Zoology of collective patterns modulated by non-reciprocal, long-range interactions

Abstract

We investigate active particles that exhibit distance-independent interactions only restricted by a field of view, which is characterized by an angle β. We show that constraining attractive interactions to a field of view leads to the emergence of a complex pattern that exhibits – depending on the value of β and initial conditions – significantly different topologies and transport properties. We find, in two dimensions, a nematic closed filament in the form of a ring that moves as a chiral active particle, a closed polar filament with one singular topological point that exhibits net polar order and moves ballistically, a structure with two singular topological points that rotates, or an open polar filament that behaves as a persistent random walk. Furthermore, we investigate the process that transforms one structure into another by slowly varying β and observe that the process is non-reversible and presents strong hysteresis. Finally, we find that in three dimensions similar patterns also emerge. The analysis sheds light on the physics of single-species active particles with long-range, non-reciprocal interactions in two and three dimensions, characterized by the absence of gas phases, and provides evidence that in these systems, topological and transport properties are closely related.

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Collective motion patterns, such as flocking or milling, observed in biological systems – including birds, fish, and sheep^1–6 – or in artificial active systems,^7–10 are almost always explained by invoking the apparent necessity of an underlying velocity alignment mechanism to reach consensus on the direction of travel, and by assuming that interactions are short-range, meaning that the interaction influence decays rapidly with distance, becoming negligible beyond a finite characteristic length scale, and thus providing the sense of locality.

Velocity alignment is a central concept in polar active fluids – e.g. in the Vicsek model¹¹ – as well as in active nematics. This alignment mechanism is inspired by the XY model, treating velocities as interacting spins. This analogy makes velocity alignment particularly appealing from a theoretical standpoint.^1,2 Despite this connection to the XY model, the intrinsic non-equilibrium nature of active systems leads to fundamental differences, such as the emergence of long-range orientational order in two-dimensions^11–13 or the presence of anomalous density fluctuations.^14,15 However, collective motion patterns, such as flocking or milling, can emerge even in the absence of velocity alignment.^16–24 Particularly relevant for applications to animal systems,^22,25,26 pedestrian models,¹⁸ and realistic vision-based models^27,28 is the notion that collective organization can emerge from simple rules based on the position, and not the velocity, of the neighbors. Moreover, a simple short-range, attraction rule has been shown to lead to the emergence of complex collective patterns, beyond standard aggregation, if and only if the interactions among identical particles break Newton's third law, i.e. the action-reaction symmetry, for example, by restricting perception via a field of view.²²

On the other hand, action-reaction symmetry breaking has been shown to play a major role in (scalar) “active mixtures” – i.e. active systems with two or more particle types – where non-reciprocal interactions between particle types lead to non-equilibrium patterns. Examples range from the emergence of traveling patterns²⁹ and non-equilibrium self-assembly³⁰ to phase coexistence³¹ and chaotic dynamics of bands for quorum-sensing interactions.³² A major theoretical interest in (scalar) active mixtures has been that the effects of non-reciprocal interactions can be understood at the hydrodynamic level by constructing Cahn–Hilliard-type models for non-reciprocal interactions.^29,33,34 However, single-species, non-reciprocal active systems – despite being ubiquitous in real-world systems such as in sheep,³⁵ birds,³⁶ starfish embryos³⁷ or robots³⁸ – remain, comparatively, unexplored. Exceptions include: observations of long-range order (LRO) in two dimensions for XY spins with vision cones,³⁹ recently found to be metastable,^40,41 order in the presence of velocity alignment with fore-aft asymmetry,⁴² collective patters of attractive active particles with vision cone,^22,25,26 including adaptive attraction⁴³ and steric repulsion^44,45 and aggregation patterns of torque-free particles.⁴⁶ Finally, hydrodynamic equations for these systems are limited to those derived for attractive active particles with a (short-ranged) vision cone^22,47 and those for particles with non-reciprocal velocity alignment.⁴⁸

Here, for the first time, we investigate the emergence of collective patterns in single-species, active systems with long-range, as opposed to short-range, non-reciprocal interactions. Importantly, the two most common assumptions of flocking models – velocity alignment mechanism and short-range interactions – are not considered in the investigated system, where active particles move at constant speed towards the position of those particles within their field of view, independently of their relative distance [Fig. 1(a)]. Note that previous models including a field of view^22,43–46 considered a finite horizon beyond which interactions do not occur, and thus effectively restricted interactions to be of short range. On the other hand, in ref. 25 and 26, distance-independent, topological interactions – where each particle interacts with a fixed number of neighbors within its visual cone – were investigated.

