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Onset of collective and cohesive motion Guillaume Grégoire and Hugues Chaté CEA – Service de Physique de l’E´tat Condensé, CEN Saclay, 91191 Gif-sur-Yvette, France and Pôle Matière et Systèmes Complexes CNRS FRE 2438, Université Paris VII, Paris, France. (Dated: February 2, 2008) 4 0 We study the onset of collective motion, with and without cohesion, of groups of noisy self- 0 propelled particles interacting locally. We find that this phase transition, in two space dimensions, 2 isalwaysdiscontinuous,includingfortheminimalmodelofVicseketal. [Phys. Rev. Lett. 75,1226 n (1995)] for which anon-trivialcritical pointwas previously advocated. Wealso show thatcohesion a is always lost near onset, as a result of theinterplay of density,velocity, and shape fluctuations. J 3 PACSnumbers: 64.60.Cn,05.70.Ln,82.20.-w,89.75.Da 1 ] Collectivemotioncanbeobservedatalmosteveryscale particles or long-range or global forces [12]. h innature,fromthefamiliarhumancrowds[1],birdflocks In this Letter, we study the onset of collective motion c e and fish schools [2], to unicellular organisms like amoe- with and without cohesion in this very general setting, m bae and bacteria [3], individual cells [4], and even at mi- trying to assess the universality of the results of Vicsek croscopic level in the dynamics of actin and tubulin fil- et al. In both cases, we find that the onset of collec- - t aments and molecular motors [5, 6]. Whereas biologists tive motion in the VM and related models is actually a t tendtobuilddetailedrepresentationsofaparticularcase, discontinuous (“first-order”) and that its apparent con- s . theubiquityofthephenomenonsuggestsunderlyinguni- tinuous character is due to strong finite-size effects. We t a versal features and thus gives weight to the bottom-up also show that without cohesion, the transition point is m modeling approach usually favored by physicists [7]. nevertheless accompanied by a non-trivial superdiffusive - Inthisrespect,thesimplemodelintroducedbyVicsek behaviorofparticleswhich,weargue,couldbemeasured d and collaborators [8] stands out because of its minimal experimentally. In the presence of cohesion, our study n o character and the a priori least-favorable conditions in reveals that the onset of collective motion is the theater c which it is defined. In the Vicsek model (VM), identical of a complex interplay between density, velocity, sound [ pointwise particles move at constant velocity and inter- and shape modes, giving rise to fascinating dynamics. 1 act locally by trying to align their direction with that of The originalVM is defined as follows: identical point- v neighbors. Remarkably,eveninthepresenceofnoiseand wise particles move synchronously at discrete timesteps 8 in the absence of leaders and global forces, orientational ∆t = 1 by a fixed distance v0. In two space dimen- 0 long-range order arises, i.e. collective motion emerges, if sions —to which we restrict ourselves in the following— 2 1 the density ofparticles is high enoughor,equivalently, if the direction of motion of particle j is just an angle θj, 0 the noise is weak enough. The existence of the ordered calculated from the previous directions of all particles k 4 phasewaslater“proved”byarenormalisation-groupap- within an interaction range r0 =1>v0∆t: 0 proach based on a phenomenological mesoscopic equa- at/ tion [9]. More recently, this work was extended to the θjt+1 =argXeiθkt+ηξjt , (1) case where the ambient fluid is taken into full account, m k∼j  yielding novel mesoscopic equations for suspensions of - where ξt is a delta-correlated white noise (ξ ∈ [−π,π]). d self-propelled particles [10]. j Thisintroducesatendencytoalignwithneighboringpar- n The nature of the non-equilibrium phase transition o ticles, with two simple limits: in the absence of noise, to collective motion, however, is not well established. c interacting particles align perfectly, quickly leading to Vicsek et al. concluded from numerical simulations in v: two andthree dimensions that it is continuous (“second- completeorientationalorder. Formaximalnoise(η =1), i particles follow random walks. The transition that nec- X order”)andcharacterizedby asetof criticalindices, but essarily lies in between these two regimes can be charac- these results remain somewhat crude [11], even though r terized by the following instantaneous order parameter: a the undeniably minimal character of the VM makes it a good candidate for representing a universality class. sirMabolereionvgerre,dfrieonmt ammissoidnegliinngtphoeinVtMofivsiecwoh,easnioonf:tewnhdeen- ϕt ≡ N1 (cid:12)(cid:12)(cid:12)XN eiθjt(cid:12)(cid:12)(cid:12) (2) puttogetherinaninfinitespace,particlesdonotstayto- (cid:12)(cid:12)j=1 (cid:12)(cid:12) (cid:12) (cid:12) getherandflyapart. Inotherwords,nocollectivemotion where N is the total number of particles. is possible in the zero-densitylimit of the VM. Recently, Varyingeitherthenoisestrengthη ortheparticleden- we have shown how one can ensure cohesion in simple sity ρ =N/L2 in periodic domains of linear size L, Vic- models derivedfrom the VM without resorting to leader sek et al. found that hϕi varies continuously across the 2 (a) (b) (a) (b) 1 <ϕ> 0.8 G 1 G 1 ρ ϕ 1 ⊥ 0 0.5 0.5 0.4 -1 L=128 0.5 0 L=256 -2 L=512 -0.5 L=1024 η x⊥ η η -3 0 -1 0 0 0.12 0.18 0.24 -512 -256 00 256 512 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 (c) (d) FIG. 1: Onset of collective motion in cohesion-less models 1 PDF η=0.190 η=0.191 (1) (original VM, circles) and (3) (vectorial noise, squares). η=0.193 Variation of order parameter ϕ (a) and Binder cumulant G (b) with the noise strength η. (v0 =0.5, L=32, ρ=2, and equivalent statistics for both models.) 0.5 ϕt transition,suggestingthe existenceofacriticalpoint[8]. 0 Studying finite-size effects, they estimated a set of scal- 0 0.05 0.1 0.15 0.2 ing exponents. Interested in assessing the universality of these results and possibly improving these estimates, FIG.2: Discontinuouscharacteroftheonsetofcollectivemo- we first introduced simple modifications of the original tion in theoriginal VM at ρ= 18. (a): G vsη at various system sizes. (b): transverse density (bottom curve) and order- VM such as changing v0 or adding a repulsive force be- parameterprofile(topcurve)intheorderedphase(L=1024, tween particles to give them a finite extent. Using the η = 0.18) (c): probability distribution function (PDF) of ϕt finite-size scaling Ansatz appropriate for XY-model like near the transition point, t ∈ [τ;500τ], here the correlation systems, domain sizes, and particle numbers similar to time is τ ≃105 [13], L=512. (d) snapshot of coarse-grained those used in [8], but with much better, well-controlled densityfieldindisorderedphaseatthreshold,ρ=2,L=256. statistics, we were only able to estimate a roughly co- Thearrowsindicatethedirectionofmotionofdense,ordered herent set of critical exponents after allowing for rather regions. strong corrections to scaling [13]. For modeling reasons, we also changed the way noise is incorporated in the system. In (1), particles make an change the order of the transition upon introducing this error when trying to take the new direction they have nonlinear term. Considering in addition the strong cor- perfectly calculated (“angular noise”). One could argue rections to scaling found with angular noise, we strived that, rather, errors are made when estimating the in- toreachlargersystemsizesinsomeofthese cases,albeit teractions, for example because of a noisy environment. atthe costof statisticalaccuracy[13]. The conclusionof This leads to change Eq.