# Flocking and Turning: a New Model for Self-organized Collective Motion

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## Abstract

Birds in a flock move in a correlated way, resulting in large polarization of velocities. A good understanding of this collective behavior exists for linear motion of the flock. Yet observing actual birds, the center of mass of the group often turns giving rise to more complicated dynamics, still keeping strong polarization of the flock. Here we propose novel dynamical equations for the collective motion of polarized animal groups that account for correlated turning including solely social forces. We exploit rotational symmetries and conservation laws of the problem to formulate a theory in terms of generalized coordinates of motion for the velocity directions akin to a Hamiltonian formulation for rotations. We explicitly derive the correspondence between this formulation and the dynamics of the individual velocities, thus obtaining a new model of collective motion. In the appropriate overdamped limit we recover the well-known Vicsek model, which dissipates rotational information and does not allow for polarized turns. Although the new model has its most vivid success in describing turning groups, its dynamics is intrinsically different from previous ones in a wide dynamical regime, while reducing to the hydrodynamic description of Toner and Tu at very large length-scales. The derived framework is therefore general and it may describe the collective motion of any strongly polarized active matter system.

## Keywords

Collective behavior Flocking Self-organization Emergent behavior Animal groups## 1 Introduction

Flocks of birds represent paradigmatic cases of emergent collective behavior, where short range interactions between individuals lead to nontrivial movement patterns on the large scale [1, 2, 3, 4, 5]. Several models have been developed by biologists [6, 7, 8, 9, 10], physicists [11, 12, 13, 14, 15, 16, 17] and control theorists [18, 19] to describe flocking behavior. Some of these models greatly improved our understanding of collective motion, offering a powerful description of the large scale properties of active systems [20, 21, 22, 23]. However, recent experiments on real flocks [24] revealed new surprising features in the way collective turns are performed that are still unaccounted for, urging for a novel theoretical explanation.

The most conspicuous feature of flocking is the presence of collective order: flocks are strongly polarized groups, where individuals all move in approximately the same direction. This kind of orientational order can be naturally explained in terms of alignment interactions between the birds velocities and basically all models of flocking include a ‘social’ force term, describing the tendency of individuals to adjust their flight direction to those of neighbors. Recently, using inference techniques on real data of flocking birds [25], it was in fact shown that simple models based on pairwise alignment topological interactions are able to explain the long-range correlations between flight directions and the large degree of coherence exhibited by natural flocks [26].

The role of alignment is emphasized in the simplest model of flocking, the Vicsek model [11]. In the Vicsek model flocking individuals are described as self-propelled particles of constant speed, where the velocity (i.e. the flight direction) of each particle is updated from one time step to the next by computing the average direction of motion of neighbors. Several variants of the model have been analyzed in the literature, where additional cohesive terms are added [9, 13, 14, 20], and interactions are chosen topologically rather than metrically (as in the original version) [17, 27]. In all these cases the structure of the dynamical equations always remains the same, where alignment forces due to neighbors directly act on the individual velocity. In other terms, the velocity evolves according to a first order Langevin equation, its time increments being determined by the social force and random noise. This kind of dynamics generates long-range order at low noise and scale-free correlations in the polarized phase, both experimentally observed features of natural flocking. Additionally, at very large scales all these models are well described by the hydrodynamic theory of flocking, whose intriguing predictions of giant density fluctuations, propagating sound modes and anomalous diffusion have been observed in a variety of active inanimate and animate fluids [12, 20, 22, 23, 28].

In Ref. [24] some of us showed that there are two crucial and connected ingredients missing in the microscopic dynamics of the Vicsek model, that must be taken into account to theoretically reproduce the dispersion law observed in turning flocks: (1) the presence of a conservation law associated to the rotational symmetry of the system; (2) the existence of a behavioral inertia mediating the effect of the social force. In this paper we introduce a new microscopic model for flocking where these two ingredients are appropriately incorporated. This model provides a unified general set of equations to describe systems of interacting self-propelled individuals, where all the salient traits of real flocking—not only order and correlations, but also turns and information propagation - are correctly reproduced. Interestingly, the Vicsek model can be retrieved as the overdamped limit of this more general model. Additionally, in the long wavelength limit the model is well described by the hydrodynamic theory of flocking of Toner and Tu [20] and is therefore consistent with its predictions on the large scale behavior.

