Hydrodynamic Equations for Flocking Models without Velocity Alignment
aa r X i v : . [ phy s i c s . b i o - ph ] D ec Hydrodynamic Equations for Flocking Models without Velocity Alignment
Fernando Peruani Universit´e Cˆote d’Azur, Laboratoire J.A. Dieudonn´e,UMR 7351 CNRS, Parc Valrose, F-06108 Nice Cedex 02, France (Dated: December 18, 2019)The spontaneous emergence of collective motion patterns is usually associated with the presenceof a velocity alignment mechanism that mediates the interactions among the moving individuals.Despite of this widespread view, it has been shown recently that several flocking behaviors canemerge in the absence of velocity alignment and as a result of short-range, position-based, attractiveforces that act inside a vision cone. Here, we derive the corresponding hydrodynamic equations ofa microscopic position-based flocking model, reviewing and extending previously reported results.In particular, we show that three distinct macroscopic collective behaviors can be observed: i)the coarsening of aggregates with no orientational order, ii) the emergence of static, elongatednematic bands, and iii) the formation of moving, locally polar structures, which we call worms.The derived hydrodynamic equations indicate that active particles interacting via position-basedinteractions belong to a distinct class of active systems fundamentally different from other activesystems, including velocity-alignment-based flocking systems.
INTRODUCTION
The emergence of self-organized patterns of activelymoving entities, from bacteria to sheep [1–7] and in-cluding human-made active systems [8–11], are system-atically explained invoking the presence of some veloc-ity alignment mechanism that mediates the interactionsamong the moving individuals. This widespread viewon collective motion patterns finds its roots in the so-called Vicsek-like models [12] extensively used to studyflocking patterns [1, 2]. The popularity of these modelsmay be related to the fact that they represent a veryappealing playground for theoretical physicists given theVicsek model’s direct connection to one of the corner-stone models of equilibrium statistical physics: the XYmodel [13]. While some nonequilibrium extensions of theXY model, including the diffusive XY spin model [14, 15],are susceptible of being mapped to their equilibriumcounterpart, flocking models with velocity alignment,such as the original time-discrete Vicsek model [12] andits continuum time version [16], are fundamentally dif-ferent. In idealized homogeneous media, these systemsexhibit long-range orientational order in two dimen-sions [12, 17, 18] and the presence of anomalous densityfluctuations [19, 20]. Although it has been recently shownthat the introduction of a few spatial heterogeneities re-stores a seemingly equilibrium-like behavior with quasi-long-range order and normal fluctuations in two dimen-sions [21, 22], important differences (in two dimensions)remain in both homogeneous and heterogeneous media:the convective transport dictated by the orientation ofthe spin seems to prevent the emergence of topologicaldefects.While the relevance of flocking models based on veloc-ity alignment is undisputed in the realm of active matterand nonequilibrium statistical physics, their systematicapplicability to explain real-world collective motion pat-
