Collective behavior in the nonreciprocal multi-species Vicsek model

TL;DR

The paper investigates collective behavior in a nonreciprocal, permutation-symmetric $Q$-species Vicsek model by introducing a constant phase shift $α$ in inter-species alignment. It develops a Boltzmann equation and a hydrodynamic theory to analyze continuum behavior, and maps a four-phase diagram featuring a chiral quasi-long-range-ordered phase, a species-separation phase with vortex cells, and a coexistence regime. Nonreciprocity induces either counter-clockwise or clockwise chirality, with small $α$ yielding XY-like QLRO and a BKT-like transition to disorder, while large $α$ drives spontaneous species separation via an unstable antisymmetric polarization mode. A coexistence phase emerges at intermediate $α$, reflecting complex coupling between density and chirality fields and signaling the need for a coupled-field theoretical description. Overall, the work provides a minimal, symmetry-preserving framework for nonreciprocal multi-species active matter and lays groundwork for extensions to study odd viscosity, species-dependent couplings, and driven pattern formation.

Abstract

We investigate collective behavior in a $Q$-species Vicsek model with a nonreciprocal velocity alignment interaction. This system is characterized by a constant phase shift $α$ in the inter-species velocity alignment rule. While the phase shift renders the interaction nonreciprocal, the system is globally invariant under any permutations of particle species, possessing Potts symmetry. The combination of Potts symmetry and nonreciprocity gives rise to a rich phase diagram. The nonreciprocal phase shift generates either counter-clockwise or clockwise chirality. Potts symmetry can be broken spontaneously. Consequently, the system exhibits four distinct phases: A species-mixed chiral phase where particles perform counter-clockwise chiral motion with quasi-long-range order, a species separation phase where Potts symmetry is broken and species-separated particles form vortex cells with clockwise chirality, a coexistence phase, and a disordered phase. We derive a Boltzmann equation and a hydrodynamic equation describing the system in the continuum limit, and present analytic arguments for the emergence of chirality and species separation.

Figures (14)

• Figure 1: Mean-field phase diagram from the linear stability analysis at $Q=3$ in the $\alpha-\kappa$ plane with fixed $\eta=0.3$ in (a) and in the $\alpha-\eta$ plane with fixed $\kappa=0.1$ in (b). The chiral angular frequency $\Omega$ in the in-phase and out-of-phase chiral states are color-coded according to the bar chart.
• Figure 2: Color plot of the order parameters (a) $m_{\rm s}$, (b) $e_{\rm s}$, (c) $\gamma_{\rm s}$, and (d) the bimodality coefficient $\beta_{\rho}$ for the three-species system of size $L=128$ with $\rho_0=2$. Phase boundaries of a chiral phase ($\circ$), a species separation phase ($\square$), and a coexistence phase ($\diamond$) are drawn.
• Figure 3: Snapshots of a particle configuration in (a) and a polarization field in (b) in the QLRO chiral phase. In (a), each dot representing a particle is color-coded according to particle species. In (b), the phase angle of a polarization field is color-coded according to the chart in the inset. Parameters: $Q=3$, $L=128$, $\rho_0=2$, $\alpha = 0.3\pi$, and $\eta=0.4$.
• Figure 4: Snapshots of a density field in (a) and a polarization field in (b) computed with the Boltzmann equation \ref{['eq:BoltzmannVector']} for the three-species system SM. The Boltzmann equation, truncated within $|k|\leq 4$, is integrated numerically using the pseudo-spectral method fornberg1998practicalboyd2001chebyshev. For numerical convergence, the Boltzmann equation is regularized with a diffusion term mahault2018outstanding. The snapshots are taken when the system, starting from a random initial configuration, reaches a steady state. The density field in (a) is color-coded using local densities $(\rho^1, \rho^2, \rho^3)$ of the three species as an RGB code. These snapshots reveal that the three species particles are mixed with $\rho^1=\rho^2=\rho^3$ and that spatial fluctuations persist. Parameters: $L_x=L_y=64$, $\eta = 0.3$, $\alpha/\pi=0.1$ and $\kappa=0.053$.
• Figure 5: Snapshots of a particle configuration in (a) and a polarization field in (b) in the SS phase. The same color scheme is used as in Fig. \ref{['fig:snapshot_chiral']}. Snapshots for different values of $Q$ are found in Fig. \ref{['fig:SS245']}. Parameters: $Q=3$, $L=128$, $\rho_0=2$, $\alpha=0.8\pi$, and $\eta=0.35$.