Topological interactions in a Boltzmann-type framework

TL;DR

This paper models particle interactions based on proximity and leader-follower dynamics, deriving a Boltzmann-like equation with spatial non-locality, using Bernstein polynomial approximations in the limit of large systems.

Contribution

It introduces a novel Boltzmann-type framework for particle interactions based on topological proximity, highlighting spatial non-locality and employing Bernstein polynomial techniques.

Findings

Derivation of a Boltzmann-like equation with spatial non-locality

Demonstration of propagation of chaos in the system

Use of Bernstein polynomials for approximation

Abstract

We consider a finite number of particles characterised by their positions and velocities. At random times a randomly chosen particle, the follower, adopts the velocity of another particle, the leader. The follower chooses its leader according to the proximity rank of the latter with respect to the former. We study the limit of a system size going to infinity and, under the assumption of propagation of chaos, show that the limit equation is akin to the Boltzmann equation. However, it exhibits a spatial non-locality instead of the classical non-locality in velocity space. This result relies on the approximation properties of Bernstein polynomials.