Decrease of entropy and emergence of order in collective dynamics

We study the hydrodynamic description of collective dynamics driven by velocity alignment. It is known that such Euler alignment systems must flock toward a limiting “flocking” velocity, provided their solutions remain globally smooth. To address this question of global existence we proceed in two steps. (i) Entropy and closure. The system lacks a closure, reflecting lack of detailed energy balance in collective dynamics. We discuss the decrease of entropy and the asymptotic behavior toward a mono-kinetic closure; and (ii) Mono-kinetic closure. We prove that global regularity persists for all time for a large class of initial conditions satisfying a critical threshold condition, which is intimately linked to the decrease of entropy. The result applies in any number of spatial dimensions, thus addressing the open question of existence beyond two dimensions.