During the last decades, there has been a growing effort to understand how complex self-organized patterns (or structures) can emerge from active particle systems when the number of particles becomes very large. Typical examples in biology include the flock of birds or the swarm of bacteria and other active cells. More recently, this modelling framework has also been applied in socio-economical contexts (opinion dynamics, wealth distribution…) or in data science and optimization with the development of so-called particle methods. Sensible modelling attempts have been based on classical tools developed in statistical physics to study inert systems and in particular on the kinetic theory of gas. The core idea is the (rigorous) derivation of PDE models from many-particle systems: this is a long-standing mathematical question tracing back to Boltzmann, but which has recently enjoyed some kind of a renaissance. In this talk I will briefly review and discuss some recent trends in the study of collective dynamics and self-organization phenomena and discuss how the behavior of many-particle systems can be inferred by looking at appropriate scaling limits. Then, I will illustrate these ideas with a system of so-called « body-oriented » particles which, in particular, demonstrates the influence of stochasticity and geometry on self-organization.
This is a joint work with Pierre Degond, Amic Frouvelle and Sara-Merino-Aceituno initiated at Imperial College London.