Stochastic active flows and interacting particle systems

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Mathematical models describing continua have been used very successfully to explain the behaviour of a wide rangeof systems beyond the liquid flows that inspired their initial development. For example, the formation of traffic jamsand the collective behaviour of large animal groups. However, these models typically take a purely macroscopicapproach to explain these phenomena. This class of PDE-based models does not capture the effects of noiseinducedfluctuations arising from finite size effects in 'fluids' composed of discrete moving agents. Compartmentbased-methods for the interaction of agents (in reaction diffusion systems for instance) allow for the incorporation ofstochastic effects and quantification of these effects on the system as a whole. This method is less convenient wheninteractions between agents are not confined to their local vicinity where only the particles in the same compartmentneed to be considered. Microscopic level models allow for the greatest level of accuracy to the underlying rules ofinteractions between particles. However, using these models to understand the evolution of the system as a whole isa difficult challenge for many systems.The first task of my PhD will be to explicitly link microscopic continuum based models of interacting stochastic agentswith macroscopic kinematic fluid models in one dimension, enabling an analysis of the effects of noise induced byfinite size effects. This will require the classification of the types of macroscopic models which can be understood asthe large particle limit of pair-wise particle interactions. I will then find a stochastic PDE for the evolution of thegeneralised system before looking at specific rules of interactions and the SPDEs which result from this. I willcompare simulations of the SPDEs for specific cases of the model with the microscopic simulations, checking foragreement with the overall density evolution and quantifying the scale of the stochastic effects.Secondly, I will look into the case of two dimensions. Initially focusing on conservative interactions between particles,I will then look at the potential for milling pattern formation observed in many swarming behaviours. For both one andtwo dimensions I will look for bi-stable systems which could be modelled using pair-wise interaction models ofparticles. The mean switching time between these two states, observed in cases such as locust swarming, could thenbe quantified from the SPDE for the density evolution.