Infinite particle systems with collective behaviour and related mesoscopic equations

We study analogs of Kuramoto and Cucker-Smale models for systems with infinite interacting particles. For such systems, we describe the way to derive the corresponding kinetic (mesoscopic) equation reflecting the statistical characteristics of the whole microscopic system in the sense of a proper scaling. This derivation has an algorithmic realization in terms of related hierarchical chains of correlation function equations. We compare the obtained mesoscopic equations for considered infinite interacting particle systems with kinetic equations derived from ordinary differential equation models of finite particle systems using the Bogoliubov–Born–Green–Kirkwood–Yvon hierarchy technique.