We study, in two space dimensions, the large-scale properties of collections
of constant-speed polar point particles interacting locally by nematic
alignment in the presence of noise. This minimal approach to self-propelled
rods allows one to deal with large numbers of particles, revealing a
phenomenology previously unseen in more complicated models, and moreover
distinctively different from both that of the purely polar case (e.g. the
Vicsek model) and of active nematics.Comment: Submitted to Phys. Rev. Let

Baer, M.

Chate, H.

Ginelli, F.

Peruani, F.

English

arXiv

International audienceWe study, in two space dimensions, the collective properties of constant-speed polar point particles interacting locally by nematic alignment in the presence of noise. This minimal approach to self-propelled rods allows one to deal with large numbers of particles, which exhibit a rich phenomenology distinctively different from all other known models for self-propelled particles. Extensive simulations reveal long-range nematic order, phase separation, and space-time chaos mediated by large-scale segregated structures

Ginelli, Francesco

Peruani, Fernando

Baer, Markus

Chaté, Hugues

HAL-CEA

Large-Scale Collective Properties of Self-Propelled RodsFrancesco Ginelli, Fernando Peruani, Markus Baer, Hugues Chate´To cite this version:Francesco Ginelli, Fernando Peruani, Markus Baer, Hugues Chate´. Large-Scale CollectiveProperties of Self-Propelled Rods. Physical Review Letters, American Physical Society, 2010,104 (18), pp.184502. <10.1103/PhysRevLett.104.184502>. <hal-00905223>HAL Id: hal-00905223https://hal.archives-ouvertes.fr/hal-00905223Submitted on 18 Nov 2013HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.arXiv:0911.1924v1  [cond-mat.soft]  10 Nov 2009Large-scale collective properties of self-propelled rodsFrancesco Ginelli,1, 2 Fernando Peruani,1 Markus Ba¨r,3 and Hugues Chate´11Service de Physique de l’E´tat Condense´, CEA–Saclay, 91191 Gif-sur-Yvette, France2Institut des Syste`mes Complexes de Paris Iˆle-de-France, 57-59 rue Lhomond, 75005 Paris, France3Physikalisch-Technische Bundesanstalt, Abbestrasse 2-12, 10587 Berlin, Germany(Dated: November 10, 2009)We study, in two space dimensions, the large-scale properties of collections of constant-speedpolar point particles interacting locally by nematic alignment in the presence of noise. This minimalapproach to self-propelled rods allows one to deal with large numbers of particles, revealing aphenomenology previously unseen in more complicated models, and moreover distinctively differentfrom both that of the purely polar case (e.g. the Vicsek model) and of active nematics.PACS numbers: 05.65.+b, 87.18.Hf, 87.18.GhCollective motion is an ubiquitous phenomenon ob-servable at all scales, in natural systems [1] as well ashuman societies [2]. The mechanisms at its origin canbe remarkably varied. For instance, they may involvethe hydrodynamic interactions mediated by the fluid inwhich bacteria swim [3], the long-range chemical signal-ing driving the formation and organization of aggregationcenters of Dictyostelium discoideum amoeba cells [4], orthe local cannibalistic interactions between marching lo-custs [5]. In spite of this diversity, one may search forpossible universal features of collective motion, a contextin which the study of “minimal” models is a crucial step.Recently, the investigation of the simplest cases, wherethe problem is reduced to the competition between a lo-cal aligning interaction and some noise, has revealed awealth of unexpected collective properties. For example,constant speed, self-propelled, polar point particles withferromagnetic interactions subjected to noise (as in theVicsek model [6]) can form a collectively moving fluctu-ating phase with long-range polar order even in two spa-tial dimensions [7], with striking properties such as spon-taneous segregation into ordered solitary bands movingin a sparse, disordered sea, or anomalous (“giant”) den-sity fluctuations [8]. In contrast, active apolar particleswith nematic