Large-scales patterns in a minimal cognitive flocking model: incidental leaders, nematic patterns, and aggregates
aa r X i v : . [ phy s i c s . b i o - ph ] D ec Large-scales patterns in a minimal cognitive flocking model:incidental leaders, nematic patterns, and aggregates
Lucas Barberis
1, 2 and Fernando Peruani ∗ Universit´e Nice Sophia Antipolis, Laboratoire J.A. Dieudonn´e,UMR 7351 CNRS, Parc Valrose, F-06108 Nice Cedex 02, France IFEG, FaMAF, CoNICET, UNC, C´ordoba, Argentina (Dated: December 18, 2019)We study a minimal cognitive flocking model, which assumes that the moving entities navigateusing exclusively the available instantaneous visual information. The model consists of active par-ticles, with no memory, that interact by a short-ranged, position-based, attractive force that actsinside a vision cone (VC) and lack velocity-velocity alignment. We show that this active system canexhibit – due to the VC that breaks Newton’s third law – various complex, large-scale, self-organizedpatterns. Depending on parameter values, we observe the emergence of aggregates or milling-likepatterns, the formation of moving – locally polar – files with particles at the front of these structuresacting as effective leaders, and the self-organization of particles into macroscopic nematic structuresleading to long-ranged nematic order. Combining simulations and non-linear field equations, weshow that position-based active models, as the one analyzed here, represent a new class of activesystems fundamentally different from other active systems, including velocity-alignment-based flock-ing systems. The reported results are of prime importance in the study, interpretation, and modelingof collective motion patterns in living and non-living active systems.
PACS numbers: 87.18.Gh, 05.65.+b, 87.18.Hf
It is believed that complex, self-organized, collectivemotion patterns observed in birds, fish, or sheep [1–6] aswell as non-living active systems [7–10] result from thepresence of a velocity alignment mechanism that medi-ates the interactions among the moving individuals. Suchvelocity alignment mechanism is at the core of the so-called Vicsek-like models [11] extensively used to studyflocking patterns [1, 2]. Intrinsically non-equilibrium,these patterns differ remarkably from those observed inequilibrium systems by the lack of both, Galilean invari-ance and momentum conservation, which allows, for in-stance, the emergence of long-range orientational orderin two-dimensions [11–13] and the presence of anomalousdensity fluctuations [14, 15].Few recent pioneering works [16–23] have challengedthe wide-spread view that behind each collective motionpattern of self-propelled entities, there is a velocity align-ment mechanism at work. Here, we explore the possibil-ity of observing flocking patterns in the absence of suchalignment. The model we analyze is a minimal cognitiveflocking model that assumes that the moving entities nav-igate using exclusively the instantaneous visual informa-tion they receive. Importantly, the moving particles haveno memory as to compute the moving direction of neigh-boring particles, in sharp contrast to standard flockingmodels [1, 2, 11]. The navigation strategy we investi-gate is based on the instantaneous position of neighbor-ing particles and not on their velocity, which makes themodel simpler from a cognitive point of view and com-putationally less intensive, providing an alternative inthe design of robotic navigation algorithms. The modelincorporates few well-known physiological and cognitiveconcepts. For instance, we assume that particles are at- (c)(b)(a)
FIG. 1: (Color online) Depending on vision cone size β (in-sets), we can observe the formation of locally polar structures,which we call “worms”, panel (a) – β = 0 .
8, or the emer-gence of aggregates/milling-like patterns as shown in panel(b) for β = 2 .
