Fluid dynamic description of flocking via Povzner–Boltzmann equation

We introduce and discuss the possible dynamics of groups of indistinguishable agents, which are interacting according to their relative positions, with the aim of deriving hydrodynamic equations. These models are developed to mimic the collective motion of groups of species such as bird flocks, fish schools, herds of quadrupeds or bacteria colonies. Our starting model for these interactions is the Povzner equation, which describes a dilute gas in which binary collisions of elastic spheres depend of their relative positions. Following the Cucker and Smale model, we will consider binary interactions between agents that are dissipative collisions in which the coefficient of restitution depends on their relative distance. Under the assumption of weak dissipation, it is shown that the Povzner equation is modified through a correction in the form of a nonlinear friction type operator. Using this correction we formally obtain from the Povzner equation in a direct way a fluid dynamic description of a system of weakly interacting agents interacting in a dissipative way, with a coefficient of restitution that depends on their relative distance.