(1) into, e.g.: thesenumericaleffortsisthatthe transitionis discontinuous in all cases, with finite-size effects being somewhat weaker at low densities. As an example, the behavior of θjt+1 =argXeiθkt +ηntjeiξjt (3) G with increasing system size shown in Fig. 2a for the k∼j  original VM at ρ = 1 reveals the characteristic fall to 8 where nt is the current number of neighbors of particle negative values. The distribution function of ϕt is bi- j modal around threshold, without any intermediate uni- j. Inthis caseof“vectorialnoise”,the onsetofcollective motion is discontinuous: for large-enough system sizes, modal regime (Fig. 2c). Thus, the continuous transition reported by Vicsek et al. is only apparent. hϕi jumps abruptly to zero as η is decreased, whereas it varies smoothly in the original VM (Fig. 1a). This In the ordered phase, the particles are organized in is perhaps best seen from the behavior of the so-called density waves moving steadily in a disordered “vapour Binder cumulant G = 1−hϕ4i/3hϕ2i2 (Fig. 1b). In the pressure”backgroundof well-defined asymptotic density case ofvectorialnoise,G falls to negative values nearηc, (Fig. 2b). These solitary waves become metastable to a the sign of a discontinuous transition, together with the long-wavelength longitudinal instability below the den- phase coexistence expected then. sity threshold ηc (defined to be located at the minimum Going from angular to vectorial noise is indeed a ofG),leadingtoanhysteresisloop. Atthresholdandbe- less innocent modification than those mentioned earlier: low, the disordered phase consists of nucleated ordered in model (3), locally-ordered regions are subjected to patches competing in space and time (Fig. 2d). weaker noise than disordered ones. However, it was un- Atthreshold,inthedisorderedphase,auniversalnon- clear to us what precisely would be the mechanism to trivial algebraic scaling law is nevertheless found: the 3 superdiffusive behavior of particles already reported by (a) (b) us in [14] is valid in all cases. Trajectories then con- <ϕ>,<n/N> ω,<ϕ>×4 sist of “flights”, occuring when a particle is caught in a 1 3 moving ordered patch, separated by normal diffusion in ln(PDF) the disordered regions. The mean square displacement 1 <ϕ>,< ∆ > 8 2 of particles hδr2i varies like tα with α=1.65(5). 4 0.5 ln t We now turn to the onset of collective motion in the rot α 1 0 presence of cohesion. As shown in [12], the cohesion of 0 0 3 6 9 a population of particles can be maintained without re- 2 2.4α α solving to long-rangeor globalinteractions. In the spirit 0 0 1.5 2 2.5 1.6 1.8 2 of the VM, and following [15], a two-body short-range interactionforcecompetingwiththe alignmenttendency FIG. 3: Onset of motion of cohesive groups in model (4) is introduced, leading to the following model: with η = 1, v0 = 0.05. (a) hϕi and n/N (normalized size of largest connected cluster) vs α (ρ = 1 , β = 20 (liquid 16 θjt+1 =argαXeiθkt +βXfjtkeiθjtk +ηntjeiξjt (4) spehta:sseo),liddagsrhoeudpli(nβes=: N84)=o4f0N96=; s4o0li9d6lipnaerst:icNles=; d1a6s3h8e4d).linIne-:  k∼j k∼j  hϕi;solid line: relativediffusionof initialy neighboringparti- where α and β control the strength of alignment and cles∆≡hn1j Pk∼j(1−rj2k(t)/rj2k(t+T))ij,t whereT ≈20N cohesion,θt isthedirectionofthevectorlinkingparticle (∆ ≃ 1 in the liquid phase, while ∆ ≃ 0 in the solid phase, jk see [12]). (b) variation with α of the maximal absolute ro- j to particle k. The interaction force between these two tation angle |ω| averaged over 100 samples of 1000 vortices particles,of amplitude fjtk, is actually repulsive upto an (N = 2048, ρ = 312, β = 30 (liquid phase)). Dashed line:hϕi intermediate equilibrium distance re, with a short-range duringthesameruns. Inset: distributionofrotationtimesat hard-coreatrc andattractiveuptotheinteractionrange thetransition with decay exponent∼1.3. r0. In the following, as in [12], we used: fjk = −41r∞rjak−−rree iiff rrcjk<<rjrkc <, ra , (5) omcecduirastethpeeaokrd(earropuanradmαet=er1f.a7llisnbFaicgk.,3lae)a.viInngcraenasiinntgerα- 1 if ra <rjk <r0 . further,hϕirisesagainandfinallyjumpstohϕi=1when  where rjk is the distance between j and k, with rc = full cohesion is recovered(for α=1.88 in Fig. 3a). This 0.2, re = 0.5, and ra = 0.8. Note that vectorial noise discontinuousjump is the true locationofthe transition: was chosen in (4), in the hope of reaching asymptotic For an infinite group, the onset of motion must occur properties more easily. abruptly near this value, as the precursory features de- The above model has three main parameters, α, β, scribed above disappear because the population