@article{publications25738,
          volume = {169},
          number = {5},
           month = {December},
          author = {Adrien Blanchet and Pierre Degond},
           title = {Kinetic models for topological nearest-neighbor interactions},
       publisher = {Kluwer},
            year = {2017},
         journal = {Journal of Statistical Physics},
           pages = {929--950},
        keywords = {rank-based interaction, spatial diffusion equation, continuity equation, concentration of measure},
             url = {https://publications.ut-capitole.fr/id/eprint/25738/},
        abstract = {We consider systems of agents interacting through topological interactions. These have been shown to play an important part in animal and human behavior. Precisely, the system 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 closest neighbor, the leader. We study the limit of a system size going to infinity and, under the assumption of propagation of chaos, show that the limit kinetic equation is a non-standard spatial diffusion equation for the particle distribution function. We also study the case wherein the particles interact with their K closest neighbors and show that the corresponding kinetic equation is the same. Finally, we prove that these models can be seen as a singular limit of the smooth rank-based model previously studied in [10]. The proofs are based on a combinatorial interpretation of the rank as well as some concentration of measure arguments.}
}