Anti-aligning Self-propelled Model of Two Species: Emergence of Self-organized Heterogeneous Aligned and Clustered Order

Abstract

Self-propelled particles with anti-aligning interactions generally do not form a polar order. However, in this Letter, we show that when multiple types of such particles coexist and interact through aligning interactions between different species, a global polar order can emerge through the formation of elongated clusters with alternating domains of each species. By developing a mean-field theory, we reveal the conditions for cluster formation and characterize the resulting patterns. Our findings highlight the critical role of inter-species interactions in the emergence of complex ordered states.

Figures (10)

• Figure 1: Behavior of the two-species AAVM at $c=1.0$. (a) Snapshots at early, middle, and late stages (from top to bottom). Type A and B particles are represented by red and blue dots, respectively. The black dashed circle indicates the interaction region of radius $R$. The black arrow indicates the forward direction of the cluster. (b) The mean squared displacement (MSD) of a particle in the steady state. Green, blue, and dashed black lines indicate the MSDs of the two-species AAVM, single-species AAVM, and Brownian motion, respectively. (c, d) Mean particle density of a single particle within the interaction radius $R$. Color maps indicate the particle density of (c) the same-type and (d) different-type particles in the steady state. The coordinates were chosen such that the center particle moves in the positive direction of the abscissa. The density is normalized by the average density of the particles within the distance $R$ (the number of type A or B particles within the interacting region divided by the area $\pi R^2$). The regions with higher and lower densities than the averaged density are represented by yellow- and black-color scales, respectively.
• Figure 2: Polar order parameter dependence on the parameter sets of (a) $(c, R)$ and (b) $(c,\tau)$. Color map indicates the polar order parameter $\phi$. The black dashed line in (a) indicates the analytically calculated transition point $c = 0.69$ (see the main text). We used $L=64$ as the system size for calculating the order parameter. The other parameters were fixed to the values described in the main text. (c) Snapshots of the simulations using the parameters indicated by the symbols in (b): $(c,\tau) = (0.9, 0.5)$ (red triangle), $(0.9, 2.81\times 10^{-2})$ (green square), and $(0.9, 1.58\times 10^{-3})$ (blue circle). (d) Number fluctuation measured using the time-averaged particle number $\langle N \rangle$ and its standard deviation $\Delta N = \sqrt{\left(N-\langle N\rangle \right)^2}$ within a given observation region. $\Delta N$ scales asymptotically as $\langle N \rangle^{\alpha}$. The same simulation data as in (b) and (c) were used.
• Figure 3: Mean-field analysis of the interaction energy. (a) Distribution of particles assumed by idealizing the numerically obtained particle density (Fig. \ref{['fig:AAVM']} (c) and (d)). Yellow and gray areas represent the regions where the same-type and different-type particles exist in high concentrations, respectively. (b) Interaction energy normalized by the area within the interaction radius, $E(\ell)/R^2$, calculated through the mean-field analysis. We calculated $E(\ell)/R^2$ by setting $c$ to $1.0$ (blue) and $0.69$ (green). The red dashed line corresponds to $\ell/R = 2/\sqrt{10}$ at which $E(\ell)$ is the minimum irrespective of $c$.
• Figure S1: Characteristics of the one-species AAVM. (a) Typical snapshot of the one-species AAVM. Black circles indicate interaction regions of radius, $R$, for a certain particle. (b,c) Comparison of the one-species AAVM and the persistent random walk model (PRWM). (b) Radial distribution function, $g(r)$. (c) Mean square displacement (MSD). The blue lines are for the one-species AAVM. The solid and dashed black lines are for PRWM with $\eta =0.1$ and $\eta =1.0$, respectively.
• Figure S2: Cluster formation in different size of systems. (a–c) Simulation snapshots at $t = 100$, $500$, $1,000$, $5,000$, $7,500$, for different total particle numbers. (a) $N_T=1,600$, $L=40$, (b) $N_T=10,000$, $L=100$, (c) $N_T=40,000$, $L=200$. The particle density is fixed at $N_T/L^2=1.0$, with parameters $R=3.0$, $c=0.9,$ and $\tau = 2.81\times 10^{-2}$. (d) Time series of the local density fluctuation $\Delta\rho$ for $N_T = 1,600$, $10,000$, $40,000$. Data from $11$ independent runs are shown for each total particle number. The trajectories corresponding to the simulations in panels (a–c) are highlighted as thick black lines. The bottom panels show the time series of global polar order parameter, $\phi(t)$.