Crystal formation in systems of pseudo-forced swarmalators

TL;DR

The paper investigates how a center-directed pseudo-force alters swarmalator dynamics by densifying assemblies and promoting crystal-like order. By introducing a Fourier-transform–based crystal-order parameter and an analytic three-body solution, it shows that increasing the pseudo-force strength lambda contracts steady-state structures and shifts the system from active or splintered phase waves to static antisynchronization. The work provides a quantitative framework to measure spatial order in swarmalator systems and reveals scaling trends, such as the inverse square-root dependence of characteristic distances on lambda at strong attraction. It also points to future directions including alternative forcing laws and multi-center forcing to explore richer crystal phases.

Abstract

Swarmalators are active agents that move in position space and exhibit internal degrees of freedom. Due to interactions of their positions and phases of oscillation, they show on the one hand swarming, similar to the effect of flocking of birds. In addition, they exhibit synchronization behavior, analogous to what has been observed in fireflies. Previous works studied scenarios in which the phases are forced externally. Here, we consider a pseudo-force that acts on the positions of the swarmalators. Due to the resulting attraction towards the center of position space, transitions from the splintered and active phase-wave state to the static antisynchronized state are found. To quantify the crystal order of swarmalators, we introduce an order parameter that is based on the Fourier transform of their positions.

Figures (7)

• Figure 1: Examples of StS and StAS steady states defined in Table \ref{['tab:standard_phases']} at time $t=10^4$ for $\lambda=0,100$ and corresponding Fourier transforms $\mathcal{F}$ of the positions, see \ref{['eq:FFT_2D']}. The larger $\lambda$, the denser the group of swarmalators (note the change of scale). (a) StS for $(K,J)=(1,0.1)$. (b) StAS for $(K,J)=(-1,0.1)$. The black squares in the plots of the Fourier transforms correspond to the excluded region when computing $\mathcal{F}_\text{max}$, see \ref{['eq:Fmax']}.
• Figure 2: Various quantities characterizing steady states of $N=300$ swarmalators averaged over $50$ realizations. (a) The smallest distances $r_\text{min}$ of any swarmalator from the origin. (b) The largest distances $r_\text{max}$ of any swarmalator from the origin. The black dashed curve corresponds to $1/\sqrt{\lambda+1}$, see end of \ref{['sec:analytics']}. (c) Order parameter $S$ of phase-position correlation, see \ref{['eq:S_W_ord_par']}. (d) Fourier-transform order parameter $\mathcal{F}_\text{max}$ quantifying crystal order, see \ref{['eq:Fmax']}. (e) Number of groups $n_g$ in SpPW class, see Appendix \ref{['sec:number_of_groups']} for a detailed description of the method. The shaded region corresponds to one standard deviation around the mean and the gray horizontal line indicates $n_g=1$. The $(K,J)$ parameters for each class of steady states are listed in \ref{['tab:standard_phases']}.
• Figure 3: Examples of StPW steady states for $(K,J)=(0,1)$ at time $t=2000$ for different values of $\lambda$ and corresponding Fourier transforms of the position, see \ref{['eq:FFT_2D']}. The larger $\lambda$, the denser the annulus of swarmalators and the more crystal-like its structure. The black squares in the plots of the Fourier transforms correspond to the excluded region when computing $\mathcal{F}_\text{max}$, see \ref{['eq:Fmax']}.
• Figure 4: Examples of SpPW and ActPW steady states at time $t=500$ for different values of $\lambda$ and corresponding phase-position correlation, see \ref{['eq:S_W_ord_par']}. The larger $\lambda$, the smaller the number of groups in the SpPW phase and the more crystal-like the structure in the ActPW phase. (a) SpPW for $(K,J)=(-0.1,1)$. (b) ActPW for $(K,J)=(-0.75,1)$.
• Figure 5: Histogram of phase difference $|\theta_{ij}|$ ($101$ bins) between swarmalators on the same triangular unit cell (triangulation) for each value of $\lambda$ taking into account $50$ realizations. (a) SpPW for $(K,J)=(-0.1,1)$. The dotted curve corresponds to $|\theta_{ij}| = 2\pi/n_g$, where $n_g$ is the number of groups, see \ref{['fig:crystal_order_parameters']}(e). (b) ActPW for $(K,J)=(-0.75,1)$. In both panels, the solid curve corresponds to the mean absolute phase difference.