interactions only exhibit quasi-long-rangenematic order in two dimensions with segregation takingthe form of a single, strongly-fluctuating, dense structurewith longitudinal order and even stronger density fluctu-ations than in the polar-ferromagnetic case [9, 10, 11].Noting that these differences reflect those in the lo-cal symmetry of particles and their interactions, a thirdsituation can be defined, intermediate between the po-lar ferromagnetic model and the apolar nematic one,that of self-propelled polar particles aligning nematically[12]. Such a mechanism is typically induced by volumeexclusion interactions, when elongated particles collid-ing almost head-on slide past each other, as illustratedschematically in Fig. 1. Thus, self-propelled polar pointparticles with apolar interactions can be conceived as aminimal model for self-propelled rods interacting by in-elastic collisions [13, 14, 15]. Other relevant situationsFIG. 1: Nematic alignment of polar particles illustrated byinelastic collisions of rods. Particles incoming at a small angle(left) align “polarly”, but those colliding almost head-on slidepast each other, maintaining their nematic alignment (right).can be found in a biological context, such as glidingmyxobacteria moving on a substrate [16], or microtubulesdriven by molecular motors grafted on a surface [17].In this Letter, we study collections of constant-speedpolar point particles interacting locally by nematic align-ment in the presence of noise. The simplicity of thismodel allows us to deal with large numbers of particles,revealing a phenomenology previously unseen in morecomplicated models sharing the same symmetries [13, 14,15]. Our study, restricted to two space dimensions, showsin particular collective properties distinctively differentfrom both those of polar-ferromagnetic case and of ac-tive nematics: only nematic order arises in spite of thepolar nature of the particles, but it seems genuinely long-ranged. Spontaneous density segregation is also observedbut it is of a different type and it splits both the (nemat-ically) ordered and the disordered phase in two. In thefollowing, we characterize these four phases and discussthe three transitions separating them.Our model consists of N point particles moving off-lattice at constant speed v0. In two dimensions, particlej is defined by its (complex) position rtj and orientationθtj , updated at discrete time steps according toθt+1j = arg∑k∼jsign[cos(θtk − θtj)]eiθtk+ ηξtj (1)rt+1j = rtj + v0 eiθt+1k , (2)where the sum is taken over all particles k within unit dis-tance of j (including j itself), and ξ is a white noise uni-2(a) (b) (c) (d) (e)I II III IVFIG. 2: (color online) (a-c) Typical steady-state snapshots at different noise values (linear size L = 2048). (a) η = 0.08, (b)η = 0.10, (c) η = 0.13, (d) η = 0.168, (e) η = 0.20. Arrows indicate the polar orientation of particles (except in (d)); only afraction of the particles are shown for clarity reasons. For a movie corresponding to (d) see [21].formly distributed in[−pi2, pi2][18]. (A continuous-timeversion of this model can be found in [19].) The systemhas two main control parameters: the noise amplitude η,and the particle density ρ = N/A, where A is the do-main area. We consider periodic boundary conditions.Polar and nematic order can be characterized by meansof the two time-dependent global scalar order parametersP (t) = |〈exp(iθtj)〉j | (polar) and S(t) = |〈exp(i2θtj)〉j |(nematic), as well as their asymptotic time averagesP = 〈P (t)〉t and S = 〈S(t)〉t.In this work, we focus on the behavior of the system forρ = 18and v0 = 1/2, varying η. We start with a brief sur-vey of the stationary states observed in a square domainof linear size L = 2048 (Figs. 2-3). Despite the polarnature of the particles, only nematic orientational orderarises at low noise strengths, while P always remains nearzero (not shown). This is in agreement with the findingsof [20]. Both the ordered and the disordered regimes aresubdivided in two phases, one that is spatially homoge-neous (Figs. 