Upscaling of interacting particles: from discrete to continuum

Description of project:
Systems of interacting particles are ubiquitous: crowds of people, birds flocking, fish swarming, dislocations in metals, vortices in superconductors are examples of discrete collections of particles – different in nature and in scale - which aggregate and have an interesting collective behaviour.

Dislocations, in particular, are defects at the microscopic scale, but collectively, at the macroscopic scale, they determine how metals deform permanently. To understand this collective behaviour one has to bridge the scales between a microscopic, individual-based description of the system, and a macroscopic, group-led one. In other words, one has to perform a discrete-to-continuum upscaling.
From the mathematical point of view, this task is extremely intriguing and challenging. Even in the static case, there are only few examples that are completely understood, mainly in the two-dimensional setting and for short-range interactions. In reality, however, dynamics plays a fundamental role, and interactions are typically long-range. The need for more realistic models has introduced countless new problems and challenges, and has led to the development of new mathematical techniques to deal with them.
A big effort has been devoted, in particular, to understanding how to couple the discrete-to-continuum upscaling with the dynamics. This is exactly the focus of this project. We will consider evolutions driven by the energy of the system and by a dissipation, and combine different types of dynamics (e.g. rate-independent or gradient-flow) with Gamma-convergence, a variational convergence that guarantees the convergence of the energy minimising configurations.

Training opportunities:
The successful applicant will receive a thorough and broad training in several modern aspects of mathematics and continuum mechanics.
In relation to the project, he/she will be trained in mathematical analysis, and develop competencies in PDEs, calculus of variations, measure theory. He/she will also have the possibility of exploring the natural connections of the project with a range of other disciplines, including probability and numerical analysis.
Being part of the Doctoral College Path to PhD, he/she will take classes from a wide range of subjects from the Taught Course Centre Programme (http://tcc.maths.ox.ac.uk), a collaboration between Bath, Bristol, Imperial, Oxford and Warwick, the Graduate Course Programme and the SaMBa Centre for Doctoral Training (http://www.bath.ac.uk/math-sci/postgraduate/samba/). Additionally, he/she will receive a Postgraduate Skills Training (http://www.bath.ac.uk/research/pgskills) aimed at developing tutoring, writing and presenting skills, among others.
The successful applicant will be part of a vibrant and lively group, and will be offered a range of regular seminars in PDEs, Continuum Mechanics, Probability, Numerical Analysis, and Landscape seminars (http://www.bath.ac.uk/math-sci/news/). Attention will be paid to make sure that he/she will be integrated in an international research network and develop his/her own collaborations and grow as an independent researcher.

Additional information:
The start of the project will coincide with a three-month visit at Bath of Amit Acharya, a leading expert in dislocations from Carnegie Mellon University. Also, there will be regular visits and exchanges with colleagues from other institutions, including Mark Peletier, Mark Geers, Markus Hütter and Ron Peerlings (Eindhoven University of Technology), Maria Giovanna Mora (University of Pavia), Gianni Dal Maso (SISSA), and Celia Reina (UPenn).

This project is in its nature interdisciplinary. Relevant examples of interacting particle systems arise in engineering, physics, biology, ecology. The regular seminar activity of the department, frequent visitors and exchanges with other institutions will create countless opportunities for interdisciplinary interactions with mathematicians and scientists working on similar problems from different perspectives.