In this talk, we review some self-organization models arising in biology and physics, in particular, Vicsek model 1999 and its mathematical re-visit by Degond and Motsch in 2007. Starting from particle system, it can be derived the kinetic models (SOK) and more macroscopic hydrodynamic models (SOH). Together and Tengfei Zhang and Yi-long Luo, we made some rigorous justifications from SOK to SOH, using a Generalized Collision Invariants (GCI)-based Hilbert expansion. Moreover, our proof is employed some geometry of S^2 and SO(3).