4, and panel (c) for β ∼ π . Parameters (a)-(c): √ D θ = 0 . L = 100, N = 10 . See [46] for movies. tracted by those particles located inside the vision cone .The vision cone (VC) results from two well-documentedfacts: i) animals have a limited field of view [18, 24, 25]– typically less than 360 degree –, which is parametrizedhere by the angle β , and ii) when navigating, objects lo-cated at far distances are ignored and the focus is puton those objects located at distances shorter than theso-called cognitive horizon [18, 24, 26] that correspondsin our model to R . The field of view ( β ) is known tovary from species to species, being for instance smaller forpredators than for preys [24, 25]. In summary, the fieldof view ( β ) combined with the cognitive horizon ( R ) de- √ ϴ β aggregates gas nematicbands worms FIG. 2: (Color online) Phase diagram: vision angle β vsangular noise intensity D θ . The color code indicates the valueof the average cluster size m ∗ in simulations with N = 10 .Vertical rectangles refer to cuts of the phase diagram shownin Fig. 3 and Fig. 4. fine the VC, which violates Newton’s third law for β < π .Notice that there exist alternative mechanisms to breakthe action-reaction symmetry to the VC [23, 27, 28] andthat the presence of non-reciprocal interactions has alsoa strong impact on the dynamics of flocking models withvelocity alignment [29–32].Here, we show that this minimal cognitive flockingmodel exhibits various large-scale self-organized patterns,depending on the size of the VC and noise intensity:aggregates or milling-like patterns of various degrees ofcomplexity, locally polar dynamical structures which wecall worms (Fig. 1), and nematic bands leading to long-ranged nematic order. Furthermore, we derive a systemof non-linear field equations to rationalize agent-basedsimulation results and show that in general position-based active models, as the one here studied, represent anew class of active systems fundamentally different fromother active systems, including velocity-alignment-basedflocking systems [11–14, 33–44]. Model definition.–
The equation of motion of the i -thparticle is given by:˙ x i = v V ( θ i ); ˙ θ i = γn i P j ∈ Ω i sin( α ij − θ i ) + √ D θ ξ i ( t ) , (1)where x i denotes the position of the particle, θ i repre-sents its moving direction, with V ( . ) ≡ (cos( . ) , sin( . )) T , v is the particle speed, γ the strength of the interac-tions, and ξ i ( t ) is a noise term such that h ξ i ( t ) i = 0 and h ξ i ( t ) ξ j ( t ′ ) i = δ i,j δ ( t − t ′ ), with the noise amplitude givenby D θ . The sum in Eq. (1) describes the projection onthe “retina” of particle i of the position of all particlesinside its VC, assuming particles are point-like, with α ij the polar angle of the vector x j − x i || x j − x i || = V ( α ij ); a proce-dure similar to the one in [19] for long-range interactions.The symbol Ω i thus denotes the set of neighbors insidethe VC of particle i , with n i its cardinal number. Par-ticles in Ω i are those that satisfy || x j − x i || ≤ R and t -3 -2 -1 M* β=π β=2.8β=1.0 β D eff / D NAP S t -2
GasWormsAggregatesTransition Region velocities positions (d)(c)(a) (b) aggregatesworm
FIG. 3: (Color online) (a) Normalized number of clusters M ∗ (normalized average cluster size m ∗ , inset) as function of time.Worms coagulate at a much faster rate than aggregates. (b)The (normalized) diffusion coefficient D eff /D NAP and polarorder S as function of β (see text). (c) Temporal evolution of h x i / (4 t ). (d) Velocities and positions of particles that forma worm: particles copy the behavior of the particle at thefront, which we refer to as the incidental leader . See [46] fora movie. Parameters: N = 10 , L = 100, √ D θ = 0 . (b) (c) β (a) NS S FIG. 4: Nematic bands. (a) The nematic order parameter S as function of β for √ D θ = 0 .
84. The inset shows S asfunction of N , with error bars obtained using 50 realizations, β = 1 .
9. (b) and (c) display simulations snapshots at thesteady state for N = 10 , L = 100, √ D θ = 0 .
84, and β =1 .
