Kinetic theory of pattern formation in a generalized multi-species Vicsek model

TL;DR

This work develops a Smoluchowski-based kinetic theory for multi-species self-propelled particles with (anti)alignment, linking microscopic dynamics to macroscopic pattern formation. By performing a Fourier-mode analysis of the Fokker-Planck equation, the authors derive an eigenvalue problem that predicts both disordered-to-ordered transitions and finite-wavelength instabilities that generate traveling stripe patterns. The theory quantitatively matches particle simulations in two-species and multi-species cyclic systems, with a characteristic stripe wavelength around λ ≈ 1.23 and distinct parity-driven stripe geometries for odd versus even numbers of species. Overall, the framework provides a general, extensible approach to understanding pattern formation in multi-species active matter and offers design principles for targeted self-organized architectures.

Abstract

The theoretical understanding of pattern formation in active systems remains a central problem of interest. Heterogeneous flocks made up of multiple species can exhibit a remarkable diversity of collective states that cannot be obtained from single-species models. In this paper, we derive a kinetic theory for multi-species systems of self-propelled particles with (anti-)alignment interactions. We summarize the numerical results for the binary system before employing linear stability analysis on the coarse-grained system. We find good agreement between theoretical predictions and particle simulations, and our kinetic theory is able to capture the correct lengthscale in the emergent coexistence phases through a Turing-Hopf instability. Extending the kinetic framework to multi-species systems with cyclic alignment interactions, we recover precisely the same emergent ordering as corresponding simulations of the microscopic model. More generally, our kinetic theory provides an extensible framework for analyzing pattern formation and collective order in multi-species active matter systems.

Figures (10)

