamic limit. However, its application to geometrically enriched particles is still at its infancy as many properties formerly needed for hydrodynamic limits are lost We will use recent mathematical breakthroughs to tackle this problem in high dimensions and will explore the so-obtained partial differential models, which have unmet new features through both mathematical analysis and numerical simulations. The tools and methods developed in the course of this project will provide a powerful framework to develop and analyze geometrically enriched particle systems in all kinds of different contexts, from biology to social sciences, and from physics to engineering and is susceptible to a wide array of applications.

Alignment to EPSRC's strategies and research areas : this project belongs to strategic theme mathematical sciences, and has strong connections with the themes healthcare technologies and ICT. The concerned research areas are mathematical biology, mathematical analysis, numerical analysis, statistics and applied probability, geometry and topology, complex fluids and rheology, continuum mechanics.

Any companies or collaborators involved : This work will be used to develop new models in tissue dynamics and/ or cancer in collaboration with biologists. It will also be used to derive new algorithms for the processing of large sets of geometrically enriched data in collaboration with data scientists.