Collective dynamics of higher-order Vicsek model emerging from local conformity interactions

TL;DR

The paper derives a Vicsek-type model with higher-order interactions arising from local conformity in a diluted active-matter framework and analyzes its collective dynamics. By starting from a weighted DADAM model and obtaining both two-body and three-body alignment terms, the authors show that higher-order interactions produce a novel bidirectional ordered phase and yield both continuous and discontinuous order–disorder transitions, signaling a universality class distinct from standard pairwise Vicsek-type models. They develop a global-coupling (mean-field) theory akin to Kuramoto-type analysis and complement it with numerical simulations to map phase behavior under deterministic and stochastic dynamics, highlighting region(s) of multistability. The findings demonstrate that local conformity can naturally induce higher-order coupling terms that qualitatively alter swarming behavior and phase transitions, providing a pathway toward refined hydrodynamic descriptions and broader applicability in active-matter systems.

Abstract

We study a system of self-propelled particles whose alignment with neighbors depends on the degree of local alignment. We show that such a local conformity interaction naturally yields a Vicsek-type model with pairwise and three-body interactions. Through numerical and approximate theoretical investigation of its deterministic and stochastic collective dynamics, we identify a novel bidirectionally ordered phase in which the particles move in opposite directions. Moreover, both continuous and discontinuous order-disorder transitions are observed, suggesting that the system belongs to a different universality class from previous models.

Figures (3)

• Figure 1: (a) Schematic illustration of the local conformity weight $w_{jk}$, which quantifies the influence of particle $k$ on particle $j$ based on the local alignment of particles around $j$. (a-1) High $w_{jk}$: particle $k$ is aligned with the local majority. (a-2) Intermediate $w_{jk}$: no dominant orientation exists locally. (a-3) Low $w_{jk}$: particle $k$ is oriented opposite to the local majority. Snapshots of the deterministic dynamics ($\eta=0$) for system size $L=32$: (b-1) homogeneous disordered phase at $\alpha=-1$, $\beta=1$; (b-2) nematic disordered phase at $\alpha=-1$, $\beta=-1$; (b-3) fully ordered phase at $\alpha=1$, $\beta=-1$; (b-4) bidirectional ordered phase at $\alpha=0$, $\beta=1$. (c-1) Probability of reaching bidirectional order from random initial conditions. (c-2) Partial phase diagram of the asymptotic state reached from random initial conditions: blue (disorder), yellow (bidirectional order), pink (full order), white (partial order). (c-3) Theoretical phase diagram in the global-coupling limit $r_0\to\infty$ (same color code); shaded regions indicate multistability.
• Figure 2: Global polar order parameter $\langle R\rangle_t$ averaged over time for different pairs $(\alpha,\beta)$ as the noise intensity $\eta$ is increased (blue dots) or decreased (red crosses), for $L=32$.
• Figure 3: Phase diagram of the higher-order Vicsek model, Eqs. \ref{['eq.vicpos']} and \ref{['eq.angthreebody']}, for fixed $\beta=1$ and $L=32$. The color scale shows the time-averaged polar order parameter $\langle R\rangle_t$ when the system is initialized with random angles (a) or nearly identical angles (b). Crosses indicate bistability between ordered and disordered phases. Blue and red lines mark the theoretical stability boundaries of disorder and order, respectively, in the global-coupling limit.