FIG. 1. (Color online) Active particles interacting by short-range, attractive force acting inside a vision cone self-organizeinto three distinct macroscopic patterns: (a) aggregates withno orientational order [ β = 2 . √ D θ = 0 . β = 1 . √ D θ = 0 . β = 1 . √ D θ = 0 . N = 10000 particles in a box of linearsize L = 100 with periodic boundary conditions with v = 1and γ = 5. terns, as well as the assumption of the existence of avelocity-alignment mechanism behind all active systemsdisplaying collective effects, has been called into ques-tion by a series of pioneering works [23–31]. In particu-lar, it has been recently shown in [32] that active parti-cles that interact only by a short-range, position-based,attractive force that acts inside a vision cone (VC) dis-play various large-scale self-organized patterns: aggre-gates, nematic bands, and moving, locally polar struc-tures referred to as worms (see Fig. 1). The resemblanceof these emerging patterns to some self-organized behav-iors found in nature [7, 26, 33], together with the sim-plicity of the model, making it amenable to analyticaltreatments, places position-based flocking models as se-rious candidates to both describe real-world active sys-tems and address fundamental theoretical questions ofnonequilibrium (active) systems. Here, we review and ex-tend the derivation of the hydrodynamic equations firstoutlined in [32]. We start by providing a definition of themicroscopic model, formulated in terms of a Langevinequation (Sect. ) to later search for a coarse-grained de-scription of the model by deriving the corresponding non-linear Fokker–Planck equation and performing a momentexpansion (Sect. ). The most subtle step in the derivationof the hydrodynamic equations is the use of local ans¨atzeto close the infinite hierarchy of equations for the ob-tained fields (Sect. ). The procedure allows us to unveilthe three distinct nontrivial macroscopic behaviors of thesystem: a) aggregate formation in the absence of orien-tation order, b) the emergence of nematic bands, and c)the appearance of locally polar structures called worms(Fig. 1). We find that (a) can be described by only onemacroscopic field, the density, while (b) and (c) requireat least two fields: density and local nematic order for(b), and density and local polar order for (c). The analy-sis indicates that position-based flocking models are fun-damentally different from other active systems, includingvelocity-alignment-based flocking systems [12, 16–19, 34–44]. MICROSCOPIC MODELEquations of motion
We consider particles moving at a constant speed,which means that any acceleration experienced by a par-ticle occurs in the direction perpendicular to its instan-taneous velocity. Given the constraint imposed on theparticles, i.e., moving at constant speed, the equation ofmotion of the i th particle in any dimension is given by¨ x i = P i ( F i + N i ) = − C ˙ x i × ˙ x i × ( F i + N i ) , (1)where we have introduced the projector operator P i = − C ˙ x i × ˙ x i × , with C = (cid:2) m i v (cid:3) − to ensure that thespeed remains constant and equal to || ˙ x i ( t = 0) || = v i .In Eq. (1) N i denotes a random force and F i an interac-tion force. Here, we focus on particles that interact viaan attractive force that acts inside a vision cone (VC)and thus define the force on particle i as: F i = ˜ γ X j ∈ Ω i x j − x i || x j − x i || , (2)where Ω i denotes the set of neighbors inside theVC of particle i and ˜ γ is a constant. Particles inthe VC are those that satisfy || x j − x i || ≤ R and x j − x i || x j − x i || . ( ˙ x i / || ˙ x i || ) > cos( β ), with β the size of the cone.This means that, by definition, the cone is oriented in thedirection given by ˙ x i ; for a sketch of the model see Fig. 2. FIG. 2. (Color online) Sketch illustrating the model definedby Eq. (1), whose dynamics in two dimensions reduces to thatgiven by Eq. (3). Particle positions are indicated by circlesand their velocities by arrows. Particles interact with particlesinside the VC. In the figure, the VC of particle i is displayed.Notice that the orientation of the VC is given by particle i ’s velocity (red arrow). In the sketch, particle i interactsexclusively with particles j and k , and only the positions of j and k , and not their velocities, are relevant for the evolutionof i . The state of particle i is given by its position x i andits velocity, which is parametrized in two dimensions by onlythe angle θ i since the dynamics keeps the speed constant. Formore details on the model, see the text. Notice that in Eq. (2) we do not divide by the numberof neighbors in contrast to the model analyzed in [32].In the following, we assume for simplicity that particlesare identical and start with the same speed, such that m i = m and v i = v for all i . Dynamics in two dimensions