divides and η, only two of which are independent. The phase into infinitely-many subgroups whose influences average diagraminthe (α,β) plane(withη =1fixedarbitrarily) themselves out. Meanwhile, cohesion is only lost at the waspresentedin[12],where,moreover,onlyneighborsin transition point in this asymptotic picture. the Voronoisenseareconsideredinthesums of(4)). For Thebreakupoflargecohesivegroupsaroundthreshold large-enoughβ, cohesionismaintained,eveninthe zero- is probably closely related to what happens in the case density limit. This “gas/liquid”transitionisfollowed,at without cohesion: the subgroups connected by filaments largerβ values, by the onset of positional(quasi-) order, may correspond to the ordered patches seen in the dis- i.e. a “liquid/solid” transition. For large α, these liquid ordered phase near threshold in Fig. 2d. The breakup orsolidcohesivegroupsmove,whereastheyremainstatic itself can be seen as resulting from the maximal effect (up to finite-size fluctuations) for small α. of acoustic modes on the shape of the group [13]. Also In the “liquid case” (intermediate β values), the on- affecting the shape dynamics are rotational modes: the setofmotionisaccompaniedbyalossofcohesion: while subgroups seen in Fig. 4a not only move but they also smallgroupssetinmotionsmoothlywithoutbreakingup rotate slowly [16]. Rotation is not steady, but intermit- (Fig.3a,dashedlines),largergroupsgraduallysubdivide tent. We recorded the rotation times and their corre- intoseveralpartsofroughlyequivalentsizelinkedbyfila- sponding angles. Extremalstatistics analysisrevealthat mentarystructures, incontrastwith their more compact the tendencytorotateismaximalatthe onsetofmotion shapes before and after onset (Fig. 4). The filaments (Fig. 3b). Moreover,at threshold, the distribution of ro- themselvesarequitestatic(Fig.4d)butaredisplacedby tationtimesisalgebraicwithadecayexponentsuchthat the subgroups which move coherently so that they even- it has no finite mean (inset of Fig. 3b). tuallybreakup,asindicatedbythedipinthenormalized Theonsetofmotionofthe “solid”groups(largeβ val- largestconnected cluster size n/N in Fig. 3a. Increasing ues) is accompanied by a loss of positional order: these α, large groups follow the same precursor of the tran- crystalsmeltnearthetransition(insetofFig.3a). Given sition as smaller groups, but when their fragmentation the above results in the liquid case, one can expect very 4 assays” consisting of grafted molecular motors such as kinesin(resp. myosin)moving filaments made oftubulin (resp. actin)mightprovidethe simplestsetting inwhich toinvestigatesuperdiffusionatonset,giventhe available observation techniques [5, 6]. [1] See, e.g.: D. Helbing, I. Farkas, and T. Vicsek, Nature 407, 487 (2000); Phys. Rev.Lett. 84, 1240 (2000). [2] J.K. Parrish and W.M. Hamner (Eds.), Three dimen- sional animals groups , (Cambridge University Press, Cambridge, 1997), and references therein. [3] J.T.Bonner,Proc.Natl.Acad.Sci.USA95,9355(1998); M.T. Laub and W.F. Loomis, Mol. Bio. of the Cell 9, 3521 (1998). FIG. 4: Typical shape of a liquid cohesive group of 16384 [4] R. Kemkemer et al., Eur.Phys. J. E 3, 101 (2000). particles(model(4),ρ= 116,β =20arrowsindicatedirection [5] F. Nédélec, Ph.D. thesis, Université Paris 11, 1998; F. of motion). 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Grégoire, H. have no specific prediction as to the onset of motion of Chaté,andY.Tu,Phys.Rev.Lett.86,556(2001);Phys. Rev. E64, 011902 (2001). these cohesive groups [17]. Without cohesion, however, [15] C. W. Reynolds,Comput. Graph. 21, 25 (1987). theuniversalsuperdiffusivebehaviorobservedinthe dis- [16] Mpeg movies are available on request to gre- ordered phase near threshold could be observed experi- [email protected]. mentally. As already suggested in [14], bacteria such as [17] In the other hand, interesting predictions were made by E. Coli might be good self-propelled particles. Human Tu and Toner [9] for theordered phase. melanocytesalsolookpromisinginthis respectasshown remarkablybythe groupofGruler[4]. Finally, “motility