9. Insets in (b) and (c) correspond to magnified views ofthe bands. See [46] for movies. x j − x i || x j − x i || · ( ˙ x i / || ˙ x i || ) > cos( β ), with β the size of the coneand its orientation given by ˙ x i . For a definition of themodel in 3D and a justification of the term sin( α ij − θ i ),see [46]. In the following we fix v = 1, set R = 1, γ = 5,and the global density ρ = N/L = 1, with N the num-ber of particles and L the linear size of the system anduse periodic boundary conditions. These parameters arein the range of the ones expected for vertebrates [45]. Phenomenology.–
The system exhibits four distinctphases – see Fig. 2 – which we refer to as i) gas phase,ii) aggregate phase, iii) worm phase, and iv) nematicphase. Phases ii) to iv) involve spontaneous phase sep-aration of the particles, while phase i) is characterizedby the absence of order and an homogeneous distribu-tion of particles in space. In the following, we studythe phase-separated phases, i.e. from ii) to iv), byperforming two vertical cuts in the phase diagram inFig. 2. The emerging macroscopic patterns are char-acterized by their level of (global) orientational orderthrough S q = | P Nj =1 exp( ı qθ j ) /N | , with q = 1 for polarorder and q = 2 for nematic order (and ı the imaginaryunit). Clustering properties are analyzed by the nor-malized number of clusters M ∗ = h M i /N and normalizecluster size m ∗ = h m i /N , defining a cluster a set of con-nected particles, where i is connected to j if j is locatedinside the VC of i . Finally, transport properties are stud-ied by looking at the behavior of the diffusion coefficient D eff = lim t →∞ P Ni =1 ( x i ( t ) − x i ( t )) / [4 N ( t − t )]. Aggregate phase.–
At large values of β , particles self-organize into aggregates of different complexity, Fig. 1,with some of these patterns comparable to the ones re-ported in [49]. The aggregates result from a phase separa-tion process (see [46] for movies) fundamentally differentfrom the one in [50, 51]. Fig. 3(a) shows the existence ofdifferent nontrivial scalings of M ∗ and m ∗ with time –as expected for active systems [52] – which suggests thatthe phase-separation process is of a different nature atlarge and intermediate values of β . Worm phase.–
At lower values of β , we observe theemergence of a new type of macroscopic structure, whichwe call worm, Fig. 1(a). This structure consists of a file ofactive particles that are locally polarly oriented. The par-ticle at the “head” of the worm – which we label “ H ” – ig-nores all other particles and becomes the effective leaderof the spontaneously formed herd of active particles. Thisis evident from the behavior of D eff as shown in Fig. 3.During the worm phase, D eff ∼ D NAP = v / (2 D θ ),with D NAP the diffusion coefficient of an ensemble ofnon-interacting active particles. This is not due to theabsence of interaction, but to the fact that all parti-cles in the worm imitate the behavior of the incidentalleader particle: the position and velocity of particle j is approximately given by x j ( t ) ∼ x H ( t − ℓ j,H /v ) and V ( θ j ( t )) ∼ V ( θ H ( t − ℓ j,H /v )), where ℓ j,H is the distancealong the worm between j and H , Fig. 3(d) and [46] fora movie. Though worms exhibit local polar order, globalpolar order drops for L ≫ v /D θ , vanishing in the ther-modynamic limit. Nematic phase.–
For larger values of D θ and β , seeFigs. 2 and 4, we find macroscopic nematic bands. Aftera complex transient where various small nematic bandsgrow in size and interconnect, the system reaches a steadystate, with one or several bands, but where only one di-rection prevails, see Fig. 4 and [46] for a movie. Thedescribed dynamics leads to the emergence of genuineglobal nematic order. Increasing the system size N , fora fixed density ρ , we observe that the nematic order S saturates, Fig. 4(a), inset. Field equations.–