In order to simplify the derivation of hydrodynamicequations, in the following we restrict the motion ofparticles to the two-dimensional plane ˆ e -ˆ e by assum-ing that at t = 0 the velocity of all particles lies onthis plane. To ensure two-dimensional motion, we ad-ditionally require that N i ( t ) lies on the plane ˆ e -ˆ e suchthat − C ˙ x i × N i = √ D θ ξ i ( t )ˆ e , with h ξ i ( t ) i = 0 and h ξ i ( t ) ξ j ( t ′ ) i = δ i,j δ ( t − t ′ ). Since we are on a planeand the speed is conserved, we can write ˙ x i = v V ( θ i )with V ( . ) ≡ (cos( . ) , sin( . )) T and thus ¨ x i = ˙ θ i v V ⊥ ( θ i ),where V ⊥ ( . ) ≡ ( − sin( . ) , cos( . )) T . Using these defini-tions, Eq. (1) can be rewritten as˙ x i = v V ( θ i ) (3a)˙ θ i = γ X j ∈ Ω i T ij + p D θ ξ i ( t ) , (3b)where the angle θ i represents the moving direction ofthe particle on the plane ˆ e -ˆ e and T ij is defined as T ij = [ V ( θ i ) × V ( α ij )] . ˆ e = sin( α ij − θ i ) with V ( α ij ) = x j − x i || x j − x i || , and γ = ˜ γ/ ( v m ). DERIVATION OF HYDRODYNAMICEQUATIONS
Since the microscopic model given by Eq. (3) has beenformulated in terms of Langevin equations, it is naturalto attempt a hydrodynamic description of the system dy-namics by deriving the corresponding nonlinear Fokker–Planck equation for p ( x , θ, t ) = h P Ni =1 δ ( x − x i ) δ ( θ − θ i ) i ,which reads ∂ t p + ∇ [ v V ( θ ) p ] = D θ ∂ θθ p − ∂ θ [ I p ] , (4)where I represents the (average) interaction experiencedby a particle located at position x and with moving di-rection θ at time t . The term I is simply defined as I = γ Z Ω( x ,θ ) d x ′ Z π dθ ′ sin( α ( x ′ − x ) − θ ) p ( x ′ , θ ′ , t )= γ Z Ω( x ,θ ) d x ′ sin( α ( x ′ − x ) − θ ) ρ ( x ′ , t ) (5)where Ω( x , θ ) corresponds to the VC for a particle lo-cated at x moving in direction θ , α ( x ′ − x ) corresponds tothe angle in polar coordinates of the vector ( x ′ − x ) / || x ′ − x || = V ( α ), and where we have introduced the definition ρ ( x , t ) = R π dθ p ( x , θ, t ). Notice that in Eq. (4) we haveassumed that p ( x , θ, x ′ , θ ′ , t ) ≃ p ( x , θ, t ) p ( x ′ , θ ′ , t ). Wecan simplify the calculations by explicitly using x ′ − x = R V ( α ), which lets us rewrite the integral over Ω( x , θ ) as I = γ Z R dR Z θ + βθ − β dα R sin( α − θ ) ρ ( x + R V ( α ) , t ) . (6)Our next step is to approximate ρ ( x + R V ( α ) , t ) ≃ P 3) ( β − sin(2 β ) / 2) and f ( β ) =( R / 6) sin ( β ). Our goal now is to obtain a descriptionof the system in terms of fields that depend on x and t , eliminating the dependence on θ . In order to do this,we multiply the left- and right-hand sides of Eq. (4) by V ( kθ ), with k ∈ N , after replacing I with Eq. (7), andintegrate over θ . For a compact notation, we introducethe following fields: P ( x , t ) = (cid:20) P x P y (cid:21) = Z π dθ V ( θ ) p ( x , θ, t ) , (8a) Q ( x , t ) = (cid:20) Q c Q s (cid:21) = Z π dθ V (2 θ ) p ( x , θ, t ) , (8b) M q ( x , t ) = (cid:20) M qc M qs (cid:21) = Z π dθ V ( qθ ) p ( x , θ, t ) , (8c)with q a natural number greater than 2. The procedureleads to the following temporal evolution of the fields: ∂ t ρ + v ∇ · P = 0 (9a) ∂ t P + v (cid:18) ∇ ρ + h ∇ T M Q i T (cid:19) = − D θ P − γg ( β )2 h M Q − ρ i ∇ ρ − γf ( β )2 M ρ [ P − M ] (9b) ∂ t Q + v h ∇ T (cid:16) M + M P (cid:17)i T = − D θ Q − γg ( β ) (cid:20) M − M TP (cid:21) ∇ ρ − γf ( β ) (cid:18) M ρ M + ρ (cid:20) Φ ρ − ∂ xy ρ (cid:21)(cid:19) , (9c)where the symbols M A denote matrices