Flocking with random non-reciprocal interactions

TL;DR

This work investigates flocking in Vicsek-like active matter where inter-particle interactions are random and non-reciprocal across many species. By combining dynamical mean-field theory for the infinite-range limit with finite-range simulations and finite-size scaling, it shows that global flocking can survive random NR couplings, while finite-size systems can host chiral or oscillatory states and finite-range systems can support coherently moving cliques. A critical exceptional point (CEP) marks the onset of chiral motion in the infinite-range limit, and phase diagrams depend on the NR parameter $\lambda$, alignment bias $J_0$, range $R$, and scaling $\alpha$. These findings offer a framework for multispecies flocking with nonrandom, complex NR interactions and suggest ways to design mobile, shape-shifting active materials using programmable non-reciprocal couplings.

Abstract

Flocking is ubiquitous in nature and emerges due to short- or long-range alignment interactions among self-propelled agents. Two unfriendly species that antialign or even interact nonreciprocally show more complex collective phenomena, ranging from parallel and antiparallel flocking over run-and-chase behavior to chiral phases. Whether flocking or any of these collective phenomena can survive in the presence of a large number of species with random nonreciprocal interactions remained elusive so far. As a first step here, the extreme case of a Vicsek-like model with fully random nonreciprocal interactions between the individual particles is considered. For infinite-range interaction, as soon as the alignment bias is of the same order as the random interactions, the ordered flocking phase occurs, but deep within this phase, the random nonreciprocal interactions can still support global chiral and oscillating states in which the collective movement direction rotates or oscillates slowly. For short-range interactions, moreover, even without alignment bias self-organized cliques emerge, in which medium-size clusters of particles that have predominantly aligning interactions meet accidentally and stay together for macroscopic times. These results may serve as a starting point for the study of multispecies flocking models with nonrandom but complex nonreciprocal interspecies interactions.

Figures (8)

• Figure 1: (a) Infinite range phase diagram for $\alpha=1$ and $\lambda=-0.5$(circle), $0$(square), and $0.5$(cross). Inset: Log-log plot of the phase diagram. Black dashed line indicates $J_0\sim N^{-1/2}$. (b) Order parameter $m(t)$ (\ref{['eq3']}) for different $J_0$ at $\lambda=0$ obtained by solving the DMFT eq.(\ref{['eq3']}). The critical point is $J_{0,c}\approx1.1$. Inset: Correleation function $C(t)$ for the same values of $J_0$ as in main figure. (c) $J_0$-$\lambda$ phase diagram for $\alpha=1/2$. Green: disordered phase, purple: flocking with relaxational dynamics(c.f. Fig. \ref{['fig:fig2']}(a)); blue: flocking region with non-relaxational dynamics (c.f. Fig. \ref{['fig:fig2']}(b-d)), c.f. (d). Note that this region shrinks with increasing $N$ (see text). Orange: static flocking. The blue line indicates the disorder-flocking phase boundary supp, the red line represents the spin glass region at $\lambda=1$. The red triangle locates the critical point from DMFT at $\lambda=0$. (d) Fraction of disorder-realizations displaying non-relaxational dynamics $p_{\text{nr}}$ as a function of $J_0$ (at $\lambda=0$). The black dashed line indicates the critical point from DMFT. $p_{\text{nr}}>0.15$ defines the blue region in (d).
• Figure 2: Emergence of non-relaxational dynamics for finite $N$ with infinite-range interactions. (a-d) Time evolution of magnetization phase $\psi(t)$ for a single realization of interaction matrix $[\mathbf J]_{ij} = J_{ij}/\sqrt N$ with $N=1024$ and $\lambda=0$: (a) flocking with diffusive $\psi(t)$, (b) vibrating oscillation, (c) oscillation, and (d) rotation(see 'movie 2'). Each line in (a-d) represents trajectories of $\psi(t)$ starting from different initial conditions. (e) Eigenvalue spectrum of the same interaction matrix $\mathbf J$ as in (a-d) and the corresponding linearized matrix $\mathbf J^\ast$, for $J_0=0$. The red dashed line indicates the circular law for $\lambda=0$Sommers1988spectrum and $z_1$ depicts the leading eigenvalue of $\mathbf J^\ast$, which locates the critical exceptional point(CEP): (f) Decreasing $J_0>z_1$ shifts the spectrum to the right, and $z_1$ hits zero at the CEP ($z_1\approx3.497$) and (g) overlap between eigenvector $\mathbf v_1$ associated with $z_1$ and the Goldstone mode $\mathbf v_0$ reaches 1, indicating two eigenvectors coalesce. (h,i) Steady-state angle configuration near the CEP. (h) For $J_0=3.48<z_1$, the outlier component $\theta_{\text{out}}$ is seperated from the others with constant angle difference $\Delta\theta = \theta_{\text{out}}-\psi$ and drives global chiral motion with constant angular velocity $\Omega$. (i) For $J_0=3.50>z_1$, all angle are perfectly aligned. (j) Probability distribution of the time averaged angular velocity $P(\Omega)$ in the steady state for fixed $J_0=2.2$ and varying $N$.
• Figure 3: Finite range interactions ($R=1$, $\lambda=0$): (a) Phase diagram for $\alpha=1$. Region between the disordered and the single-cluster phase indicates long range order $m>0$ without condensation. Inset: Configuration snapshot ($L=1024$) of growing clusters in the single cluster region at the point marked with green star in (a). (b,c) Time evolution of magnetization $m(t)$ and the relative size of the largest cluster $n_c(t)$ at the red and green point marked in (a). (d) As in (a) for $\alpha=1/2$. Inset: Configuraton snapshot in the ordered region at the orange triangle in (c). (e) Binder cumulant $G=1-\langle m^4\rangle/(3\langle m^2\rangle^2)$ at the transition ($\rho=2$).
• Figure 4: Clique formation for finite range interactions ($R=1$, $\lambda=0$, $\alpha=1/2$, $\rho=0.25$). (a) Schematic diagram of the clique detecting algorithm Buchin2015trajectory. Each set $\{\cdots\}$ represents connected components and arrow depicts particle flows between connected components in different time. The colored regions indicate three potential cliques: $\{2,3,7\}$(blue), $\{0,1,4\}$(red), and $\{6,8\}$(green). (b) Time evolution of the number of interacting neighbors of a specific particle and the size of the largest clique containing it. (c) Snapshot of a particle configuration for $J_0=0$. Cliques are represented with colors according to its size. Inset: Zoomed-in snapshot of the region indicated by the gray square. (d) $P(n,\tau)$ at $J_0=0$ averaged over many disorder realization. Inset: $\langle J_+(n,\tau)\rangle$ at the same parameters as (d). (e-f) $P(n)$ for different $J_0$ values and its characteristic cluster size $n^\ast$. (g-h) $P(\tau)$ for different $J_0$ values and its characteristic lifetime $\tau^\ast$.
• Figure S1: Finite-size scaling analysis for $\lambda=0$ case. (a) Order parameter susceptibility $\chi$ for $\alpha=1$. (b-c) Scaling collapse of Binder cumulant $G$ for (b) $\alpha=1$ with rescaled $J_0\to J_0\sqrt N$, and (c) $\alpha=1/2$. $J_{0,c}=1.1$ is the DMFT critical point and $\nu=5/2$ is estimated. (d) Scaling collapse of magnetization $m$ for $\alpha=1/2$ with $\beta=1/2$.