Fig. 1 (a) Scheme of the model: the vision cone of the black particle is the blue region defined by the angle 2β. (b) Phase diagram varying the angle β and the noise intensity D. The inset displays the probability – at the position indicated by the circle and starting from random initial conditions – of observing various patterns (that are color coded). Color code: worms (red), 2-twist (green), 3-twist (blue), ring (violet), planet (orange), amoeba (brown) and cloud (yellow); for more details see ref. 49.

In the investigated active system, long-range interactions lead to two new fundamental properties – not present in systems with short-range interactions – that strongly impact the emergent dynamics. First, despite the restriction imposed by the vision cone, a single particle can only lose contact with other particles for a short, characteristic time. As a result, the system remains cohesive even in the limit of vanishing density and for all noise intensities. In other words, a gas phase of freely expanding particles in open space does not exist. In sharp contrast, active particles with short-range interactions always exhibit a gas phase. Second, long-range interactions allow particles to form a single stable complex pattern whose size and topology result from the system dynamics and thus are not determined/constrained by a characteristic interaction length.

In short, our minimal framework – that is not intended as a realistic vision-based model – provides insight into the complex dynamics that emerges in collectives with long-range, non-reciprocal attraction. The results obtained may help to understand collective behavior in animal groups and robot swarms.

Model

We consider a system of N active particles that move at constant speed. In any spatial dimension, the equation of motion of the i-th particle, with position x_i, is given by:

ẍ_i = Ô(ẋ_i)F_i + Ô(ẋ_i) ° N_i. (1)

is the interaction force: particle i is attracted to all particles j within its field of view. The term Ω_i denotes the set of all particles present in the field of view of i at time t [see Fig. 1(a)], and n_i is its cardinality (the number of neighbors of i). The parameter γ is the relaxation constant. The projector operator Ô(ẋ_i) applied to a vector A is defined by . The operator Ô ensures that ‖ẋ_i‖ remains constant by eliminating the component of A parallel to ẋ_i and leaving only the component perpendicular to it. The term N_i denotes a vector noise with components N_iα(t) (α ∈ {x,y,z}) such that 〈N_iα(t)N_iβ(t′)〉 = 2Dδ_αβδ(t − t′). In Ô(ẋ_i) ° N_i, the noise is interpreted in the Stratonovich sense. As initial condition, we set ‖ẋ_i(t = 0)‖ = v₀, and thus the speed of all particles remains v₀ for t > 0.

In the following, we focus on two dimensions. The behavior of the system in three dimensions is discussed later. The active model given in eqn (1) reduces in 2D (for details on the derivation, see ref. 49) to:

ẋ_i = v₀ê[θ_i], (2)

where, in eqn (2), θ_i encodes the velocity direction using ê[·] ≡ cos(·)x̂ + sin(·)ŷ, while in eqn (3), α_ij is the polar angle associated with the vector (x_j − x_i)/‖x_j − x_i‖ = ê[α_ij], and ξ_i(t) is a white noise with 〈ξ_i(t)〉 = 0 and 〈ξ_i(t)ξ_j(t′)〉 = δ_ijδ(t − t′). The set Ω_i is defined by the following condition: any particle j such that ê[α_ij]·ê[θ_i] > cos(β) belongs to the field of view of the particle i, which does not involve any restriction on the distance between i and j, as occurs in ref. 22. Note that β controls the size of the vision cone. Since the equations can be adimensionalized by choosing an adequate length and time scale, we fix v₀ = 1 and γ = 5 without loss of generality.

General remarks

The force F_i in eqn (1) can be expressed as with . One of the fixed points of eqn (1) is reached at the minimum of this potential and corresponds to a vanishing interparticle distance. However, note that the constant speed constraint, among other features of the system dynamics, prevents particles from collapsing into a single point.