defined usingthe 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 . FROM LOCAL SOLUTIONS TO CLOSUREASSUMPTIONS The system of Eq. (9), owing to the presence of higher-order fields, specifically M and M , does not representa closed system of equations. If we derive equations for ∂ t M and ∂ t M , we will quickly find that they dependon M and M . In short, we have an infinite hierarchy ofequations. In order to work with Eq. (9), we are forcedto find suitable closure assumptions. We will make useof local solution ans¨atze to express higher-order fieldsin terms of ρ , P , and Q and obtain a closed system ofequations. θ p ( θ ) p( θ ) sim.p N ( θ ) theo.0 θ p ( θ ) p( θ ) sim.p F ( θ ) theo. (a) (b)(c) a bi a bi (d) FIG. 3. (Color online) Emergence of local (orientational) or-der from a given configuration of particles in space: (a) Localpolar order ( β = 0 . √ D θ = 0 . 18) and (b) local nematicorder ( β = 1 . √ D θ = 0 . i is shown. Particles a and b are the nearest neigh-bors, in distance, of particle i . By reorienting the VC, particle i can interact with either particle a or b . Notice that the ideal-ized arrangement of particles analyzed here, a straight line ofparticles, is used only as an illustration. Any long-lived, elon-gated distribution of particles in space will lead to either polaror nematic (local) orientational order. (c) [(d)] Angular dis-tribution obtained from simulations of the scenario depictedin (a) [(b)] (solid black curve), which is compared with p F ( θ )[ p N ( θ )] (dashed red curve), Eq. (27) [Eq. (15)]. When no local (orientational) order is possible We start with a trivial limit. For β = 0 it is evidentthat f = g = 0 and no local orientational order is pos-sible, i.e., P = Q = . We are left then with a simplesystem of non-interacting active particles characterizedby a diffusion coefficient D NAP = v / (2 D θ ). Our nextstep is to study the opposite situation, i.e., β = π , whichcorresponds to isotropic attractive interactions. In thislimit, f = 0 but g > 0, which implies that particles in-teract among themselves via a standard short-range at-tractive force. Such interactions cannot lead to polar ornematic local orientational order. The only relevant fieldin this scenario is ρ ( x , t ) and our goal is to find an effec-tive equation for ∂ t ρ . Given the absence of orientationalorder, and using the faster relaxation of Q with respectto P , we ignore Eq. (9c) by assuming that Q = ∂ t Q = .If we have to obtain a nontrivial dynamics, we cannotsimply discard Eq. (9b), but assume only that ∂ t P = .By substituting this assumption into Eq. (9b), we findthat P has to satisfy v ∇ ρ = − D θ P + γg ρ ∇ ρ , (10) from which we obtain an expression for P that we insertinto Eq. (9a) to arrive at ∂ t ρ = − v D θ ∇ [ − v ∇ ρ + γgρ ∇ ρ ] . (11)From Eq. (11) we learn that a homogeneous spatialdistribution of particles becomes linearly unstable when c = D NAP − v γgρ / (2 D θ ) < 0. This result is obtainedby substituting ρ = ρ + ǫ δρ [ x , t ] into Eq. (11), with ρ a constant, ǫ ≪ δρ the perturbation function, andkeeping terms linear in ǫ . If we use as a perturbation δρ = e λt e i k . x , we can easily understand that the disper-sion relation of the linearized system is not well behaved.This problem is fixed by going one order further inthe Taylor expansion of Eq. (6), which adds the term γ πR [cos( θ )( ∂ xxy ρ + ∂ yyy ρ ) − sin( θ )( ∂ xxx ρ + ∂ yyx ρ )] toEq. (7). By incorporating third order derivatives, it iseasy to show that the dispersion relation is of the form λ = − c k − c k , where c = v γπR ρ / > 0, whichindicates that the dispersion relation (of the linearizedsystem) is qualitatively similar to that of a Cahn-Hilliardequation. In summary, for β ∼ π we expect the sys-tem to undergo phase separation following standardcoarsening for sufficiently large systems. In simulations,deviations from this behavior