Table Of Content

Collective Motion with Anticipation: Flocking, Spinning, and Swarming Alexandre Morin,1 Jean-Baptiste Caussin,1 Christophe Eloy,2 and Denis Bartolo1 1Laboratoire de Physique de l’Ecole Normale Supérieure de Lyon, Université de Lyon, CNRS, 46, allée d’Italie, 69007 Lyon, France 2Aix Marseille Université, CNRS, Centrale Marseille, IRPHE UMR 7342, 13384, Marseille, France Weinvestigatethecollectivedynamicsofself-propelledparticlesabletoprobeandanticipatethe orientation of their neighbors. We show that a simple anticipation strategy hinders the emergence of homogeneous flocking patterns. Yet, anticipation promotes two other forms of self-organization: collective spinning and swarming. In the spinning phase, all particles follow synchronous circular 5 orbits, whileinthe swarmingphase, thepopulationcondensatesintoasinglecompactswarmthat 1 cruises coherently without requiring any cohesive interactions. We quantitatively characterize and 0 rationalize these phases of polar active matter and discuss potential applications to the design of 2 swarming robots. n a PACSnumbers: 05.65.+b,87.16.Uv,47.54.-r J 1 1 I. INTRODUCTION liquid phase akin to the flocking pattern observed without anticipation. Finally, at the onset of the flocking-to- ] spinningtransition,thepopulationformsstablecompact h Over the last 20 years physicists have devoted signif- swarms despite the absence of any attractive couplings. c icant efforts to elucidate the numerous dynamical pat- e The paper is organized as follows: we first introduce ternsobservedinpopulationsoflivingorganisms. Flock- m the equations of motion of the interacting self-propelled ing patterns are a prominent example. They refer to the particles. The alignement and anticipation rules are de- - t self-organizationofanensembleofmotileindividualsinto scribed. We then discuss the emergence of the three or- a a homogeneous group undergoing coherent directed mo- t dered phases (spinning, flocking, and swarming phases). s tion. As initially suggested by Vicsek et al. [1], simple They are quantitatively characterized, explained and . t alignment interactions are sufficient to trigger the emer- compared to the conventional patterns of polar active a genceofflockingpatterns. Alinement-inducedflockshave m matter. We close this paper from an engineering per- beendemonstratedtheoretically,numericallyandinsyn- spective. We show that minimal anticipation rules can - thetic experiments [2–7]. In addition, quantitative anal- d provide an effective and safe design strategy to interrupt n ysis of animal trajectories [8–11] support that the large thedirectedmotionofaflockwithoutshuttingdownthe o scale features of some animal flocks can be rationalized propulsion mechanism at the individual level. c bysimplebehavioralrulesattheindividuallevel,includ- [ ing velocity-alignment interactions. However, inferring a 1 modelfromactualdatarequiresaminimalsetofhypoth- II. MODEL: ALIGNMENT AND v esisonthepossibleformoftheinteractions[12],oronthe ANTICIPATION 8 statistical properties of the observables [8]. Until now, 6 the overwhelming majority of the available models has We consider an ensemble of N self-propelled particles 4 been merely restrained to couplings between the instan- 2 in a two-dimensional space. The particle i, located at taneouspositionsandorientationsofneighboringindivid- 0 r (t), moves at a speed v along the unit vector pˆ that i 0 i . uals (see [10, 13, 14] for noteworthy exceptions). Inves- makes an angle θ(pˆ )≡θ with the x-axis 1 i i tigating alternative dynamical rules may provide further 0 5 insightintothecollectivedynamicsofmotileindividuals. r˙i(t)=v0pî. (1) 1 Here,wegeneralizetheconventionaldescriptionofpo- : Theequationofmotionoftheirinstantaneousorientation