2(a,e)), and one where spontaneous densitysegregation occurs, leading to high-density ordered bandsalong which the particles move back and forth (Figs. 2(b-d)). A total of four phases is thus observed, labeled I toIV by increasing noise strength hereafter. Phases I and IIare nematically ordered, phases III and IV are disordered.Below, we study these four phases more quantitatively.Phase I, present at the lowest η values, is orderedand spatially homogeneous (Fig. 2a). Its nematic or-der, which arises quickly from any initial condition, isdue to the existence, at any time, of two subpopulationsof particles that migrate in opposite directions (Fig. 4a).Statistically of equal size, they constantly exchange par-ticles, those which “turn around”. These events occurat exponentially-distributed times τ (Fig. 4b). Increas-ing system size, the nematic order parameter S is almostconstant, decaying slower than a power law (Fig. 4c).A good fit of this decay is given by an algebraic ap-proach to a constant asymptotic value S∗. Thus, our dataseem to indicate the existence of true long-range nematicorder. (Quasi-long-range order, expected classically fortwo-dimensional nematic phases, is characterized by analgebraic decay of S.) A discussion of this striking factis given below. Finally, as expected on general groundsfor homogeneous ordered phases of active particles [10],phase I exhibits so-called giant number fluctuations: thefluctuations ∆n2 = 〈(n − 〈n〉)2〉 of the average numberof particles 〈n〉 = ρℓ2 contained in a square of linear sizeℓ follow the power law ∆n ∼ 〈n〉α with α > 12(Fig. 4d).Our estimate of α is compatible to that measured forpolarly ordered phases α = 0.8 [8].Phase II differs from phase I by the presence, inthe steady-state, of a low-density disordered region. Inlarge-enough systems (104-105 particles for the parame-ters used here), a narrow, low density channel emerges(Fig. 2b) when increasing η from phase I. It becomeswider at larger η values, so that one can then speak of ahigh-density ordered band, typically oriented along oneof the main axes of the box, amidst a disordered sea(Fig. 2c). Particles travel along the high-density band,turning around or leaving the band from time to time.Within the band, nematic order with properties similar0.01 0.02 0.03η210-310-210-1100SI IIIIIIV∆SFIG. 3: (color online) Nematic order parameter S (in black)and its rms fluctuations ∆S (in red) as function of the squarednoise amplitude η2 for a square domain of linear size L =2048. Here, and throughout the paper, time-averages are overat least 106 timesteps.3102 104L0.800.840.87S(c)102 104 106<n>102104 ∆n(d)0 pi 2piθ00.5P(θ)(a)LS - S*0 30000 60000τ10-610-3 P(τ)(b)χFIG. 4: (color online) Phase I (homogeneous nematic order,η = 0.095). (a) Polar orientation probability distribution ina system of size L = 2048. (b) Distribution of particles tran-sition times τ between the two peaks of (a) for three differ-ent system sizes L = 512, 1024, and 2048 (black, red, andblue lines respectively). (c) Nematic order parameter S vssystem size L in square domains. The vertical red dashedline marks the persistence length χ ≈ 4400 (see text). In-set: S − S∗ = 0.813063 vs L (red dashed line: L−2/3 decay).(d) Number fluctuations ∆n as a function of average particlenumber 〈n〉 (see text) in a system of size L = 4096 (dashedline: algebraic growth with exponent 0.8).to those of phase I is found (slow decay of S with systemsize, giant number fluctuations). The (rescaled) bandpossesses a well-defined profile with sharper and sharperedges as L increases (Fig. 5a). The fraction area Ω oc-cupied by the band is thus asymptotically independentof system size, and it decreases continuously as the noisestrength η increases (Fig. 5b).In phase III, spontaneous segregation into bands stilloccurs (for large-enough domains), however these thin-ner bands are unstable and constantly bend, break, re-form, and merge, in an unending spectacular display ofspace-time chaos (Fig. 2d) [21]. Correspondingly, S(t)fluctuates strongly (Fig. 3) and on very large time scales(Figs. 6a). Nevertheless, these fluctuations behave nor-mally (i.e. decrease like 1/√N , Fig. 6b). Thus, thespace-time chaos self-averages, making phase III a bonafide disordered phase, albeit one with huge correlationlengths and times.Phase IV, observed for the highest noise strengths, ex-hibits local and global disorder on small length- and time-scales, and is spatially homogeneous (Fig. 2e).We now discuss briefly the nature of the three transi-0 0.5 1 x/L00.2ρ⊥00.51S⊥(a)0.1 0.15η00.51Ω(b)I II IIIFIG. 