are expected as long asthe characteristic distance between aggregation centersis smaller or comparable to v D − θ . In the presence of local nematic order For β < π we can conceive the existence of particleconfigurations leading to some kind of orientational or-der. Logically, only stable configurations are relevanthere. Given the proposed microscopic equations, thereare two relevant particle configurations to be considered:i) an elongated “band” with particles moving along itin both directions and ii) a line of particles where allparticles move in the same direction, i.e., where parti-cles follow each other. These two configurations emergespontaneously in simulations of the microscopic model(see Fig. 1).Our first step is to understand that if we fix particlesin space on an elongated high-density structure and ap-ply Eq. (3b), we obtain an asymptotic local distribution p ( θ ) displaying nematic symmetry. It is important tostress that the idealized configurations shown in Fig. 3serve as an illustration of a generic mechanism leadingto orientation order. The arguments put forward belowhold true for any long-lived spatial distribution of par-ticles that displays high accumulation of particles alonga given direction, and where each particle may interactwith multiple particles simultaneously. Thus, for sim-plicity and without loss of generality we focus on theidealized situation depicted in Fig. 3(b). Let us start bysimulating the dynamics of ˙ θ i as given by Eq. (3b). Inthis configuration, particle i interacts for some time withparticle a , some time with particle b , and some time withneither of them, depending on the orientation of its VC.The dynamics of θ i is then given by˙ θ i = γ X j = { a,b } sin( α i j − θ i ) h j ( θ ) + p D θ ξ i ( t ) , (12)where α i a = π denotes the polar angle of the vector( x a − x i ) / || x a − x i || = V ( α i a ). Similarly, α i b = 0 isassociated with the vector ( x b − x i ) / || x b − x i || = V ( α i b ),and the VC of particle i is described via the twoauxiliary functions h a ( θ i ) and h b ( θ i ), which are definedin such a way that h a ( θ i ) = 1 when particle a lieswithin the VC of i , and 0 otherwise, while h b ( θ i ) = 1when particle b is located inside the VC of i , and 0otherwise. It is easy to verify that the asymptoticdistribution of θ i can be approximated by p a ( θ ) ∼N hP j e γDθ cos( α a j − θ ) h j ( θ ) + e γDθ cos( β ) Π j (1 − h j ( θ )) i ,with N a normalization constant. For simplicity, inthe following we focus on large values of β and D θ andapproximate the dynamics of θ i by˙ θ i = γ sin (2( α − θ i )) + p D θ ξ i ( t ) , (13)with α either 0 or π to apply the equation to the config-uration sketched in Fig. 3(b). The advantage of Eq. (13)is that we ignore the difficulties associated with the VC.Its associated Fokker–Planck equation reads ∂ t p = γ∂ θ [sin (2( α − θ )) p ] + D θ ∂ θθ p , (14)whose steady-state solution is the von Mises distribution p N ( θ ) = N e γ Dθ cos(2( α − θ )) = N e γ Dθ V (2 θ ) · V (2 α ) , (15)where N is again a normalization constant. It is evidentthat p a ( θ ) and p N ( θ ) share the same symmetry. Giventhe many approximations performed to arrive at p N ( θ ),it is far from evident that Eq. (15) provides a reasonabledescription of the dynamics defined by Eq. (12). Fig-ure 3(b) shows that p N ( θ ) is a reasonable approximationof the distribution p ( θ ) obtained from direct simulationsusing Eq. (12).From the previous arguments we have learned that ifparticles are arranged in an elongated, high-density spa-tial configuration, we can expect local nematic order Q to emerge. Notice that Eq. (15) allows us to establishthat Q ∝ V (2 α ). We use this knowledge to conceive theclosure of the derived field equations, i.e., Eq. (9). As-suming that the dynamics of θ is faster than the spatialdynamics, we expect that locally the distribution