Kinetic equations and self-organized band formations

TL;DR

The paper addresses how simple local alignment rules in Vicsek-type models can lead to self-organized band patterns and develops a kinetic framework to study this phenomenon. It derives kinetic equations for the Vicsek and Degond-Frouvelle-Liu dynamics, analyzes homogeneous phase behavior via a gradient-flow entropy structure, and designs a structure-preserving splitting scheme with adaptive collision timesteps that preserves mass, positivity, and entropy dissipation. In homogeneous settings the equilibria are von Mises distributions and the free-energy decreases over time, while in nonuniform settings the Degond-Frouvelle-Liu dynamics produce stable bands, contrasting with the original Vicsek model in the same regime. The work provides a robust numerical tool for exploring long-time band dynamics in active matter, clarifying the interplay between transport and alignment and enabling kinetic-level insights beyond particle simulations.

Abstract

Self-organization is an ubiquitous phenomenon in nature which can be observed in a variety of different contexts and scales, with examples ranging from fish schools, swarms of birds or locusts, to flocks of bacteria. The observation of such global patterns can often be reproduced in models based on simple interactions between neighboring particles. In this paper we focus on two particular interaction dynamics closely related to the one described in the seminal paper of Vicsek and collaborators. After reviewing the current state of the art in the subject, we study a numerical scheme for the kinetic equation associated with the Vicsek models which has the specificity of reproducing many physical properties of the continuous models, like the preservation of energy and positivity and the diminution of an entropy functional. We describe a stable pattern of bands emerging in the dynamics proposed by Degond-Frouvelle-Liu dynamics and give some insights about their formation.

Figures (18)

• Figure 1: Illustration of the simulation of the Vicsek model \ref{['eq:micro']} at two different time. We observe the formation of a vertical band. See table \ref{['tab:param_micro']} for the parameters used.
• Figure 2: Density $\rho$ and average velocity $u$ in the $x$-direction at $t=52$. Where the density $\rho$ is larger, the speed $u$ increases.
• Figure 3: Entropy $\int M \log M$ for $M$ von Mises distribution as a function of the length $|\int\omega M|$. The curve increases quadratically near $|{\bf j}|=0$.
• Figure 4: Left: for low value of the diffusion coefficient $\sigma$, the minimizers for both free energies are von Mises distribution. Right: when the diffusion coefficient $\sigma$ exceeds a certain threshold, the uniform distribution, i.e. ${\bf j}=0$, becomes the minimizer for the DFL dynamics.
• Figure 5: Comparison of the standard and adaptive methods for the collision operator (Vicsek). Parameters are: $\mu=1.0$, $\sigma=0.2$, $\rho=1.0$, $\Delta t=8.458\cdot 10^{-7}$ (standard), $\Delta t=0.1$ (adaptive), $T=1.0$. The initial condition is $f_0(\theta)=\rho\left(1+\frac{1}{5}\underset{k=1}{\overset{5}{\sum}}\cos(p_k\theta)\right)$ where $p_1=1$ and $p_k$ is the prime following $p_{k-1}$.

Theorems & Definitions (7)

• proposition 1
• proof
• lemma 1
• proof
• proposition 2
• proof
• remark 1