Long-term dynamics in interacting particle systems

In this research project, we study the mathematics behind interacting particle systems, which are models that describe the collective behaviour of interacting agents influencing each other's actions.
Our primary goal is to deepen the mathematical understanding of these models, particularly in their long-term dynamics. We seek to extend current knowledge of this research area, particularly of models which have not been analysed in full detail before. The research methodology is based on techniques from mathematical analysis. In particular, we use the powerful tools of bifurcation theory, partial differential equations, stochastic differential equations and mean field theory to model the behaviour of interacting agents. Current work specifically includes analysing how this behaviour changes as the type of interaction between agents varies, and studying this type of system on random networks, which would allow us to extend existing results to a much wider class of models.
The significance of our work extends beyond theoretical mathematics. Interacting particle systems find applications in various fields, from the movement of gases and liquid crystals to models of biological neurons, to social dynamics like opinion formation. Even phenomena like pedestrian dynamics and the collective movements of bird flocks fall within the scope of our study. By understanding the underlying mathematics, we aim to make these models more applicable to real-world scenarios.
Our research aligns seamlessly with EPSRC's theme of mathematical analysis. By deepening our understanding of interacting particle systems, we also contribute to applications in areas such as mathematical biology and physics, and, due to the wide range of potential applications, to the EPSRC's five strategic themes.