A qualitative understanding of thelarge-scale behavior of the system can be obtained interms of p ( x , θ, t ) = h P Ni =1 δ ( x − x i ) δ ( θ − θ i ) i . The evolu-tion of p ( x , θ, t ) is given by the corresponding non-linearFokker-Planck equation of Eq. (1) [48]: ∂ t p + ∇ [ v V ( θ ) p ] = D θ ∂ θθ p − ∂ θ [ I p ] , (2)where we have assumed that p ( x , θ, x ′ , θ ′ , t ) ≃ p ( x , θ, t ) p ( x ′ , θ ′ , t ) in order to define, after some simplecalculations [46], an average interaction term: I = Γ Z R dR Z θ + βθ − β dαR sin( α − θ ) ρ ( x + R V ( α ) , t ) , (3)where Γ ≃ γ/ (1 + βR ρ ) and ρ ( x , t ) = L [1] the coarse-grained one-particle density, with L [( . )] an averaging op-erator defined as L [( . )] ≡ R π dθ ( . ) p ( x , θ, t ). To an-alyze the behavior of Eq. (2), in addition to ρ ( x , t )we introduce the following fields: local polar order P ≡ ( P x , P y ) T = L [ V ( θ )], local nematic order Q ≡ ( Q c , Q s ) T = L [ V (2 θ )], and higher order fields are de-noted by M k ≡ ( M kc , M ks ) T = L [ V ( kθ )] where k > k -th order field, and recast theequation as: ∂ t ρ + v ∇ P = 0 (4a) ∂ t P + v (cid:18) ∇ ρ + h ∇ T M Q i T (cid:19) = − D θ P − Γ g ( β )2 h M Q − ρ i ∇ ρ − Γ f ( β )2 M ρ [ P − M ] (4b) ∂ t Q + v h ∇ T (cid:16) M + M P (cid:17)i T = − D θ Q − Γ g ( β ) (cid:20) M − M TP (cid:21) ∇ ρ − Γ f ( β ) (cid:18) M ρ M + ρ (cid:20) Φ ρ − ∂ xy ρ (cid:21)(cid:19) , (4c)where the symbols M A denote matrices defined using the auxiliary matrices E = (cid:20) − (cid:21) , E = (cid:20) (cid:21) , E = (cid:20) − (cid:21) , and the unity matrix as: M Q = Q c E + Q s E , M = M c E + M s E , M P = P x E + P y E , M ρ = Φ ρ/ E − ∂ xy ρ , and M ρ = ∂ xy ρ E − Φ ρ/ E .In addition, we have defined Φ ρ as Φ ρ = ∂ yy ρ − ∂ xx ρ and the terms g ( β ) and f ( β ) are, respectively, the firstand second non-zero terms in the expansion of I withrespect R , that read g ( β ) = ( R /
3) ( β − sin(2 β ) /
2) and f ( β ) = ( R /
6) sin ( β ). Equations (4), due to the k > β ∼ π , i.e. (quasi) isotropic interactions, f ≃ ∂ t ρ = − v D θ ∇ h − v ∇ ρ + Γ g ( β ) ρ ∇ ρ + O ( R ) i , (5)where O ( R ) contains spatial third order derivatives withrespect to ρ . From Eq. (5) we learn that an homogenousspatial distribution of particles – assume ρ = ρ + ǫδρ ,with ρ a constant and ǫδρ a small perturbation – be-comes linearly unstable when c = v Γ g ( β ) ρ / (2 D θ ) − D NAP > β = π , the dispersion relation is of the form λ = c k − c k , where c = v Γ πR ρ / > k thewavenumber associate to the perturbation.For intermediate values of β and D θ agent-based sim-ulations display nematic patterns. Let us then assumethat locally the distribution of θ is given by p ( x , θ, t ) ≃ ( ρ/ π ) exp[(2 /ρ ) Q . V (2 θ )] (see [46] for a derivation). Asdirect consequence of this local ansatz we find that P = M = , M c = (cid:0) Q c − Q s (cid:1) / (2 ρ ) and M c = Q c Q s /ρ .Since under this assumption M can be expressed interms of ρ and Q , Eqs. (4) define a closed set of equations.Now we look for the stationary states of the resulting sys-tem. This implies that all partial temporal derivatives ofthe fields vanish. For simplicity, but without loss of gen-erality, let us assume that Q s = 0 and that the systemis invariant in the ˆ x direction as in Fig. 4.c, and thusderivatives in x vanish. Inserting all this into Eqs. (4),we arrive to: ∂ y ( ρ − Q c ) = − Γ g ( β ) v ∂ y ρ ( ρ + Q c ) (6a) Q c = − Γ f ( β )8 D θ ∂ yy ρ (cid:18) ρ − Q c ρ (cid:19) , (6b)where Q c and ρ are functions of y . By linearizing this sys-tem of equations – assume ρ = ρ + ǫδρ and Q c = ǫδQ c ,with ρ a constant and ǫδρ and ǫδQ c perturbations inthe density and nematic order, respectively, and