of θ willfollow the functional form suggested by Eq. (15), whichwe write generically as p ( x , θ, t ) = N e w V (2 θ ) · Q , (16)where N as well as w may depend on ρ . Notice that tosimplify the notation we have not written the dependence of ρ and Q on x and t . Expressions for N and w can beobtained by self-consistency since, by definition, p ( x , θ, t )has to obey Z π dθ p ( x , θ, t ) ∼ N π = ρ , (17)and thus N = ρ π , while from the definition of Q we find Z π dθ V (2 θ ) p ( x , θ, t ) = (18) Z π dθ V (2 θ ) N e w V (2 θ ) · Q ∼ Q , which leads to w = ρ . All this means that our localansatz reads p ( x , θ, t ) = ρ ( x , t )2 π e ρ ( x ,t ) Q ( x ,t ) · V (2 θ ) . (19)This approximation is valid close to the onset of localorder, i.e., when || Q /ρ || is small. With p ( x , θ, t ) at hand,we can compute all the remaining fields: M and M . Bysymmetry, it is easy to verify that M ( x , t ) ∼ . Strictlyspeaking, we can show that M is of order higher than O ( Q ). The only remaining field to analyze is M ( x , t ).By subsituting the local ansatz into its definition, we find M ( x , t ) = Z π dθ V (4 θ ) ρ ( x , t )2 π e ρ ( x ,t ) Q ( x ,t ) · V (2 θ ) = 1 ρ ( x , t ) (cid:20) ( Q c ( x , t ) − Q s ( x , t ) ) Q c ( x , t ) Q s ( x , t ) (cid:21) . (20)By neglecting M and using Eq. (20), Eq. (9) becomesa closed system. Furthermore, the system dynamics canbe reduced to the evolution of only two fields: ρ and Q . In order to do this, we require ∂ t P = 0 at all times,which allows the fusing of Eqs. (9a) and (9b), and keepthe leading-order terms in Eq. (9c). This procedure leadsto: ∂ t ρ = (21a) v ∇ · (cid:20) −M I (cid:18) v (cid:18) ∇ ρ + h ∇ T M Q i T (cid:19) + γg h M Q − ρ i ∇ ρ (cid:19)(cid:21) ∂ t Q + 4 D θ Q = (21b) − γf (cid:18) M ρ ρ (cid:20) (cid:0) Q c − Q s (cid:1) Q c Q s (cid:21) + ρ (cid:20) Φ ρ − ∂ xy ρ (cid:21)(cid:19) , where M I is the inverse of D θ + γf M ρ .Since our goal is to look for static self-organized ne-matic patterns, we do not need to consider the tem-poral evolution of the fields. Moreover, we search forsteady-state solutions and thus set all partial tempo-ral derivatives equal to zero. We apply this conditionto Eq. (9). Given that all directions are equivalent,without loss of generality we assume that nematic or-der occurs along the ˆ x -axis, i.e., Q s = 0. This impliesthat the pattern is invariant along the ˆ x -axis, an as-sumption consistent with the nematic bands found inagent-based simulations [Fig. 1(b)]. As a consequenceof such invariance, all derivatives with respect to x van-ish and fields cannot depend on x , which, together withthe assumption of nematic order along the ˆ x -axis, yields p ( x , θ ) = ρ ( y )2 π e Q c ( y ) cos(2 θ ) /ρ ( y ) . We have already pointedout that the presence of local nematic order implies that M ∼ , and from Eq. (20) we learn that M c = Q c ( y ) ρ ( y ) and M s = 0. Under these assumptions, it is easy toverify that Eq. (9a) is automatically satisfied as it occursfor the equations for P x and Q s [see Eqs. (9b) and (9c),respectively]. We are left with the equation for P y and Q c , which reads v ∂ y [ ρ − Q c ] = γ g ( β ) ∂ y ρ ρ + Q c − D θ Q c − γ f ( β ) ∂ yy ρ (cid:18) ρ − Q c ρ (cid:19) . (22b)These equations can be expressed as the following first-order ordinary differential equation (ODE) system: ∂ y z = − Q c (cid:20) b (cid:18) ρ − Q c ρ (cid:19)(cid:21) − (23a) ∂ y ρ = z (23b) ∂ y Q c = (1 − a ( ρ + Q c )) z , (23c)where we have introduced the auxiliary field z , given byEq. (23b), and the constants a = γgv and b = γf D θ . Ournext step is to linearize either Eq. (22) or Eq. (23) using ρ ( y ) = ρ + ǫδρ ( y ) and Q c = ǫδQ c ( y ), with ρ a con-stant that represents a linear density and ǫ a perturba-tion parameter such that ǫ ≪ 1, with δρ ( y ) and δQ c ( y )the perturbation functions to be determined. Keepingthe linear-order terms with