v laractivematter[3]. Weconsiderthedynamicsofmotile defines their anticipation and alignement rules: i particles, whichalignstheirvelocitywiththeorientation X ar otifonth.eiNraniveieglyh,boarnsticbiypaatniotniciwpaotuinldgbteheeirxpreocttaetdiontoalymieold- θ˙i(t)=−τ1(cid:10)sin[θi−(θj +ασj)](cid:11)j∈Ωi +(cid:112)2ηξi(t), (2) more robust flocks. In striking contrast, we show that simple anticipation rules result in much richer collective whereαisascalarparameterandσ ≡θ˙ /|θ˙ |isthesign j j j behaviors: Firstly, when motile particles strongly antic- of the angular velocity. The particles have a finite inter- ipate the orientational changes, the population simulta- actionradiusR,andΩ denotestheensembleofparticles i neously breaks a continuous and a discrete symmetry. interacting with the ith particle. We henceforth refer to The system self-organizes into a spinning state where all σ as the spin of the particle j. This quantity is akin j the particles follow closed circular orbits in a synchro- to the spin variable defined in [14] up to a multiplicative nized fashion. Secondly, a combination of alignment and factor, whichisthelocalcurvatureoftheparticletrajec- moderateanticipationresultsintheemergenceofapolar- tory. In Eq. 2, the angular noises ξ ’s are uncorrelated i 2 Gaussian random variables of unit variance, and τ is an a b orientational relaxation time. For the sake of simplicity, unitsarechosensothatτ =1andR=1,andthecontrol parameters left are N, v , α, η, and the system size L. 0 Equations 1 and 2 have a simple physical meaning. When α = 0, they reduce to the continuous-time ver- sion of the seminal Vicsek model [15]: the particles in- teract via effective torques that promote alignment with the instantaneous orientation of the neighboring particles. Note that neither the momentum nor the angular c d momentum is conserved in Eqs. 1 and 2. When α > 0, the mean orientation of the neighbors is anticipated in a simple fashion. If the orientation of the jth particle rotates in the clockwise (resp. anti-clockwise) direction, an effective torque promotes the alignment along the direction θ −α (resp. θ +α). α is a constant angle used j j to anticipate the future orientation of the neighboring particles. In order to gain a better insight into this interaction 1 e scheme,weexpandthesinefunctionsanduseelementary algebra to recast Eq. 2 into θ˙i(t)=−cosα(cid:10)sin(θi−θj)(cid:11)j∈Ωi (3) Π0.5 (cid:10) (cid:104) π (cid:105)(cid:11) (cid:112) −sinα sin θ −(θ +σ ) + 2ηξ (t), i j j2 j∈Ωi i which is the linear superposition of two models: a 0 continuous-time Vicsek model and a α = π model. The 0 0.5 1 2 η second term on the r.h.s of Eq. 3 promotes alignment withtheaccelerationofthesurroundingparticles. Asthe FIG.1: Coloronline. Continuous-timeVicsekmodel(α=0). magnitude of the translational velocity is constant, the (a)Superimposedplotofsubsequentparticlepositionsinthe accelerationofaparticleisperpendiculartoitsdirection isotropic phase (η = 0.5). The color accounts for the spin ofmotion: θ(∂tpî)=θi+σiπ2. Equation3isqualitatively value. Red: σ = −1, and black: σ = +1. (b) Instantaneous similartothediscretetimemodelintroducedin[13]. We orientations and positions of the particles in the isotropic revisit here the impact of the velocity-acceleration cou- phase (η = 0.5). To improve clarity, only one half of all pling from a different perspective and further character- particles is shown. (c) and (d) Same plots for the flocking ize the resulting phase behavior. state (η = 10−2). (e) Polarization plotted as a function of The special status of the α = 0 and α = π models the magnitude of the angular noise. For all the pictures and 2 plots: ρ=8, v =0.5 and L=16 (N =2048). prompts us to characterize them separately. However, a 0 comprehensive numerical characterization would require tovarysystematicallythreeindependentparametersthat are the propulsion speed v , the particle density and the 0 model by setting σ (t) = sign[θ (t)−θ (t−δt)], where noise magnitude η. In this paper we set v =1/2, which i i i 0 the time step δt is 10−2τ in all the simulations. is a conventional value that facilitates comparisons with previouspolaractivemattermodels[16,17]. Ifnotmen- tioned otherwise, we provide results corresponding to a III. RESULTS AND DISCUSSION particle density ρ=8, corresponding to N =2048 particles. Smaller density values yield qualitatively identical results, but require longer convergence times for finite A. Case α=0: Alignment-induced flocking. values of α. We therefore focus solely on the impact of the anticipation angle α and the angular noise η. Here,webrieflyrecallthephenomenologyofthemodel Whenα(cid:54)=0Eq. 2and3areimplicitasσ dependson whenα=0,whichisequivalenttoacontinuous-timever- i the time derivative of θ . We solve Eqs. 1 and 3 numeri- sionoftheclassicalVicsekmodel[15]. Forhighnoiseval- i callyusingperiodicboundaryconditionsinasquaresim- ues,theself-propelledparticlesundergouncorrelatedper- ulation box of size L and a forward Euler method. This sistentrandomwalks. Thepopulationformsanisotropic method requires specifying the discretization scheme for andhomogeneousgasdepictedinFigs.1a,b. Decreasing the spin variables. Aiming at describing the impact of the noise value below η = 0.45, the rotational symme- 0 anticipation, we have naturally defined the discretized tryoftheparticleorientationsisspontaneouslybroken: a 3 macroscopicafractionofthepopulationbpropelinthesame a b direction along straight trajectories, see Figs. 1c,d, and Supplementary Movie 1) [18]. This flocking transition, is quantitatively captured by the sharp variation of the orderparameterΠ≡|(cid:104)pˆ (cid:105) |(themeanpolarization), in i i,t Fig. 1e. Note however that, unlike what is found in simulations of discrete time Vicsek models (in the dilute limit)[16,17c],wedonotobservetheprdopagationofband- 0.8 c 0.4d shape excitations at the onset of collective motion. The reason for this discrepancy is purely technical. Based 0.6 0.2 on previous numerical observations [17] using an angu- Σ lar noise as in Eq. 3, the emergence of localized bands is Π , 0.4 (cid:89)(t) 0 expected to occur only in extremely large systems. The relatively small size of our largest simulations (L = 16, 0.2 −0.2 N =2048)explainswhysolitonicbandsdonotformhere 0 at the onset of1collective motion.e 0 0η.5 −31 −0.40 5000 10000 Thesimulationsinthecaseα=0reproducethesalient x 10 t features of the Vicsek flocking transition, and therefore 3e 0.42 f validate our numerical scheme. Π0.5 0.4 B. Case α= π: Anticipation-induced spinning. (cid:89)P((t))2 δ r0.38 2 1 0 0.36 0 0.5 1 Wenowinvestigatetheηanticipationmodelcorrespond- ing to α= π2. Not surprisingly, in the high noise regime, −01 −0.5 0 0.5 1 0.340 0.5 1 all the particles behave like uncorrelated persistent ran- (cid:89)(t) η −3 x 10 dom walkers again, and form a homogeneous isotropic phase with no preferred spin, Figs. 1a and 1b. How- FIG. 2: Color online. Case α= π. (a)Typical trajectories in ever, reducing the noise amplitude below ηπ =7×10−4, the spinning phase (η=4×10−42). (b) Snapshot of the posi- 2 two symmetries are spontaneously broken: the continu- tionandorientationofthemotileparticlesforthesamenoise ous rotational symmetry of the particle orientation, to- value. The color code accounts for the instantaneous spin as gether with the discrete Z symmetry of the spin vari- in Fig. 1 (η = 4×10−4). (c) Variations