For β = π, it can be shown that particles are attracted to a (weighted) center of mass. In this reciprocal limit, i.e. β = π, the spatial distribution of particles stays asymptotically confined within a radius 0 < ε < ∞, with ε = ε(D,γ,v₀). Consequently, there is no gas phase with particles freely expanding in open space. In contrast, a system of active particles with short-range interactions possesses such a gas phase.

This result also holds for 0 < β < π, i.e. when particles display a blind angle, and thus interactions are non-reciprocal. Specifically, we argue that freely diffusive particles cannot exist for 0 < β ≤ π and D > 0: a particle taken away from a high-density swarm – independently of the morphology of the collective – tries to move back to it. Assume that a particle is in position x_A and a swarm is localized at position x_B. For simplicity, let us consider that x_B is constant. If x_B is not in the field of view of A, then θ_A will follow a purely diffusive dynamics. With certainty, in finite time τ – with 〈τ〉 ≤ (π − β)/D) in 2D – B will be in the field of view of A and will start to move towards B. In summary, A performs a (persistent) random walk when B is not in its field of view, while it moves ballistically towards B when B is detected. For any positive value of D, the dynamics is such that it prevents A to become independent of B and thus inhibits the free expansions of the spatial distribution of particles. Thus, we can ensure the absence of a gas phase for our system of active particles with long-range interactions in 2 and 3 dimensions.

Emergent self-organized patterns

As commented above, starting from a random initial spatial distribution, particles self-organize into a single pattern/structure. Interestingly, the emergent structure can display various topologies [see Fig. 2]. As in the game of life,⁵⁰ the initial configuration of the particles plays a crucial role in the collective pattern that we observe [see Fig. 1(b)]. Moreover, we find that for a fixed parameter set, several attraction basins – one for each collective pattern – coexist. Importantly, the transport properties of the center of mass (CM) of these patterns differ from one structure to another: the CM can diffuse, move ballistically (for a very long characteristic time), or the structure rotates. Interestingly, there is a clear connection between the topology of the pattern and its transport properties.

Fig. 2 Zoology of collective moving patterns as the non-reciprocity index H is increased: (a) a cloud (β = π), (b) a ring (β = 2.45), (c) a 2-twist (β = 2.06), (d) a 3-twist (β = 1.93), and (e) a worm (β = 1.54). For all patterns the noise amplitude is . The red points – in (c) and (d) – indicate singular topological points and the red arrows the resulting local polar order at those points. For more information and illustrative videos of each emergent collective pattern, see ref. 49.

A fundamental aspect of these complex structures is that they can only emerge in the absence of the action-reaction symmetry, which is broken by the field of view.⁵¹ The level of non-reciprocity is then a key feature and it differs from structure to structure. To characterize how non-reciprocal interactions are, we introduce the non-reciprocity index H, defined as:

where, at time t, the adjacency matrix element is defined as A_i,j = 1 if particle j is in the vision cone of i and 0 otherwise. K is a normalization constant (K = N(N − 1)/2) which ensures that H(t) ∈ [0,1]. When all interactions are reciprocal, e.g. for β = π, A_i,j(t) = A_j,i(t) for all (i,j), and thus H(t) = 0. On the other hand, for non-reciprocal interactions |A_i,j(t) − A_j,i(t)| = 1. If this applies to all pairs (i,j), then H(t) = 1. We characterize emergent self-organized structures by computing H = 〈H(t)〉_t, with 〈⋯〉_t denoting the temporal average. Notably, the value of H depends not only on the vision cone angle β, but also on the topology of the structure. We further analyze the interaction network A_i,j(t) by measuring the temporal evolution of the 1-norm between the adjacency matrix at time t₀ and at time t > t₀: .

Furthermore, we study the phase portrait [θ_i,[small theta, Greek, dot above]_i] and the transport properties of the collective pattern. To do that, we compute the temporal evolution of the CM, defined as , its average squared δ²(t) = 〈(x_CM(t₀ + t) − x_CM(t₀))²〉_t0, the polarization , and its correlation C(t) = 〈P(t₀ + t)·P(t₀)〉_t0. Note that ẋ_CM = v₀P(t) and thus , implying that .

In the following, we describe the most representative structures and their main properties for increasing non-reciprocity index H; see ref. 49 for illustrative videos of each collective pattern and more technical information.

