Active pattern formation emergent from single-species nonreciprocity

TL;DR

The paper addresses pattern formation in a single-species system with nonreciprocal interactions that break actio and reactio symmetry. It derives Active Model N from microscopic vision-cone forces and torques, producing coupled density and polarization dynamics (with a slaved nematic tensor $\mathbf{Q}$) and explicit coefficients $B_i$ in a nondimensional form. It uncovers self-traveling, PT-symmetry-breaking patterns, notably active branches and the interwoven 'active yarn' that combines micro- and bulk phase separation and travels opposite to the mean polarization $\langle \mathbf{P} \rangle$, and maps a rich phase diagram including a reciprocal limit that recovers standard flocking. These results extend active-matter theory to single-species nonreciprocity and offer design principles for programmable matter in artificial systems.

Abstract

Nonreciprocal interactions violating Newton's third law are common in a plethora of nonequilibrium situations ranging from predator-prey systems to the swarming of birds and effective colloidal interactions under flow. While many recent studies have focused on two species with nonreciprocal coupling, less is examined for the basic single-component system breaking the actio and reactio equality of force within the same species. Here, we systematically derive a field theory for the case of single-species nonreciprocal interactions from the microscopic particle dynamics, leading to a generic continuum model termed Active Model N (N denoting nonreciprocity). We explore the rich dynamics of pattern formation in this nonreciprocal system and the emergence of self-traveling states with persistent variation and flowing of active branched patterns. One particular new characteristic pattern is an interwoven self-knitting "yarn" structure with a unique feature of simultaneous development of micro- and bulk phase separations. The growth dynamics of a "ball-of-wool" active droplet towards these self-knitted yarn or branched states exhibits a crossover between different scaling behaviors. The mechanism underlying this distinct class of active phase separation is attributed to the interplay between nonreciprocity and competition of interparticle forces. Our predictions can be applied to various biological and artificial active matter systems controlled by single-species nonreciprocity.

Figures (8)

• Figure 1: Schematic of nonreciprocal single-species vision-cone model. An agent 1 with orientation $\mathbf{\hat{u}_1}$ sees and interacts with other agents within a vision cone of full opening angle $2\vartheta$. Agent 2 is included in the perception zone of agent 1, such that there is a resulting nonreciprocal force acting on agent 1 which is directed along $\mathbf{r} = \mathbf{r}_2 - \mathbf{r}_1$. There is also a torque turning its orientation to be aligned with the orientation $\mathbf{\hat{u}_2}$ of the second agent. In this example, agent 2 with its own vision cone does not see agent 1 and thus there is no interaction between them that influences the motion of agent 2. Likewise, agent 3 is not visible for agent 1 such that there is no mutual interaction between agents 1 and 3 that influences the motion of either of them.
• Figure 2: (a)--(e) Simulation snapshots for various active patterns that emerge at average particle density $\bar{\rho}=1.5$, including active stripes at $\vartheta=30^\circ$ ((a) and Supplemental Video 1), active branches at $\vartheta=90^\circ$ ((b) and Supplemental Video 2), active yarn at $\vartheta=140^\circ$ ((c) and Supplemental Video 3) and $160^\circ$ ((d) and Supplemental Video 4), and a homogeneous flocking phase with vortex defects of polarization field $\mathbf{P}$ at the reciprocal limit $\vartheta=180^\circ$ ((e) and Supplemental Video 5). In (a)--(d) with patterns induced by nonreciprocity, the spatial profiles of particle density $\rho$ are presented, and the local distribution of vector field $\mathbf{P}$ are indicated as arrows in the insets which are enlarged portions of the boxed regions. The patterns self-travel opposite to the average direction of $\mathbf{P}$ (Supplemental Videos 1-5). In (e) the spatial profile of polarization magnitude $|\mathbf{P}|$ is shown. (f) The phase diagram of $\bar{\rho}$ vs $\vartheta$, with solid curves evaluated from the analytical results of bifurcation analysis and the symbols identified via outcomes of numerical simulations (those giving disordered or homogeneous phases are not shown).
• Figure 3: (a) Visualization of a single particle with orientation $\hat{\mathbf{u}}$ and vision-cone opening angle $2\vartheta$. Via interacting with other nearby particles within its field of vision, it experiences strong short-range repulsive (red) and weak long-range attractive (green) forces, as well as aligning torques (blue). The individual forces and torques (light arrows) result in a net force and a net torque (bold arrows). (b) On a microscopic level, spontaneous motion originates from the net repulsive forces in the bulk. Each particle experiences a net repulsion by the particles in front of it (within its vision cone) but no force from those behind, leading to collective motion in the direction of the bold violet arrow on the right. (c) Visualization of the microscopic mechanism for branch merging. If two branches cross, the particles belonging to different branches start seeing each other and, due to the aligning torques, acquire the same orientation. As a result, they travel within one merged branch.
• Figure 4: Dynamical processes of active pattern formation. Snapshots of density profile at different time stages of system evolution for two sample average densities $\bar{\rho}=0.7$ ((a), (c)) and $\bar{\rho}=1$ ((b), (d)), illustrating the rich dynamics of phase-separated fibers/strands which leads to the persistent variation of active branches ((a), (b) with $\vartheta=110^\circ$, $90^\circ$; Supplemental Videos 6 and 7) or the self-knitting of active yarn ((c), (d) with $\vartheta=130^\circ$; Supplemental Video 8).
• Figure 5: Time variation of active yarn pattern. During the nonreciprocity-induced self migration, the orientation and morphology of active yarn can vary with time, as illustrated in this sample simulation at $\bar{\rho}=1.5$ and $\vartheta=150^\circ$ (see also Supplemental Video 9).