Density dependent diffusion models for the interaction of particle ensembles with boundaries

TL;DR

This paper develops a macroscopic density-dependent diffusion model from microscopic particle interactions, capturing flock behavior and boundary interactions through an advection-diffusion equation with a specialized numerical solution.

Contribution

It introduces a novel macroscopic model with density-dependent diffusion derived from microscopic particle dynamics, including a numerical algorithm for solving the transition phases.

Findings

The model accurately describes particle flock interactions with boundaries and obstacles.

The advection-diffusion equation transitions between hyperbolic and parabolic states during interactions.

Numerical simulations demonstrate the model's effectiveness in various scenarios.

Abstract

The transition from a microscopic model for the movement of many particles to a macroscopic continuum model for a density flow is studied. The microscopic model for the free flow is completely deterministic, described by an interaction potential that leads to a coherent motion where all particles move in the same direction with the same speed known as a flock. Interaction of the flock with boundaries, obstacles and other flocks leads to a temporary destruction of the coherent motion that macroscopically can be modeled through density dependent diffusion. The resulting macroscopic model is an advection-diffusion equation for the particle density whose diffusion coefficient is density dependent. Examples describing i) the interaction of material flow on a conveyor belt with an obstacle that redirects or restricts the material flow and ii) the interaction of flocks (of fish or birds) with boundaries and iii) the scattering of two flocks as they bounce off each other are discussed. In each case, the advection-diffusion equation is strictly hyperbolic before and after the interaction while the interaction phase is described by a parabolic equation. A numerical algorithm to solve the advection-diffusion equation through the transition is presented.