Blanchet A, Degond P. Kinetic Models for Topological Nearest-Neighbor Interactions.
JOURNAL OF STATISTICAL PHYSICS 2017;
169:929-950. [PMID:
32009675 PMCID:
PMC6959382 DOI:
10.1007/s10955-017-1882-z]
[Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 03/09/2017] [Accepted: 09/18/2017] [Indexed: 06/10/2023]
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 Blanchet and Degond (J Stat Phys 163:41-60, 2016). The proofs are based on a combinatorial interpretation of the rank as well as some concentration of measure arguments.
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