Flocking in weakly nonreciprocal mixtures

Abstract

We show that weakly nonreciprocal alignment leads to large-scale structure formation in flocking mixtures. By combining numerical simulations of a binary Vicsek model and the analysis of coarse-grained continuum equations, we demonstrate that nonreciprocity destabilizes the ordered phase formed by mutually aligning or anti-aligning species in a large part of the phase diagram. For aligning populations, this instability results in one species condensing in a single band that travels within a homogeneous liquid of the other species. When interactions are anti-aligning, both species self-assemble into polar clusters with large-scale chaotic dynamics. In both cases, the emergence of structures is accompanied by the demixing of the two species, despite the absence of repulsive interactions. Our theoretical analysis allows us to elucidate the origin of the instability, and show that it is generic to nonreciprocal flocks.

Figures (4)

• Figure 1: Phase behaviour of weakly nonreciprocal flocks. (a,b) Stylized finite-size phase diagrams in the $(\Delta\chi,\eta)$ plane for aligning (a) and anti-aligning (b) intraspecies interactions. (c)-(h) Representative snapshots of the observed phases. Particles are colour-coded according to their species (orientation) in the left (right) panel (caption below (d)), while the corresponding points in (a,b) are indicated with solid symbols. Additional details on the construction of (a,b) and simulation parameters are provided in supplement.
• Figure 2: Characterisation of the single band phase. (a,b) $\rho^{\textsl{a}}$(a) and $\rho^{\textsl{b}}$(b) profiles averaged along the band and over time for different system sizes. Inset of (b): the component of the local polarity profiles along the global order. (c) Density histograms at $L = 1024$. The band densities $\rho^{\textsl{a},\textsl{b}}_{*}$ (vertical dashed lines) are evaluated from the position of the relevant peaks. (d,e) Band width (d) and density (e) as functions of system size. Symbols show the numerical data and solid lines indicate fits (see text). (f,g) The densities in the band (f) and its normalized width (g) as functions of $\Delta\chi$ for various system sizes. The horizontal grey line in (f) indicates $\rho_0$. In all panels, $\eta =0.3$ and for (a-e) $\Delta\chi = 0.4$.
• Figure 3: Phase diagrams in the $(\Delta\chi, \eta)$ plane obtained from the linear stability of the stationary homogeneous solutions of Eqs. \ref{['hydro']} in the flocking (a) and anti-flocking (b) regimes. Stable disordered, flocking and anti-flocking regions appear in pink, blue and gray, respectively. The other regions indicate an instability, and are colored according to the orientation of the most unstable wave vector with respect to the direction of order. The solid cyan lines mark the onset of order, while the pink lines enclose regions where the matrix $\mathbb{M}_{\rm h}(\bm q)$ in Eq. \ref{['mat_ensl']} predicts an instability. Inset of (b): Growth rate of the flocking (blue) and anti-flocking (red) instabilities as functions of the wave number $q = |\bm q|$ along the most unstable direction.
• Figure A1: Instability of the laning pattern for mutually anti-aligning species. (a) Exemplary time series of the global polar order $|\bm \Pi^{\textsl{a}}|$ in the laning pattern for two system sizes. (b) The corresponding distributions. Parameters: $\eta = 0.30$ and $\Delta\chi = 0.35$.