The Vicsek-Kuramoto model in collective dynamics: macroscopic equations and pattern formation

TL;DR

This work develops a Vicsek–Kuramoto model that couples local orientation alignment with angular-velocity relaxation for self-propelled agents. Starting from an individual-based model, the authors derive a mean-field Fokker–Planck equation and perform a hydrodynamic scaling to obtain a closed macroscopic Euler-type system using Generalized Collisional Invariants. The macroscopic system governs density, mean orientation, and average angular velocity, and is complemented by a detailed analysis of equilibria and particular solutions. Numerical simulations at both particle and continuum scales reveal rotating clusters, traveling orientation waves, and global synchronization, clarifying the regime of validity for the macroscopic description and its limitations in capturing all particle-scale phenomena.

Abstract

In this work, we investigate an individual-based model (IBM) for self-propelled agents interacting locally on a plane. Agents are characterized by their position, the angle determining their direction of motion, and their angular velocity. The dynamics combine features of the well-known Vicsek and Kuramoto models, which describe collective dynamics and synchronization, respectively. The evolution of the directions of motion follows a Vicsek model, where agents align their orientations with the mean orientation of their neighbors, subject to some noise. Similarly, the angular velocities relax towards the average angular velocity of the neighboring agents, also subject to noise. From the IBM we derive the corresponding kinetic equation in the limit of a large number of agents and formally obtain the macroscopic equations through a macroscopic (hydrodynamic) limit. Numerical simulations of the IBM reveal a variety of patterns, including rotating clusters, traveling orientation waves, and globally synchronized rotational motion. A qualitative comparison with simulations of the macroscopic system show the ability of the macroscopic model to reproduce some emergent behavior of the IBM.

Figures (4)

• Figure 1: Results of microscopic simulations for varying values of the alignment parameters $k_{\theta}$ and $k_{\omega}$, with all other parameters fixed as reported in Table \ref{['tab:parameters_micro']}. Each simulation involves $15{,}000$ agents interacting within an interaction radius $R = 2$. The columns (from left to right) correspond to increasing values of the angular velocity alignment strength $k_{\omega} = 1, 11, 51, 61, 81$, while the rows (from bottom to top) correspond to increasing values of the directional alignment strength $k_{\theta} = 1, 11, 21, 61, 71$. The colormap, shown on the right hand side of the figure, provides the correspondence between orientation angles and colors. We can distinguish at least three types of patterns: rotating clusters (framed in light blue), traveling waves in orientation (framed in red), synchronised behaviour (framed in purple). Videos are available in the appendix.
• Figure 2: Sequence of snapshots showing the emergence of a traveling wave in orientation for alignment parameters $k_\theta = 21$ and $k_\omega = 81$. The system evolves towards a state with nearly uniform spatial density, while the orientation field propagates with a wave-like motion across the domain. The corresponding video can be found in Appendix \ref{['appendix:videos']} (Video 2).
• Figure 3: Sequence of snapshots illustrating the steady-state dynamics for alignment parameters $k_\theta = k_\omega = 71$. To highlight the synchronised behavior, we display multiple time frames from the same simulation. The particles rotate collectively as a rigid body, with the orientation field evolving in time across the entire domain. This global rotation characterizes the synchronised regime, where the direction of motion changes uniformly for all the particles in the domain, while the distribution of particles remains uniform in space. The corresponding video can be found in Appendix \ref{['appendix:videos']} (Video 3).
• Figure 4: Macroscopic simulation of the Vicsek--Kuramoto system with $k_\theta / \alpha^2 = 8$ starting from random initial data. (a) Snapshot at early time $t = 0.20$. The color scale represents the density $\rho(x,t)$ and it shows the formation of localized rotating clusters. These are transient structures similar to those observed in the microscopic simulations with small $k_\theta$ and $k_\omega$, corresponding to weak orientational and angular velocity alignment. The video of the simulation is included in Appendix \ref{['appendix:videos']} (Video 5) (b) Same configuration as in (a), with the color scale representing the average orientation angle of each cell grid between $[-\pi, \pi]$. Arrows indicate the local mean orientation field $\Omega(x,t)$. The video of the simulation is included in Appendix \ref{['appendix:videos']} (Video 6) (c) Snapshot of Video 5 at time $t = 95.0$. The density $\rho$ becomes nearly uniform in space, and the system reaches a rotating regime driven by a non-zero mean average angular velocity. (d) Snapshot of Video 6 at time t =95, with colorbar indicating the angle. The global coherent rotation of the direction field confirms the emergence of a synchronised state, in qualitative agreement with the behavior observed in the microscopic simulations of Video 4.

Theorems & Definitions (17)

• Lemma 2.1: Expansion for localized interactions
• proof
• Theorem 3.1
• Remark 3.2
• Lemma 3.3
• proof
• Proposition 3.4
• Corollary 3.5: Solutions for constant initial data
• proof : Proof of Prop. \ref{['prop:solutions_constant_density']}
• Remark 3.6