*inertial spin model*:

These equations might look unfamiliar and difficult to interpret, but we will show that they are quite natural. Most of the paper is devoted to explaining how the equations can be derived from general symmetry considerations, what is their meaning, what are the interesting limiting cases (the Vicsek model and the deterministic limit), and what are the predictions of the model for information propagation in the ordered phase. We hope the reader will go through this whole exercise, but for those less interested in the details, the last section of the paper presents in a self-contained way numerical simulations of model (1) and illustrates its behavior.

## 2 Setting up the Stage

Our aim in this work is to introduce a model of flocking able to describe collective turning and information propagation in flocks. Since these phenomena occur on finite scales of space and time, we need to go beyond hydrodynamics. As a consequence, the universality of the long length scale behavior is lost, and a degree of arbitrariness arises in the way we chose the terms of the microscopic dynamical equations. One way to address this issue is to propose a model and check a posteriori its validity. The reader can look at Eq. (1) as our guess, and then directly skip to the last section, where we show with numerical simulations that model (1) correctly describes the experimental results of [24].

In the following sections we will follow a longer route, and show that it is possible to construct a microscopic model of collective motion starting from effective variables, symmetries, conservation laws and Hamiltonian dynamics. This may seem a bit formal when we think about systems as complex and highly dissipative as real flocks. However, our point will be that all the nontrivial dissipative nature of animals’ active motion can be captured by an effective constraint on their velocity. Once we find the right set of canonical variables enforcing this constraint, we are able to give a Hamiltonian description of flocks. In a second part of the paper, we then generalize this description to include non-Hamiltonian contributions and noise.

In Sect. 2.1 we want to make this formal description easier to understand by going back to a very simple (in fact, quite trivial) example, showing that the use of effective variables and symmetries in presence of a constraint is quite standard in physics.

For the sake of simplicity we will assume in this section that bird velocities lie on a plane. Although this is true in most experimental cases [24], it is not an essential mathematical condition, and in the next sections we will analyze the fully three-dimensional case. We will also disregard the reciprocal movement of birds (which is very low during real turns [24]) when deriving analytically an expression for the dispersion law. Later in Sect. 6 we then perform numerical simulations of model (1) where the complete dynamics of positions, velocities and spins is implemented, and network rearrangements are fully taken into account.

### 2.1 Circular Motion in the External Space of Position

The message of this repetition of classical mechanics is that, regardless of the complexity by which the constraint (3) is enforced in practice, once we introduce the right canonical variables we can completely forget about these complications. Additionally, the symmetry of the Hamiltonian with respect to the transformation parametrized by the effective coordinate ensures the conservation of the generator of the transformation itself, that is the momentum.

### 2.2 Circular Motion in the Internal Space of Velocity

*internal*space of velocity, while \(\theta \) was the parameter of the rotation in the

*external*space of positions. This difference is crucial: the generator of the rotation parametrized by \(\varphi \) is

*not*the orbital angular momentum \(l\), but rather the internal angular momentum, or spin, \(s\). The variables \((\varphi ,s)\) are the canonically conjugated pair enforcing constraint (10), exactly as \((\theta , l)\) are those enforcing constraint (3). For this reason, the single-particle Hamiltonian in absence of forces other than those enforcing the constraint is given by

### 2.3 From One to Many Birds

*a priori*equally likely at the individual and collective level, it is reasonable to expect that interactions between individuals will respect this symmetry. This means that \(V\) (and therefore the Hamiltonian) is invariant under a global rotation of all the velocity vectors \(\vec {v}_i\), that is under the transformation \(\varphi _i \rightarrow \varphi _i +\delta \varphi \). Thanks to Noether’s theorem, this global symmetry implies a global conservation law. Although the individual spin \(s_i\) is now

