From Global Flocking to Local Clustering: Interplay between Velocity Alignment and Visual Perception of Active Particles

Abstract

Collective behavior in biological systems was first captured by the Vicsek model, in which particles align their velocities in the average direction of neighbors, leading to coherent motion and showing an order-disorder transition. However, in many complex environments, the interactions are non-reciprocal, lacking an action-reaction symmetry. Using framework of the Vicsek model, we implement non-reciprocity by restricting interactions to neighbors located inside a finite vision cone, for a particle by limiting its set of interacting neighbors which fall within a vision-cone, providing a minimal description for cognitive perception. Using detailed numerical simulations, we explore the clustering and flocking behavior due to competition between noise and limited visual perception in the presence of alignment interaction. For low noise, with reduction in the vision angle the system shows transition from a global coherent motion to locally ordered small-sized clusters. This behavior is confirmed through the steady-state distributions of velocity components and their fluctuation relative to the global mean. This is also characterized using a polar order-parameter and a two-point velocity correlation function. Interestingly, at small vision angles, particles exhibit strong short-range correlations within clusters even in the absence of any global coherence. Time-evolution of the related correlation functions illustrate the pathways towards the emergence of such structures. The time dependence of the average cluster size and the length-scale calculated from the two-point velocity correlation show scaling behavior and indicate that the emergence of density field clustering is a consequence of the velocity-field coherence. Any kind of ordering and clustering disappear in the limit of high noise and low vision-angle regime.

Figures (19)

• Figure 1: (a) Schematic representation showing non-reciprocal interaction between any two species, say, $i$ and $j$. Inter-particle forces $\vec{F}_{ij} \ne \vec{F}_{ij}$ accounts for such non-reciprocity. (b) Implementation of non-reciprocity via the vision cone with half-angle $\alpha$ within which $i$-th particle (position $\vec{r}_i$ and velocity $\vec{v}_{i}$) interacts with the neighbors within cut-off distance $r_{int}$ (not to scale). For clarity, interacting particles including the $i$-th one is shown by green color. Other particles (marked with cyan) within $r_{\text{int}}$ which are not within the vision cone and do not contribute to the interaction. (c) Enlarged version to show that the regions of interaction which falls within the vision cone become different for particles $i$ and $j$. Due to limitation in visual perception the interacting neighbors for a particle are always in its heading direction and thus creates anisotropy in interaction.
• Figure 2: Typical steady-state snapshots of the velocity field of the particles for different values of the vision-angle $\alpha$ and noise strength $\eta$, as mentioned. Different colors (shown in the bar) identify the regions of different orientations of the particles. For $\alpha=\pi$, i.e., with full vision angle, snapshots for different values of $\eta$ look similar to the ones for original Vicsek model.
• Figure 3: Heatmap of $\langle v_{a} \rangle$ in $\alpha - \eta$ parameter space. The color bar represents the magnitude of $\langle v_a\rangle$ which varies between $0$ to $1$. The points for simulations are marked by circles.
• Figure 4: Snapshots showing deviation of the velocity field $\delta \vec{v}_i$ with $\eta=0.05$ for $\alpha=\pi/4$ in (a) and $\alpha=\pi$ in (b). For better visualization magnitudes of $|\delta \vec{v}_i|$ are set to $1$. (c)-(e) Plots of normalized distributions $P(|z|)$ versus $|z|$ (in units of $v_0=0.01$) with $z$ corresponding to $v_x$ and $\delta v_x$, respectively, for different choices of $\alpha$ and $\eta$, as mentioned. All data are with $N=2560$ and $L=32$.
• Figure 5: (a) Plots of velocity correlation function $C_v(r)$ versus $r$ in the steady state for different noise strengths $\eta$, as mentioned, for full angle $\alpha=\pi$. (b)-(d) Plots of $C_v(r)$ for different values of $\eta$ as mentioned. In each frame, comparative plots of $C_v(r)$ are shown for different values of $\alpha$. All the presented data are for $N=2560$ and $L=32$.