Kinetic models for topological nearest-neighbor interactions - Blanchet,A; Degond,PAA (Journal of Statistical Physics, 2017)
We consider systems of agents interacting through topological interactions. Thesehave been shown to play an important part in animal and human behavior. Precisely, thesystem consists of a finite number of particles characterized by their positions and velocities.At random times a randomly chosen particle, the follower, adopts the velocity of its closestneighbor, the leader. We study the limit of a system size going to infinity and, under theassumption of propagation of chaos, show that the limit kinetic equation is a non-standardspatial diffusion equation for the particle distribution function. We also study the case whereinthe particles interact with their K closest neighbors and show that the corresponding kineticequation is the same. Finally, we prove that these models can be seen as a singular limitof the smooth rank-based model previously studied in Blanchet and Degond (J Stat Phys163:41–60, 2016). The proofs are based on a combinatorial interpretation of the rank as wellas some concentration of measure arguments.