of the order pa- 2 able. As seen see Figs. 2a,b, a macroscopic fraction of rameters Π (light/black symbols) and Σ (dark/red symbols) the population synchronously follows persistent orbital with respect to the noise amplitude η. (d) Time series of the instantaneous global spin Σ(t), and (e) related probability trajectories with a constant radius, see also Supplemen- distribution that displays a clear bimodal structure. (f) The tary Movie 2 [18]. spatialcondensationatthetransitionisclearlyseenfromthe This alternative type of collective motion is well cap- variationsofδr withη,thehorizontallinecorrespondstothe tured by two order parameters: the global polarization value of δr for a uniform distribution of particles in a square ΠandtheglobalspinΣ, whichisdefinedastheabsolute box. All the simulations correspond to ρ = 8, v = 0.5 and 0 value of the most probable instantaneous average spin L=16 (N =2048). (cid:104)σ (t)(cid:105) . These two quantities sharply increase as η goes i i below ηπ as shown in Fig. 2c. It is also worth noting 2 that, the temporal variations of (cid:104)σ (t)(cid:105) is intermittent i i over time scales that are much larger than the individ- verges to 1 as η goes to 0 (Fig. 2c). At any given time, a ual relaxation time τ regardless of the noise amplitude, small yet finite fraction of particles does not follow syn- see Fig. 2d. The rare intermittent events correspond to chronous circular trajectories even for vanishingly small the interruption of the circular trajectories. The parti- noise. These unusual strong fluctuations in the weak- cles randomize their orientation at the same time and noise limit, together with the circular motion of the par- subsequentlyself-organizeagaintorotatesynchronously, ticle, willbebetterunderstoodinthenextsection, when possibly along a different direction. This long-time dy- inspecting the phase diagram of the general α model. namicsresultsinabimodalstructureofP((cid:104)σ (t)(cid:105) ),seen Last, we stress that at the onset of global spinning i i in Fig. 2e, and a posteriori justifies defining the spin or- motion,thespatialdistributionoftheparticlesishetero- derparameterasthemodulusofthemostprobablevalue geneous. Thepopulationconcentratesintoalargedenser of Σ(t). regionsurroundedbyamorediluteensembleofspinning In addition the individual circular trajectories ob- particles. Thisspontaneousbreakingofthetranslational served in this spinning phase are not robust: they con- invariance is quantified by measuring the variations of tinuously form and disrupt, and neither Π nor Σ con- δr ≡ |(cid:104)r −(cid:104)r (cid:105) (cid:105) | as a function of the angular noise, i i i i,t 4 a c e 2 g Isotropic π/2 1.5 3π/8 ψ Spinning α κ 1 π/4 Flocking π/8 0.5 0 0 −5 −4 −3 −2 −1 0 −5 −4 −3 −2 −1 0 10 10 10 10 10 10 10 10 10 10 10 10 η η 1 π/4 2 b d f h 3π/16 1.5 Σ Π , 0.5 ψ π/8 κ 1 π/16 0.5 0 0 0 −5 −4 −3 −2 −1 0 π/8 π/4 3π/8 π/2 0 π/6 π/3 π/2 10 10 10 10 10 α η α FIG. 3: Color online. Alignment and anticipation for finite α. (a) Phase diagram showing the region where the isotropic, the flocking and the spinning phases are stable. (b) Sharp variations of the polarization and spin order parameters with η (same color code as in Fig. 2). (c) Exemples of particle trajectories for α=0.8π/2 in the flocking state, for η=10−2. (d) Variations of the angle ψ between the polarization and the principal axis of the simulation box, as a function of the anticipation angle α. (e) Typical spinning trajectories observed for η = 7×10−4. (f) Instantaneous position and orientation of one half of the particles, with the same color code for the spin variable. The entire population rotates in the same direction. (g) Variation of thecurvatureofthespinningtrajectorieswiththenoiseamplitudeforα=0.4π. (h)Variationofthecurvatureofthespinning trajectories as a function of α in the zero-noise limit. Full line: analytical prediction κ=sinα/v τ. For all simulations, ρ=8, 0 v =0.5, and L=16 (N =2048). 