Cloud

For β = π interactions are reciprocal, and thus H = 0. The particles orbit around the CM and the pattern appears as a roundish cloud [Fig. 2]. The resulting interaction network is a fully connected static network [Fig. 3, row a)]. The phase portrait [θ_i,[small theta, Greek, dot above]_i] is homogeneously covered [Fig. 3, row a)]. Particles move along elongated 8-shaped trajectories around the CM.⁴⁹ This cloud of particles has a vanishing polarization. Fluctuations, due to D > 0, lead asymptotically to the diffusive behavior of the CM [see Fig. 4].

Fig. 3 Columns correspond to the phase portrait (I), the interaction network (II), and the interaction network dynamics (III) of the collective pattern. From top to bottom: clouds (a), rings (b), 2-twists (c), 3-twists (d) and worms (e). Vision angle values β are given in Fig. 2. Noise amplitude . For more details, see ref. 49.

Fig. 4 Transport properties: trajectories of the CM for different collective patterns (a), mean quadratic distance δ² of CM vs. time (b), modulus of the polarization |P| vs. time (c) and autocorrelation C of the direction of the polarization (d). Color code: worms (black lines), 2-twist (red), 3-twist (green), ring (blue) and cloud (orange). Vision angle values β are given in Fig. 2. Noise amplitude .

Ring

When β is slightly smaller than π and such that H ∼ 0.19, particles self-organize into ring patterns [Fig. 2]. The interaction network remains almost fully connected and exhibits a clear oscillatory dynamics [Fig. 3, row b)]. Rings consist of 50% of the particles, homogeneously distributed along the ring, rotating clockwise, while the other 50% rotating counter-clockwise. This implies that locally, in a small segment of the ring, half of the particles move in one direction and the other half in the opposite direction. The local polar order vanishes, and thus there is zero global polarization. The ring is a closed nematic filament. This is evident by looking at the phase portrait that shows that [small theta, Greek, dot above]_i is 0.7γ or −0.7γ, while θ_i is homogeneously distributed over [0,2π) [Fig. 3, row b)]. The period of a particle turning around the ring is , and the period exhibited by the interaction network is, as expected, half of this value. Since the radius R obeys v₀ = [small theta, Greek, dot above]_iR, then . The presence of a blind angle implies that each particle does not move towards the CM, but to a point slightly displaced away from the CM. Furthermore, we observe that the CM rotates. This rotation is noisy because of the angular fluctuations experienced by each particle. We find that the behavior of the CM is consistent with a chiral random particle model such that ẋ_CM = v₀P = v_Rê(θ_R) and , with v_R, Ω_R, and D_R constant and η(t) a white noise [Fig. 4]. Asymptotically, the behavior of CM is diffusive with diffusivity⁵², which is much smaller than the diffusivity of the cloud.

2-Twist

Further decreasing the value of β, several stable complex patterns can emerge depending on the initial condition. One of them is an 8-shaped pattern, which we call the 2-twist pattern [Fig. 2], commonly observed at non-reciprocity index H ∼ 0.55 and displaying a relatively complex interaction network with intermediate connectivity values [Fig. 3, row c)]. The 2-twist pattern consists of particles moving along this 8-shaped orbit always in the same direction, implying that the particles exhibit local polar order. The time a particle takes to move along this orbit sets the period observed in the periodicity of the interaction network. The phase portrait corresponds to a non-trivial closed orbit that reflects a complex oscillatory behavior of θ_i as the particle moves along the structure.

The remarkable feature of this pattern is the presence of a singular topological point where the derivative of the polarization along the structure exhibits a discontinuity [see Fig. 2]. This point corresponds to the crossing of two segments of the polarized filament. Note that closed polar filaments with no crossing cannot display (global) polar order. However, if the self-organized structure has a crossing, i.e. a singular topological point, the structure can exhibit non-zero polar order. The polar order displayed by the structure [see Fig. 4] is given by the polar order at the singular topological point. As indicated above, ẋ_CM = v₀P. The high temporal correlation value displayed by polar order P implies that the CM moves ballistically for a long characteristic time [Fig. 4]. Arguably, angular fluctuations should render CM motion asymptotically diffusive, but the persistence time seems to be extremely large, to the point that we failed to observe it in simulations.