5: (color online) Phase II (stable bands) (a) Rescaledtransverse profiles in square domains of linear size L = 512(black), 1024 (red), and 2048 (blue) at η = 0.14. (Data av-eraged over the longitudinal direction and time, translated tobe centered at the same location.) Bottom: density profiles.Top: nematic order parameter profiles. (b) Surface fraction Ωas a function of noise amplitude η (defined here as the widthat mid-height of the rescaled S profile).tions that separate the four observed phases (details willappear elsewhere [22]) . The I–II transition, located atηI−II ≃ 0.098(2), is characterized by the emergence of anarrow low-density disordered channel. Within phase II,the emergence of these structures from disordered initialconditions is reminiscent of a nucleation process. Eventhough the emerging channels might occupy an arbitrar-ily small proportion of space near the transition (Ω ∼ 1for η >∼ ηI−II), they seem to possess a minimum absolutewidth. These facts suggest a discontinuous I–II transi-tion. The transition between phase II and III, locatednear ηII−III ≃ 0.163(1), constitutes the order-disordertransition of the model. As mentioned above, it resem-bles a long-but-finite wavelength instability of the band(see, e.g., Fig. 6c) and does not appear as a fluctuation-driven phase transition. The disorder-disorder transitionbetween phases III and IV occurs near ηIII−IV ≃ 0.169(1),where the instantaneous order parameter S(t) exhibits abistable behavior between a low value, fast fluctuatingstate typical of phase IV and a larger amplitude, slowlyfluctuating one characteristic of phase III. This bistabil-ity, leading to a bimodal order parameter distribution,suggests a discontinuous phase transition.At this point, the most crucial question is perhaps thatof the stability of the nematic order observed in phasesI and II. Indeed, much of what we described above forlarge but finite systems relies on our conclusion of possi-ble truly long-range (asymptotic) order (Fig. 4c). On theone hand, one could argue that the exponential distribu-tions of flight times between the two opposite polar orien-tations (Fig. 4b) define a finite persistence time τ and acorresponding finite persistence lengthscale χ ≈ v0τ (in-dicated by the blue dashed line in Fig. 4c). Therefore, atscales much larger than χ, the polar nature of our parti-cles could become irrelevant, and the system would then40 106 106 t00.10.2 S(t)(a)105 106 N0.1S(b)(c)FIG. 6: Phase III (unstable bands, η = 0168). (a) Typicalnematic order parameter time series for a system of linearsize L = 2048. (b) S vs N in square domains of increasingsizes. (The dashed line marks a 1/√N decay.) (c) Snapshotof coarse-grained density field during the growth of the insta-bility of an initially straight band in a 2048 × 512 domain.behave like a fully nematic one, with only quasi-long-range order. As of now, we have been able to probe sys-tems sizes up to three or four times the persistence lengthχ. So far, as shown in Fig. 4c, these systems comprisingup to twenty million particles show no sign of breakdownof order. On the other hand, χ is a single-particle quan-tity. Even though it is finite and system size independent,particles travel in rather dense polar packets which haveflights longer than χ. Indeed, the giant density fluctu-ations reported (Fig. 4d) indicate that denser, more or-dered, and hence probably longer-lived packets occur inlarger systems. Unfortunately, packets’ flight times arehard to define and measure [22]. But should this “polarpacket lifetime” diverge with system size, then one wouldhave a mechanism opening the door for the emergence oftrue long-range nematic order. To summarize this dis-cussion, nematic order could break down for sizes muchlarger than χ, but our data (Figs. 4c,d) and the argumentabove suggest the picture of two opposite polar compo-nents each with true long-range order (as in fully-polarmodels [23]) summing up to true nematic order.Further work is thus needed, but most of our resultsare rather robust. For instance, the introduction of somesoft-core short-range repulsion between particles does notmodify our main findings [22]. Thus, these are not dueto the pointwise nature of the particles, and should alsobe observed in previous, more detailed models of self-propelled rods if sufficiently-large populations are con-sidered. We note also that our results, and in particularthe instability and space-time chaotic motion of the spon-taneously segregated bands (phases II and III) [21], arereminiscent of the streaming and swirling regime whichcharacterizes the aggregation of myxobacteria [16, 24]and thus our model could prove relevant in this context.At a more general level, our findings reveal unex-pected emergent behavior among even the simplest situ-ations giving rise to collective motion. Our model of self-propelled polar objects aligning nematically stands out asa member of a universality class distinct from both thatof the Vicsek model [6, 7, 8] and of active nematics [9].Thus, in this out-of-equilibrium context, the symmetriesof the moving particles and of their interactions must beconsidered separately and are both relevant ingredients.We thank J. Toner and S. Ramaswamy for fruitful dis-cussions. This work was partially funded by the FrenchANR projects “Morphoscale” and “Panurge”, and theGerman DFG grants DE842/2, SFB 555, and GRK 1558.[1] Three Dimensional Animals Groups, edited by J.K. Par-rish and W.M Hamner (Cambridge University Press,Cambridge, England, 1997), and references therein.[2] D. Helbing, I. Farkas, and T. Vicsek, Nature (London)407, 487 (2000).[3] A. Sokolov, et al., Phys. Rev. Lett. 98, 158102 (2007).[4] E. Ben-Jacob et al., Adv. Phys. 49, 395 (2000).[5] P. Romanczuk, I.D. Couzin, and L. Schimansky-Geier,Phys. Rev. Lett. 102, 010602 (2009).[6] T. Vicsek et al., Phys. Rev. Lett. 75, 1226 (1995).[7] J. Toner, and Y. Tu, Phys. Rev. Lett. 75, 4326 (1995);Phys. Rev. E 58, 4828 (1998).[8] H. Chate´, et al., Phys. Rev. E 77, 046113 (2008);G. Gre´goire and H. Chate´, Phys. Rev. Lett. 92, 025702(2004).[9] H. Chate´, F. Ginelli, and R. Montagne, Phys. Rev. Lett.96, 180602 (2006).[10] S. Ramaswamy, R.A. Simha, and J. Toner, Europhys.Lett. 62, 196 (2003).[11] S. Mishra and S. Ramaswamy, Phys. Rev. Lett. 97,090602 (2006).[12] The reverse case of active nematic particles with a ferro-magnetic interaction does not make sense.[13] F. Peruani, A. Deutsch, and M. Ba¨r, Phys. Rev. E 74,030904(R) (2006).[14] A. Kudrolli et al., Phys. Rev. Lett. 100, 058001 (2008).[15] H.H. Wensink and H. Lo¨wen, Phys. Rev. E 78, 031409(2008).[16] L. Jelsbak and L. Sogaard-Andersen, Proc. Natl. Acad.Sci. USA 99, 2032 (2002); D. Kaiser, Nat. Rev. Microbiol.1, 45 (2003).[17] F. Ziebert et al., Eur. Phys. J. E 28, 401 (2009).[18] Note that the sign operation in Eq. (1) makes it invariantunder the rotation θtk → θtk+π of any of the neighboringparticles, i.e. the interaction is nematically symmetric.[19] F. Peruani, A. Deutsch, and M. Ba¨r, Eur. Phys. J-Spec.Top.,157, 111 (2008).[20] A. Baskaran and M.C. Marchetti, Phys. Rev. E 77,011920 (2008); Phys. Rev. Lett. 101, 268101 (2008).[21] See EPAPS Document No. XXX for a movie.[22] F. Peruani, F. Ginelli, and H. Chate´, to be published.[23] This is also suggested by our observation that the density5fluctuations are governed by the same exponent as in thepolar case (Fig. 4d).[24] D.R. Zusman, et al., Nature (London) 5, 862 (2007), andreferences therein.

Large-scale collective properties of self-propelled rods

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