Kinetic theory of particle interactions mediated by dynamical networks

TL;DR

This paper derives and analyzes kinetic equations for particles interacting via a dynamic network, simplifying to a macroscopic model and studying stability and phase transitions, especially for Hookean potentials.

Contribution

It introduces a multiscale derivation from microscopic to macroscopic models for particle-network interactions, including stability analysis and bifurcation characterization.

Findings

Derived coupled kinetic equations for particles and links.

Simplified to a macroscopic aggregation-diffusion model under fast remodelling.

Identified stability conditions and phase transition criteria for the system.

Abstract

We provide a detailed multiscale analysis of a system of particles interacting through a dynamical network of links. Starting from a microscopic model, via the mean field limit, we formally derive coupled kinetic equations for the particle and link densities, following the approach of [Degond et al., M3AS, 2016]. Assuming that the process of remodelling the network is very fast, we simplify the description to a macroscopic model taking the form of single aggregation-diffusion equation for the density of particles. We analyze qualitatively this equation, addressing the stability of a homogeneous distribution of particles for a general potential. For the Hookean potential we obtain a precise condition for the phase transition, and, using the central manifold reduction, we characterize the type of bifurcation at the instability onset.