keepinglinear order terms in ǫ – it becomes evident that δQ c ∝ ∂ yy δρ and the system reduces to ( aρ − / ( bρ ) z = ∂ yy z , with z = ∂ y δρ , a = Γ g/v and b = Γ f / (8 D θ ), whose solu-tions for ρ − < Q c ∝ ρ ,parallel to each other and equally spaced, i.e. there is awell-defined wave length. These solutions are consistentwith the nematic bands in Fig. 4.Finally, we have observed in agent-based simulationslocally polar patterns (worms). Let us assume then that p ( x , θ, t ) ≃ ( ρ/ π ) exp[(2 /ρ ) P . V ( θ )] [46]. Under this as-sumption, it is possible to show that static polar bands – i.e. static straight worms – cannot exist. The field equa-tions suggest that polar structures never reach a steadystate as observed in simulations, see Fig. 1(a) and 3(c). Concluding remarks.–
The derived field equations,combined with the presented numerical study, showthat there exist fundamental differences between velocityalignment-based models, including polar fluids [12, 13,33, 53], active nematics [14, 35, 36], and self-propelledrods [37–40, 42–44, 54], on the one hand, and position-based models as the one analyzed here on the otherhand. An evident and fundamental difference, revealedby Eqs. (4) and confirmed in simulations, is that position-based models cannot develop either polar or nematicordered phases that are spatially homogeneous – cf.with the well reported spatially-homogeneous orderedphases in (velocity-alignment) flocking models, such asthe celebrated Toner-Tu polar phase [12, 13, 33, 53] andhomogeneous nematic phase [14, 38–40, 54]. In con-trast, in position-based active models (orientational) or-der emerges always, even at short-scales, associated todensity instabilities. In addition, worms display localpolar order that is parallel (locally) to the band, witha highly dynamical center band line that prevents polarlong-range order (LRO) to emerge, in striking differencewith polar bands in Vicsek models, where polar order isorthogonal to the band and long-ranged [33, 53, 55]. Onthe other hand, it has been argued that nematic bands,reported in self-propelled rods [40] are unstable [54]. Thisis again in sharp contrast to the nematic bands reportedhere that remain stable in the thermodynamical limit.In summary, position-based active systems as the onepresented here belong to a new universality active classfundamentally different to any previously reported activematter classes [11–14, 33–44, 53].Our results could be relevant to study and interpretcollective patterns in animal groups. They indicate thatseveral flocking patterns observed in nature, such as themilling-like patterns found in fish [4], the file formationreported in sheep herds [6], and the emergence of nematicbands in human crowds and ants [18] could result fromsimple navigation strategies that do not require mem-ory, use exclusively the instantaneous position of neigh-boring particle, and limit the interaction neighborhoodby a vision cone. These concepts, that lead to navi-gation strategies that are computationally less intensivethan those based on velocity alignment, could help inthe design of new robotic navigation algorithms, as forinstance for phototactic robots [63]. Extensions of thisminimal cognitive model could find applications in otheractive systems such as chemophoretic particles [23, 28]and chemotactic colloids [56–59] and organisms [60–62],among other examples where aggregations patterns havebeen reported. This could require either taking R → ∞ or replacing the interaction cut-off by a slowly decayingfunction of the distance, as well as specializing the modelfor β = π , which corresponds to the limit of isotropic in-teractions, though asymmetric interactions may also berealistic [23, 28, 60]. Acknowledgement
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