• Figure 1: Phase diagram from particle simulations of the microscopic model with additive sine interactions. We solve Eq. (\ref{['eq:langevin']}) with the following parameters: $N=20000$, $\rho=100$, and $D_r=0.2$. Phases were classified numerically, with data averaged over 10 independent realizations for each data point. See details of the order parameters and classification in lardet2025 and Appendix \ref{['sec:appendix_order_params']}. Here, we list the phases observed and the parameters used to obtain the associated example snapshots: (a) Nematic stripes, demixed (dark-green plus), $J_{AA}=-0.12$, $J_{AB}=0.06$; (b) Flocking stripes (dark-blue star), $J_{AA}=-0.08$, $J_{AB}=0.08$; (c) Parallel flocking (purple upward triangle), $J_{AA}=-0.04$, $J_{AB}=0.12$; (d) Nematic stripes, mixed (light-green hexagon), $J_{AA}=-0.12$, $J_{AB}=-0.06$; (e) Antiparallel flocking stripes (light-blue diamond), $J_{AA}=-0.08$, $J_{AB}=-0.12$; (f) Antiparallel flocking (pink downward triangle), $J_{AA}=-0.02$, $J_{AB}=-0.12$. We also observed a disordered state (orange circle), independent flocking (pink square), and independent nematic ordering (khaki cross). Phase boundaries were added manually as a visual guide.
• Figure 2: Linear stability analysis results. (a) Phase diagram of linear stability around the disordered homogeneous solution. Yellow indicates the disordered state is stable [$\operatorname{Re}(\mu(k))\leq 0 \ \forall k$]; purple indicates a flocking instability [$\max\{\operatorname{Re}(\mu(k))\}>0$ at $k_{c}=0$]; blue is a finite wavelength instability [$\max\{\operatorname{Re}(\mu(k))\}>0$ at $k_{c}>0$]. (b)--(d) Growth rates from the disordered state linear stability analysis alongside snapshots from numerical simulations with the same parameters. (b) Disordered stable ($J_{AA}=-0.05$, $J_{AB}=0.02$). (c) Flocking instability ($J_{AA}=-0.04$, $J_{AB}=0.10$). (d) Finite wavelength instability ($J_{AA}=-0.10$, $J_{AB}=0.10$) with $k_c\approx5.1$ indicated with a vertical gray dashed line. There is also a zoomed in snapshot showing the stripe wavelength $\lambda\approx1.23\sigma_I$ in the particle simulations. (e) Phase diagram of linear stability around the ordered homogeneous solution (both parallel and perpendicular to direction of order). Purple indicates the parallel or antiparallel flocking state is stable [$\operatorname{Re}(\mu(k))\leq0 \ \forall k$]; blue is a short-wavelength instability [$\max \operatorname{Re}(\mu(k))>0$ at $k_{c}>2$]; gray indicates no uniform ordered solution from the Fokker-Planck equation exists to perturb around. (f)--(i) Growth rates from the ordered state perturbation parallel to direction of travel with $\mathbf{k}=(k,0)$ alongside snapshots from numerical simulations with the same parameters. (f) Parallel flocking state is stable ($J_{AA}=-0.02$, $J_{AB}=0.08$). (g) Antiparallel flocking state is stable ($J_{AA}=-0.02$, $J_{AB}=-0.08$). (h) Finite wavelength instability from parallel ordered state ($J_{AA}=-0.10$, $J_{AB}=0.11$). (i) Finite wavelength instability from the antiparallel ordered state ($J_{AA}=-0.10$, $J_{AB}=-0.11$). Parameters for linear stability analysis: $N_c=50$, $D_r=0.2$, $v_0=1$, $\sigma=1$, $\rho=100$. The microscopic simulations were performed with $N=2\times 10^4$ particles.
• Figure 3: Comparison of linear stability analysis and particle-based simulations. The background colors represent the results of the linear stability analysis combining information from the disordered and ordered state perturbations. Yellow indicates that the homogeneous disordered state is linearly stable to perturbations. Purple indicates that the homogeneous parallel flocking or antiparallel flocking state is stable to perturbations from any direction. Blue indicates a finite wavelength instability in any direction, either from the homogeneous disordered or ordered state. The numerical phase classifications are superimposed as scatter points, obtained from particle simulations of the microscopic equations \ref{['eq:langevin']}. See the caption of Fig. \ref{['fig:phase_diagram_AS_microscopic']} for a description of the phase symbols and lardet2025 for details of the order parameters used for classification. The parameters kept constant in each phase diagram are: (a) $D_r=0.2$, $\rho=100$; (b) $J_{AA}=-0.08$, $\rho=100$; (c) $J_{AA}=-0.08$, $D_r=0.2$. For the linear stability analysis $N_c=50$ and for the particle simulations $N=2\times10^4$.
• Figure 4: Analysis of the flocking stripes phase with $m=5$ (left) and $m=6$ (right) species. (a), (f) Schematics of the cyclic interactions according to Eq. \ref{['eq:cyclic_couplings']}. The black arrows represent the interspecies alignment and the inverted chevron arrows represent the intraspecies anti-alignment. In (a) the gray arrows denote the ordering of the stripes in simulations. In (b) the blue and red triangles denote the grouping of species in the stripes. (b), (g) Snapshots of simulations of the microscopic equations Eq. \ref{['eq:langevin']} with parameters $N=2\times10^4$, $\rho=100$, $D_r=0.2$, and $J=0.05m$. (c), (h) Growth rates obtained from linear stability analysis of the disordered state, using the same parameters used in the simulations for (b), (g). Both show a finite wavelength instability with $k_c\approx5$. The growth rate associated with the largest eigenvalue is highlighted in red with the largest eigenvalue given by $\mu^*(k_c)$. (d), (i) Time-averaged density profiles, projected along the direction of motion, are shown for simulations of the microscopic equations with a mean-sine alignment in the flocking stripes phase ($\rho=100$, $D_r=0.2$, $J=10(m+1)$). (e), (j) Coefficients of the eigenvector associated with $\mu^*(k_c)$ in the complex plane. The largest magnitude coefficients are colored according to the species they correspond to.
• Figure 5: Wavelength of flocking stripes in particle-based simulations of the microscopic equations. (a) Snapshot of a system with flocking stripes and (b) its structure factor. The location of the peak $\mathbf{k}^*$ is highlighted in red. (c) Histogram of the same-species stripe wavelength $\lambda=2/|\mathbf{k}^*|$ for systems classified as parallel flocking stripes in Fig. \ref{['fig:phase_diagram_AS_microscopic']}. The mean wavelength of $\lambda=1.22$ is shown as a vertical dashed black line.