0 Fig. 2f . This spatial heterogeneity emerges in the ab- conditions, the global polarization does not align with senceofexplicitcouplingsbetweenthepositionaldegrees one of the principal axis of the simulation box. In con- of freedom, which is a very persistent feature of all the trast, it makes a finite angle, ψ, that is set by the antic- active-particle models involving any form of short-range ipation angle α, Figs. 3c, and 3d. interactions [19]. When further reducing the noise amplitude below η = η (α),theflockself-organizesintoaspinningphase. As FS opposedtothecasewhereα=π/2examinedinprevious C. Case α>0: From flocking to spinning and section, we show in Fig. 3b that Σ increases sharply and swarming. saturates to 1 together with Π, as η goes to 0 deep in the spinning phase. The individual circular trajectories We now discuss the general case, for which the antic- are now robust. They do not disrupt and reform inter- ipation angle α can take any positive value. The col- mittently; all the particles endlessly follow synchronous lective behavior of the population is summarized by a circular orbits, see Figs. 3e, and 3f. phase diagram that divides the (η,α) plane into three In the spinning phase, the curvature κ of the particle regions as shown in Fig. 3. The boundaries of the phase orbitshasaconstantvalue,Fig.3g. Thisresultcanbeex- diagram correspond to the points where the order pa- plained by considering the zero-noise limit. When η =0, rameters, Π or Σ, go from zero to a finite value. When a perfectly polarized state with all particles aligned is a 0<α(cid:28)π/2,forveryhighnoisevaluesthesystemforms solution of Eq. 3 for any value of α, provided that the anisotropicandhomogeneousgas. Reducingη, thepop- particles rotate at the same angular velocity θ˙ = sinα. ulation undergoes two subsequent phase transitions to- As the particle velocity v is a constant, the correspond- 0 ward collective motion. A flocking transition occurs at ing trajectories are circles of curvature κ = sinα/v τ. 0 η =η (α): the polarization increases sharply while the As seen in Fig. 3h, this mean-field argument correctly GF global spin remains vanishingly small, Fig. 3b. However, accounts for our numerical results when 0<α<π/2. unlike all the Vicsek-like models with periodic boundary Figure 3a shows that the phase behavior of the pop- a c e g 2 Isotropic π/2 1.5 3π/8 ψ α Spinning κ 1 π/4 Flocking 0.5 π/8 0 0 −5 −4 −3 −2 −1 0 −5 −4 −3 −2 −1 0 10 10 10 10 10 10 10 10 10 10 10 10 η η b dπ/4 f h 2 1 3π/16 1.5 Σ Π , 0.5 ψ π/8 κ 1 0.5 π/16 0 0 0 0 π/6 π/3 π/2 −5 −4 −3 −2 −1 0 π/8 π/4 3π/8 π/2 α 10 10 10 10 10 α η 5 ulation is qualitatively different when α approaches and 0.5 a b exceeds π/2. The polar-liquid/flocking phase, does not exist anymore above α ∼0.9(π/2). The system under- T 0.4 goes a single transition from an isotropic to a spinning strtiactrei.tiFcaolrpαoTin∼t,0w.h9(eπre/2t)hethfleotcrkainngsi,ttiohneospccinunrsintghraonudgthhae δ r0.3 isotropic phases coexist. In addition, above α the spin- T ning phase displays marked qualitative differences from 0.2 the one characterized above for small values of α: the polarization and the spin do not saturate to 1 as η → 0 0.1 −4 −2 0 10 10 10 (aspreviouslynotedforα=π/2). Inadditionthecurva- η tureofthetrajectoriesdeviatesfromthenaivemean-field prediction, Fig. 3h. FIG. 4: Color online. Swarming at the onset of collective In order to elucidate this rich phase behavior, we now spinning. (a) The variations