*not*constant, due to the force \(\partial V/\partial \varphi _i\), the

*total*spin, \(S=\sum _i s_i\), is conserved. This is clear from Hamilton’s second equation,

It is very important to understand that in the case of many particles, a uniform rotation of the polar angle \(\theta \) and a uniform rotation of the phase \(\varphi \) give two completely different results. The \(\theta \)-rotation, generated by the orbital angular momentum \(l\), acts on the positions of the points and therefore it gives rise to *parallel paths* trajectories: these all have the same origin as a centre of rotation, but different radii of curvature. On the other hand, the \(\varphi \)-rotation, generated by the spin \(s\), acts on the velocities of the points, giving rise to *equal radius* trajectories: these have different centres of rotation, but the same radius of curvature. Turning flocks of birds are known to move along equal radius trajectories [24, 31], as this is the only way to keep cohesion at constant speed. This experimental observation confirms that phase and spin are indeed the correct canonical variables.

Let us summarize this introductory Section. We describe birds’ motion through a set of effective conjugated variables, the phase \(\varphi _i\) and the spin \(s_i\), that automatically enforce the constraint of constant speed, \(|\vec v_i \, |=v_0\). This allows us to disregard the highly complex and dissipative mechanism enforcing this constraint and to switch to a Hamiltonian description. Once the speed is constant, the only transformation that the velocity can undergo is a rotation, which is parametrized by the phase. The generator of this rotation is the spin (internal angular momentum), so that in presence of rotationally symmetric forces the total spin of the system is conserved. The way real birds turn is of course complicated, involving the details of flight aerodynamics. However, as long as birds are able to control their motion keeping their speed fixed and performing equal radius turning—as they do—they can be described in a simple way within this framework.

## 3 Deterministic Equations for the Velocity and the Spin

We have written the dynamical equations for the phase and the spin, which are the canonical variables, but not for the velocity. However, in order to actually run a simulation of a system of self-propelled particles, we need to know how the velocities evolve in time. Because the velocity is *not* a canonical variable, the update equations for \(\vec v_i\) will be nontrivial. Moreover, we need to study the fully three-dimensional case, in order to have a model as general as possible. In this Section we will write the dynamical equations for the velocity, in absence of noise and dissipation, which amounts to describing a deterministic flock. Noise and dissipation will be introduced later on.

### 3.1 From the Phase to the Velocity

### 3.2 Spin Conservation and Information Propagation

Another crucial consequence of the presence of a global continuous symmetry is that the generator of the symmetry is a constant of motion of the dynamics. In our case, this implies that the global spin \(\vec {S}=\sum _i\vec {s}_i\) is conserved. Interestingly, for a polarized flock this conservation law also has local implications (i.e. at the level of individual particles): if a strong misalignment occurs between a particle \(i\) and its neighbors, this causes an excess of curvature and an excitation of the local spin \(\vec {s}_i\); since the global spin is a conserved quantity, the local excitation cannot be re-absorbed or dissipated, but must be carried away. For this reason we expect propagating modes in this system.

To better understand how this occurs, let us consider a polarized flock of particles all moving approximately in the same direction. We stick to the simpler planar case for convenience, but the argument is general. For an ordered flock, the rotational symmetry is spontaneously broken since a specific global direction of motion is selected by the system. Still, a residual symmetry remains in the space orthogonal to this direction: we can choose the phase \(\varphi _i\) of each particle as the angle with respect to the global flock velocity, and use Eq. (15) to describe the evolution of such phases.