respect to ǫ , it is possible toshow that the linear system reduces to ∂ yy ˜ z = aρ − bρ ˜ z , (24)where ˜ z = ∂ y δρ . This reduction is possible because δQ c = − bρ ∂ yy δρ in the linearized system. It is evidentthat Eq. (24) admits trigonometric functions as solutionswhen aρ − < 0. This implies that we expect the pres-ence of multiple nematic parallel bands, where densityand nematic order are closely related: Q c ( y ) ∝ ρ ( y ).In summary, by assuming the presence of local nematicorder, we obtained a closed system of field equations andshowed that this system of equations has steady-state so-lutions. Furthermore, we indicated that these solutionsare consistent with the presence of multiple parallel ne-matic bands observed in agent-based simulations. In the presence of local polar order Particles arranged in elongated spatial configurationscan also exhibit (transient [45]) local polar order. Inagent-based simulations it becomes evident that elon-gated particle configurations with polar order exhibitlong-lived, dynamical structures that we refer to asworms. Our first goal is to understand how an elongatedconfiguration of particles can induce a local distributionof θ displaying polar symmetry. We start by looking atthe configuration shown in Fig. 3(a). We stress that theidealized configuration depicted in Fig. 3(a) only servesas an illustration of a generic orientational order mecha-nism. To further simplify the argument we ignore particle a and express the dynamics of θ i as˙ θ i = γ sin( α i b − θ i ) h b ( θ ) + p D θ ξ i ( t ) . (25)The associated Fokker–Planck equation of Eq. (25) – for h b ( θ ) = 1 for all θ – reads: ∂ t p ( θ, t ) = − ∂ θ [ γ sin( α i b − θ ) p − D θ ∂ θ p ] . (26)The steady-state solution of Eq. (26), denoted by p ( θ ),takes the form p F ( θ ) = N e γDθ cos( θ ) = N e γDθ V ( θ ) · V ( α i b ) , (27)where N is again a normalization constant. Now,we observe that in the idealized image depicted inFig. 3(a), the nearest neighbors of i share the sameorientation (see arrows) and so the local polar order P is parallel to V ( α i b ). The previous assumption al-lows us to express the solution given by Eq. (27) as p ( θ, t ) = N e w V ( θ ) · P ; see comment below on the esti-mation of N and w . If we analyze the problem withthe original definition of h b ( θ ), we find that p ( θ, t ) ≃N h e γDθ V ( θ ) · P h b ( θ ) + e γDθ V ( β ) · P (1 − h b ( θ )) i . Figure 3(a)shows that Eq. (27) is a good approximation of this ex-pression. Thus, we adopt the functional form given byEq. (27) as the local ansatz for the distribution of θ . Af-ter fixing N and w by requiring R dθp ( θ, t ) = ρ ( x , θ, t )and R dθ V ( θ ) p ( θ, t ) = P ( x , θ, t ), we find p ( x , θ, t ) = ρ ( x , t )2 π e ρ ( x ,t ) P ( x ,t ) · V ( θ ) . (28)It is important to understand that in this argument wehave not considered the motion of particles. We knowthat for a static, elongated spatial configuration of parti-cles, polar order can only be observed during a transient.However, here the spatial configuration of particles is alsoevolving. In particular, the temporal evolution of thespatial configuration of particles may be such that polarorder is maintained. Thus, adopting as the local ansatzEq. (28), we compute Q and M as Q ( x , t ) = 1 ρ (cid:20) ( P x − P y ) P y P x (cid:21) (29a) M ( x , t ) = 16 ρ (cid:20) P x − P x P y P y P x − P y (cid:21) , (29b)where we have not explicitly written the dependence on x and t for the field ρ , P x , and P y . Equation (29), to-gether with Eq. (9), allows us to obtain a closed systemof equations for ρ and P , where we have to use the defi-nition of M given above, the definition of M ρ providedbelow Eq. (9), and the following definition of M Q : M Q = 1 ρ (cid:20) ( P x − P y ) P x P y P x P y − ( P x − P y ) (cid:21) . Let us now investigate the possibility of having static,straight, percolating