of δr echo the condensation of investigate the linear stability of the two ordered states. theparticlesintoasinglecompactswarmundergoingdirected Let us first consider an ensemble of particles forming a motion. (b) Snapshot of the instantaneous particle positions polar liquid oriented along θ = 0. Far from a transition and orientations in the swarming state where Π=0.977 and point, the angles weakly deviate from θi = 0 and the Σ = 0.063. α = 0.8π/2, η = 3×10−3, ρ = 8, v0 = 0.5 and angular dynamics reduces to: L=16 (N =2048). θ˙ (t)∼sinα(cid:104)σ (cid:105) −cosα(cid:104)θ −θ (cid:105) +(cid:112)2ηξ (4) i i j∈Ωi i j j∈Ωi i In this polarized but non-spinning state, the individual in a narrow range of the noise amplitude η, typically a spins undergo uncorrelated fluctuations. Therefore, for few percent above η . Unlike the band-shape patterns FS any finite interaction radius, the first term on the right- observed in the standard Vicsek models, these compact hand side acts as an additional angular noise that im- swarms are not surrounded by a sea of isotropic parti- pedes the velocity alignment, in stark contrast with the cles, but freely propagate in an empty space. To our intuitive picture that one could have about the effect knowledge, this condensation is the first evidence of the of anticipation. The magnitude of this effective angu- self-organisationofmotileparticlesintocompactswarms lar noise increases with α which qualitatively explains without any attractive interactions. why the flocking transition occurs for smaller values of η We close this section by pointing two questions that as α increases. In addition, it readily follows from Eq. 4 readily arise from the above analysis. First the nature that the effective angular stiffness, cosα, is negative for of the transition between the four phases (gas, flocking, α > π/2. Hence the homogeneous polar-liquid phase is spinning and swarming states) remains to be elucidated linearly unstable to angular fluctuations. In agreement (critical, firstorder, orsharpcross-over). Answeringthis withthephasediagramshowninFig. 3a,nopolar-liquid questionwouldrequirestudyingmuchlargersystemsand phase exists for α>π/2. a systematic finite-size analysis [17], which goes beyond Let us now repeat the same analysis for the spin- thescopeofthepresentpaper. Asecondnaturalquestion ning phase. Introducing the angular fluctuations δθi ≡ concerns the robustness of these phases with respect to θi ±tsinα deep in the ordered phase, the orientational the noise structure. We focused here on the case of an dynamics reduces to: angular noise. Whereas the present results are expected to hold for a vectorial noise as well, the impact of more δθ˙i(t)∼−cosα(cid:104)δθi−δθj(cid:105)j∈Ωi +(cid:112)2ηξi. (5) complex multiplicative noises, such as the one found to be relevant to locust swarms in [22], remains an open Againwefindthattheorientationalfluctuationsaresta- question. bilized for α<π/2 and amplified otherwise. This stability analysis is consistent with the strong fluctuations of the individual trajectories found for α = π/2: the spinning state is linearly unstable for large anticipation an- IV. CONCLUSION AND PERSPECTIVES gles. The ordered spinning phase numerically found for α (cid:38) π/2 is therefore stabilized by the nonlinear nature We conclude this paper by addressing potential ap- of the spin-angle interactions and cannot be captured by plications of our findings to multi-agent robotics. In a linear-stability analysis alone. this context, biomimetic strategies have always been ap- The flocking and the spinning phases are separated pealing to achieve collective intelligence. However man- by another dynamical state where the polar-liquid