Stability and phase transitions in mean field interacting particle systems
File(s)
Author(s)
Bertoli, Benedetta
Type
Thesis or dissertation
Abstract
Large collections of interacting agents frequently exhibit collective behaviour such as spontaneous synchronisation, pattern formation, and transitions between ordered and disordered states. This thesis studies such phenomena through McKean–Vlasov equations, the nonlinear partial differential equations arising as mean field limits of stochastic interacting particle systems. The central questions are when non-uniform stationary states exist, what determines their stability, and how phase transitions depend on the interaction potential and the underlying network structure. We first study McKean–Vlasov equations on the torus with multichromatic interaction potentials involving multiple Fourier modes. Motivated by models from biophysics and neuroscience, we carry out a spectral stability analysis of stationary states, derive explicit formulas for critical inverse temperatures at which the uniform state loses stability, and study the stability of multipeak stationary states near the phase transition. These results are complemented by numerical simulations of both the McKean–Vlasov PDE and the underlying stochastic particle system. We then extend this analysis to interacting particle systems on random graphs, using the theory of graphons. We derive explicit synchronisation thresholds for a variety of network topologies, develop a self-consistency formulation tracking secondary bifurcations of multipeak states, and use the interaction energy as an order parameter to study the influence of network topology on phase transitions and dynamical metastability. Finally, we study collective motion in systems of self-propelled agents subject to both a rotational bias and an external orienting field. We analyse how these competing effects determine when collective order emerges, deriving explicit thresholds for the onset of synchronisation. Overall, this thesis contributes new analytical and numerical results on the stability of stationary states and the nature of phase transitions in mean field interacting particle systems, across interaction structures and network topologies motivated by applications in physics and biology.
Version
Open Access
Date Issued
2026-04-08
Date Awarded
2026-07-01
Copyright Statement
Attribution-NonCommercial 4.0 International Licence (CC BY-NC)
License URL
Advisor
Pavliotis, Greg
Sponsor
Engineering and Physical Sciences Research Council
Grant Number
EP/W523872/1
Publisher Department
Department of Mathematics
Publisher Institution
Imperial College London
Qualification Level
Doctoral
Qualification Name
Doctor of Philosophy (PhD)