polar bands. Since all directionsshould be equivalent, for simplicity and without loss ofgenerality, we assume that the polar order is along the x -axis and the pattern is invariant along x . This impliesthat all derivatives with respect to x and time vanish; thelatter is due to the fact that we look for static patterns.All this together means that P y = 0, P x = P x ( y ), and ρ = ρ ( y ). By inserting this into Eq. (9), we find thatEq. (9a) is automatically satisfied, while from Eq. (9b)we obtain0 = − D θ P x − γf (cid:20) − ρ P x (cid:21) ∂ yy ρ P x (31a) ∂ y ρ − ∂ y (cid:18) P x ρ (cid:19) = γgv ∂ y ρ (cid:18) P x ρ − ρ (cid:19) . (31b)From Eq. (31a) we express P x ρ as a function of ρ andderivatives of ρ . The next step is to insert the resultingexpression into Eq. (31b). After performing an expansionin ρ , we find that Eq. (31) has no solution. This provesthat percolating, static polar patterns are not a solutionof Eq. (9). We stress that this result does not precludethe existence of dynamic locally polar structures such asthe dynamic worms observed in agent-based simulations.In summary, from the hydrodynamic equations we learnthat while dynamical, locally polar structures can exist,percolating, static polar bands leading to global polarorder – i.e., structures similar to the obtained nematicbands discussed above but polar – cannot emerge. CONCLUSIONS The derived hydrodynamic equations reveal that activeparticles interacting only by short-range, position-based,attractive interactions can exhibit various complex col-lective motion patterns if Newton’s third law is brokenby using a vision cone. For isotropic, and thus recip-rocal interactions, i.e., β = π , we have shown that for sufficiently small D θ values, the system undergoes phaseseparation with a classical, equilibrium-like, coarseningdynamics. It is during this phase that we observe theformation of aggregates with no orientation order. Thisbehavior is expected to be representative of what hap-pens in the vicinity of isotropic interactions. For β < π ,interactions are nonreciprocal, and for small D θ and β values, the absence of Newton’s third law leads to inter-esting effects. In particular, we have seen that the accu-mulation of particles along a given direction in space (i.e., the formation of a high-density stripe), can induce ei-ther polar or nematic local orientational order. We madeuse of the proposed local distributions of orientations forthe polar and nematic cases to obtain suitable closures ofthe derived hydrodynamic equations. We learned that forlocally polar structures – worms – there is no static, per-colating polar band, which indicates the absence of globalpolar order. For locally nematic structures, on the otherhand, we managed to find static patterns, which corre-spond to elongated structures leading to global nematicorder: nematic bands. All these observations are (qual-itatively) consistent with observations with agent-basedsimulations.One important message from the derived hydrody-namic equations is that position-based flocking modelsbelong to a distinct active class and are fundamentallydifferent from velocity-alignment-based flocking models,including the so-called polar fluids [17, 18, 34, 46], activenematics [19, 36, 37], and self-propelled rods [16, 38–40, 42–44, 47]. For instance, in position-based flockingmodels such as the one analyzed here, the orientationalorder that emerges is always associated with density in-stabilities, and polar or nematic spatially homogeneousordered phases cannot exist. This is in sharp contrastwith the spatially homogeneous Toner–Tu polar phasein (velocity-alignment-based) polar fluids [17, 18, 34, 46]and the spatially homogeneous nematic phases reportedin [16, 19, 39, 40, 47].Several fundamental questions remain open forposition-based flocking models. 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