phase madeprogramableunitscouldexploitavirtuallyinfinite condenses into compact swarms undergoing coherent di- number of alternative strategies to accomplish emergent rected motion. This behavior is characterized and illus- tasks. In this paper, we have demonstrated that sim- trated in Figs. 4a and 4b respectively, see also Sup- ple artificial interaction can be effectively used to self- plementary Movie 4 [18]. This swarming phase exists organize an entire population. By coupling the spin and 6 the orientation of motile particles, three symmetries can moving coherently could stop and explore the neighbor- be spontaneously broken (rotation, translation and spin- hoodofagivenspotwithouthavingtoworryaboutpos- reversal) thereby resulting in three spatiotemporal pat- sible collisions between the motile units. Experimental terns: homogeneousandcoherentdirectedmotion(flock- investigations along these lines are now needed to ad- ing),synchronouscircularmotion(spinning)andconden- dress the relevance of these simple interaction rules to sation into a compact group propagating in a coherent devise functional robotic swarms. fashion (swarming). In addition, the transition from a swarmingtoaspinningstatecanbetriggeredbyaminute increaseoftheanticipationangleαthatcharacterizesthe spin-orientationcoupling. Thereforethissharptransition Acknowledgments could provide a useful design rule to arrest and disperse a motile swarm without having to stop the individual propulsion. Since all the particles undergo synchronous DB acknowledges support from Institut Universitaire circular orbits in the spinning state, a swarm of particles de France, and ANR project MiTra. [1] T. Vicsek, A. Cziro´k, E. Ben-Jacob, I. Cohen, and O. 021908 (2009). Shochet, Phys. Rev. Lett. 75, 1226 (1995). [14] A. Cavagna, .L Del Castello, I. Giardina, T. Grigera, A. [2] T.VicsekandA.Zafeiris.PhysicsReports517,71(2012). Jelic,S.Melillo,T.Mora,L.Parisi,E.Silvestri,M.Viale [3] M.C.Marchetti,J.F.Joanny,S.Ramaswamy,T.B.Liv- and A. M. Walczak, arXiv:1403:1202 (2014). erpool, J. Prost, M. Rao, and R.A. Simha, Rev. Mod. [15] F.Peruani,A.DeutschandM.Bar,Eur.Phys.J.–Spec. Phys. 85, 1143 (2013). Top. 157, 111–122 (2008). [4] A. Bricard, J.B. Caussin, N. Desreumaux, O. Dauchot, [16] G. Grégoire, and H. Chaté, Phys. Rev. Lett. 92, 025702 and D. Bartolo, Nature, 503, 95 (2013). (2004). [5] J. Deseigne, O. Dauchot and H. Chaté, Phys. Rev. Lett. [17] H. Chaté, F. Ginelli, G. Grégoire, and F. Raynaud, 105, 098001 (2010). Phys. Rev. E 77, 046113 (2008). H. Chaté, F. Ginelli, [6] V. Schaller, C. Weber, C. Semmrich, E. Frey, and A.R. G.Grégoire,F.Peruani,andF.Raynaud.,Eur. Phys. J. Bausch, Nature 467, 73 (2010). B. 64, 451 (2008) [7] N. Kumar, H. Somi, S. Ramaswamy and A.K. Sood, [18] Supplementary videos showing the three non-trivial arXiv:1402.4262 (2014). phases of the self-propelled particles. [8] A.CavagnaandI.Giardina,Ann.Rev.Cond.Mat.Phys. [19] J.-B.Caussin,A.Solon,A.Peshkov,H.Chaté,T.Daux- 5 183–207 (2014). ois,J.Tailleur,V.VitelliandD.Bartolo,Phys.Rev.Lett. [9] A. Berdahl, C. Torney, C. C. Ioannou, J. J. Faria and I. 112, 148102 (2014). D. Couzin, Science 339, 574–576 (2013). [20] C. W. Reynolds, Computer Graphics 21, 25–34 (1987). [10] D.S.Calovi,U.Lopez,S.Ngo,C.Sire,H.ChatéandG. [21] J. Toner, and Y. Tu, Phys. Rev. Lett. 75, 4326 (1995). Theraulaz, New J. Phys. 16, 015026 (2014). J. Toner, Y. Tu, and M. Ulm, Phys. Rev. Lett. 80, 4819 [11] C.K.HemelrijkandH.Hildenbrandt,Interface Focus2, (1998) 726–737 (2012). [22] C. A. Yates et al Proc. Natl. Acad. Sci. USA 106, [12] U. Lopez, J. Gautrais, I.D. Couzin and G. Theraulaz, 54645469 (2009). Interface Focus 2, 693–707 (2012). [13] P. Szabò, M. Nagy and T. Vicsek, Phys. Rev. E 79,