On reduced SPDE models for stochastic Cucker-Smale flocking dynamics

Abstract

Flocking refers to a phenomenon in collective motion where particles align their velocities with each other, commonly modelled using Cucker-Smale dynamics. In this work, we introduce and analyse two general reduced SPDE models for stochastic Cucker-Smale particle dynamics that encompass environmental and i.i.d.-type noises. Our models are computationally cheaper alternatives to the underlying particle system and preserve the flocking behaviour. To derive our reduced models, we consider suitable empirical particle densities which depend on position and time, but feature velocity dependency in a multiplicative fashion: this setup crucially allows for a very intuitive and convenient notion of flocking at the SPDE level. In $d=1$ we prove flocking of the reduced SPDE models, and, for the environmental noise, we also quantify the discrepancy between the reduced SPDE and mollified particle system. We also discuss the technical modifications required to extend the results to higher dimensions. We then provide extensive numerical experiments to verify our theoretical results in $d=1$ and to explore the behaviour of our models in $d=2$.