Kinetic Models for Topological Nearest-Neighbor Interactions

Adrien Blanchet,Pierre Degond

doi:10.1007/s10955-017-1882-z

Adrien Blanchet, Pierre Degond

Open Access

https://doi.org/10.1007/s10955-017-1882-z

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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.

Highlights

In the literature on animal behavior including fish [3], birds [40] and even pedestrians [34], interactions between individuals are often assumed to be strongly dependent on their relative distance
We show that we can pass from the non-local integral equation issued from the smooth rank-based dynamics to the nonlinear spatial diffusion equation for the nearest-neighbor dynamics by a process involving a singular concentration of the kernel of the integral equation
We have considered a model in which particles interact with their K nearest neighbors, for a fixed value of K, showing that the corresponding kinetic model is the same as in the nearest neighbor interaction case

Summary

Introduction

In the literature on animal behavior including fish [3], birds [40] and even pedestrians [34], interactions between individuals are often assumed to be strongly dependent on their relative distance. It has recently been demonstrated that individuals in bird flocks interact with their nearest neighbors irrespective of their distance [5,18]. Degond in [5] claim that each bird interacts with between six to eight of its closest neighbors. The authors coined the term of “topological interaction” to refer to such interaction mechanisms and ”topological distance” to refer to how many other individuals were closer. Even though the reality of this topological interaction has been debated [26], it seems to receive consensus following reports that self-propelled particle models based on topological interactions successfully reproduce the observed experimental features [11,14,30]

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