3-Twist

For parameter values where the 2-twist pattern is observed, we can also find another structure that displays not one, but two singular topological points [Fig. 2]. We call this structure 3-twist. The non-reciprocity index H ∼ 0.55 is close to the one measured in 2-twist patterns. The interaction network presents a connectivity similar to the 2-twist case [Fig. 3, row d)] but oscillates with a longer period (these patterns are longer and the particles take longer time to move around them). Only half of the phase portrait is occupied by a structured spatial coverage determined by the wave-like dynamics of the particles trajectories (details in ref. 49). Importantly, the 3-twist pattern corresponds to a closed polar chain of particles that displays two crossings, i.e. topological singular points. The magnitude of polarization at these two points (|P_sp|) is the same and, given the topology of the structure, the local polar order, at each point, is opposite to the other. This implies that polar order is nearly zero and that the velocity of the CM vanishes. On the other hand, these topological singular points are separated by a distance ℓ and since each of them moves with speed v₀|P_sp|, the structure rotates around its CM. The angular velocity of this rotation is proportional to v₀|P_sp|/ℓ. While particles move along the 3-twist structure, the oscillations of θ_i, combined with the rotation of the structure itself,⁴⁹ lead to a phase portrait pattern that fills half of the plane [Fig. 3, row d)]. Finally, while the average velocity of the CM is 0, angular fluctuations in the equations of motion lead asymptotically to diffusive behavior of the CM [Fig. 4].

Worm

At lower values of β, other distinct self-organized patterns with non-reciprocal index H ∼ 1 emerge: open polar filaments, which correspond to files of active particles following each other [Fig. 2]. We call this collective pattern worm. The resulting interaction network is static, highly non-reciprocal and hierarchical [Fig. 3, row e)]. The particles interact with those in front of them in the filament. Thus, the particle in the front does not interact with anybody, the second particle in the filament with the particle at the front, the third particle with the second and first particle, and so on. In this structure 〈[small theta, Greek, dot above]_i〉 = 0 and, for all i, θ_i(t) ∼ θ_*(t), where θ_*(t) is the angular variable of the particle in front of the worm at time t. Thus, the phase portrait is approximately reduced to a fixed point. Worms display high polar order with |P| ∼ 1. At a given time t, we can roughly assume that P(t)‖ê[θ_*(t)]. This means that the stochastic dynamics of θ_* is followed by all particles and therefore the CM moves ballistically during a characteristic persistent time D⁻¹ [Fig. 4]. Only on much longer timescales can motion be recognized as diffusive, as is expected for an isolated particle.

Pattern metamorphosis

We have already mentioned that, for a fixed parameter set, it is possible to see the emergence of different patterns depending on the initial conditions. Now, we explore whether we can go from one collective pattern to another by slowly varying the value of β. First, we observe that, starting with the same pattern, by repeating the same protocol where β(t) changes from β(t₁) = β₁ to β(t_f) = β_f, we go through different patterns. The observed sequence of patterns is not fixed and depends on the realization of the noise between t₁ and t_f. For instance, starting with a worm, we can go to a cloud passing by a 3-twist, or by a ring, or a 2-twist and then a 3-twist [see Fig. 5(a)]. On the other hand, we can start with a ring and reach a cloud by increasing the value of β and, reversely, start with a cloud and get to a worm by passing through a ring that emerges at a different β value. We can quantify these results estimating the probabilities of finding a configuration at the end of the time interval corresponding to a given β value. It is worth noting that collective patterns survive the slow tuning of the field of view angle even outside the typical range in which these patterns appears, from random initial conditions, in the phase diagram of Fig. 1(b).

Fig. 5 Evolution of a collective motion pattern as the field of view β is varied. (a) Above: Transition from a worm to a cloud pattern. As β is varied, different patterns can emerge. There is no unique sequence of patterns. Below: Transition from a ring to a cloud. (b) The sequence of patterns observed varying β₁ → β₂ is different from β₂ → β₁, indicating hysteresis effects. (c) Probabilities of finding the considered collective patterns tuning the β value for the cases displayed in (a) – from a worm to a cloud, increasing β – and in (b) – decreasing β –. Same color code of Fig. 1. For more details, see ref. 49.

From these numerical experiments, we learn that there is no reversibility and that there are strong hysteretic effects: if we start from a given pattern A at β₀ and, by tuning β₀ → β₁, we reach pattern B at β₁, this does not imply that starting from pattern B and performing the reverse transformation β₁ → β₀, we will end up with pattern A.

Robustness of results and model extensions

The diversity of complex patterns generated by non-reciprocal attraction highlights the relevance of such a simple rule as a pattern-forming mechanism for flocking. Importantly, this richness of flocking behaviors is not observed when particles interact through standard velocity-alignment mechanisms. The minimal model explored here involves several idealizations (or simplifications): particles experience no repulsive forces, they are treated as point masses, and the analysis has so far been restricted to 2D. Although we consider simplifications essential for gaining theoretical insight, in the following, we examine to which extent the reported results depend on these specific model assumptions.

First, we investigate the effect of including a short-range repulsive force. Provided that the repulsive core – which can be associated with the particle size – is sufficiently small, all patterns shown in Fig. 2 remain stable. Stronger repulsion leads to broader polar filaments and can cause the nematic ring to reorganize into two opposite polar filaments. Supplementary figures and details of the repulsive force implementation are provided in ref. 49. Note that, although we do not explicitly account for visual occlusion, the force F_i involves an average over the n_i neighbors of the particle i. Consequently, the dynamics of the particle i is identical whether it has a single neighbor or multiple neighbors aligned behind one another. Furthermore, visual occlusion becomes relevant only when particles come into very close proximity, a situation that can generally be avoided by introducing an effective repulsion that keeps particles separated.

Second, we examine the impact of the spatial dimension on the system dynamics. Using eqn (1) we investigate the emergent collective behavior in 3D. Details of the implementation are provided in ref. 49. As the field of view β is decreased, we observe cloud-like patterns for reciprocal interactions (β = π), nematic ring patterns for β ∼ 2.2, closed elongated (locally) polar loops for β ∼ 1.9, and worm-like patterns when β < 1.5 – see Fig. 6. Interestingly, the 2-twist and 3-twist patterns observed in 2D appear to be replaced by elongated polar loops, suggesting the absence of singular topological points in three dimensions. Even if a more detailed study of the 3D structures is required, the comparison of Fig. 2 and 6 suggests that collective patterns in 2D and 3D are qualitatively similar.

Fig. 6 Zoology of emergent collective patterns in 3D as function of the non-reciprocity index H. Patterns correspond, from left to right, to a cloud (β = π), a ring (β = 2.06), a polar, elongated, closed loop (β = 1.9), and a worm (β = 1.5). For all patterns D = 0.05, except for closed polar loop where, to allow a better visualization of the pattern, D = 0.0072. All other parameter values as in Fig. 1–5.

Conclusions

Here, we have studied active particles with long-range interactions restricted only by a field of view. In sharp contrast to active systems with short-range interaction, the system does not possess a gas phase. Importantly, we have shown that the combined effect of attraction and a field of view leads particles to self-organize within a manifold of reduced dynamical dimensions, i.e., a complex collective pattern in dimensions 2 and 3. Furthermore, we have revealed the existence of a fundamental interplay between the topology and the transport properties of the emergent structures, whose morphology is regulated by the degree of non-reciprocity. The obtained results shed light on the physics of single-species, non-reciprocal active particles and may help elucidate emergent, complex collective behaviors observed in animal groups and swarm robotics. An analytical understanding of the complex patterns that emerge in a system of active particles with non-reciprocal interactions – beyond the numerical study performed here – remains a major theoretical challenge for future research.

Conflicts of interest

There are no conflicts to declare.

Data availability

Supplementary information is available. See DOI: https://doi.org/10.1039/d5sm00657k.

This paper does not use original empirical data. The numerical processed data produced for the analysis are available in the manuscript. E.B. can be contacted to request the raw data from this study.

Acknowledgements

E. B. was partially supported by CNPq (Grant No. 305008/2021-8 and 305984/2024-1) and FAPERJ (Grant No. 260003/005762/2024). F. P. acknowledges financial support from C.Y. Initiative of Excellence (grant Investissements d'Avenir ANR-16-IDEX-0008), INEX 2021 Ambition Project CollInt, Labex MME-DII, projects 2021-258 and 2021-297, and projects ANR-22-CE30-0038 “push–pull” and (ANR-NSF) ANR-24-CE95-0002 “MotDis”.

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