The field equations we have written for \(\varphi (\vec {r},t)\) and \(s(\vec {r},t)\) in this Section make one important approximation, namely that the interaction matrix \(n_{ij}\) is fixed in time. Without this approximation we cannot use the continuous Laplacian, nor obtain d’Alembert’s equation. This condition is, in general, violated, and this is what marks the difference between a lattice model and an off-lattice model of moving particles, where \(n_{ij}= n_{ij}(t)\) depends on time. However, if the time scale of propagation in d’Alembert’s equation (29) is shorter than the typical time scale of reshuffling of the network \(n_{ij}\), then the equations are justified. This is indeed the case in natural flocks of starlings [24], where phenomena like collective turns happen on a time scale indeed significantly shorter than the time needed for an individual to change its interacting neighbours [32]. Nevertheless, at the end of this work we will run numerical simulations of the full model with varying \(n_{ij}(t)\) and show that when \(c_s\) is large enough, the propagation phenomena are quantitatively in agreement with the field equations we have written above.

### 3.3 Dynamical Universality Class

The relationship between symmetries of the system, equations of motions and conservation laws is a well known feature of Hamiltonian dynamics and a particular manifestation of Noether’s theorem. The presence of a continuous symmetry implies the existence of a conserved quantity, and the concomitant linear propagation of local disturbances. What is novel in our analysis is that we have studied these properties in the context of the dynamics of flight directions of flocking particles.

Interestingly, the same structure of equations and dispersion law that we have introduced for flocks also hold in other kinds of systems, where a non-conserved order parameter (the flight direction in the case of flocks) co-evolve with a conserved quantity (e.g. the spin/curvature). In fact, Eqs. (26–28) are formally identical and belong to the same dynamical universality class as the planar ferromagnet [33] and superfluid Helium [34, 35] (model F in the Halperin-Hohenberg classification [36]). In liquid Helium, for example, the quantum phase plays the role of the flight direction, the (conserved) superfluid component plays the role of the spin, \(\vec {j}\) is the superfluid current, and the propagating wave is the so-called ‘second sound’ [34, 35, 37]. In the same way, the full three-dimensional model Eqs. (22–24) is equivalent to the isotropic antiferromagnet (model G dynamical universality class), with exactly the same equations, the staggered magnetization playing the role of the flight direction, and the (conserved) total magnetization playing the role of the spin.

In both these cases (model F and model G of [36]), propagating modes and a linear dispersion law are the consequence of the rotational symmetry and of the conservation of the spin deriving from it. This is not a generic feature of spin models. For example, in the case of the isotropic ferromagnet, the conserved field is the order parameter itself and this conservation does *not* derive from any symmetry of the Hamiltonian. In this case the linear propagation of the phase is lost, and a dissipative, diffusive dispersion law holds [36]. Therefore, the essential ingredient of linear spin propagation is not simply the presence of a conserved quantity, but rather the fact that this conserved quantity must be the generator of a spontaneously broken continuous symmetry. This condition is the true hallmark of superfluid transport, namely of the undamped and linear propagation of the spin. In the case of Helium, the spin is mapped onto the superfluid component, which propagates through second sound [36], whereas in the isotropic antiferromagnet one has actual spin superfluid propagation [36]. In flocks, spin carries the turning information.

## 4 A New Model of Collective Motion

The deterministic inertial spin model we have introduced in the previous Section conserves spin (through the continuity equation), but it does so a bit too effectively, as in absence of forces the angular velocity remains forever constant. Here we want to cure this anomaly. An example may help: although linear momentum conservation and inertia is essential to propagate standard sound in a crystal, we certainly do not expect sound to propagate forever and for infinite distance, as some dissipation will eventually dampen the signal. The same must be true for second sound, which is carried by spin conservation. Therefore, we must introduce a dissipative term coupled to \(\vec s\). Moreover, the dissipative term will be associated to a noise term, making the model stochastic.

It is important to understand that the dissipation we need to introduce is *not* the standard linear one, proportional to \(\dot{\vec x}\). Standard linear dissipation would drive the modulus of the velocity to zero, while this cannot happen here: the constraint \(|\vec v_i| = v_0\) must be automatically enforced by our equations. Instead, spin dissipation must account for the natural and rather intuitive fact that in absence of spin injection and of interaction with neighbours, a bird is unlikely to turn forever with constant angular velocity. As we shall see, regulating the strength of this dissipative term will make the difference between propagating and nonpropagating information.

### 4.1 The Inertial Spin Model

The inertial spin model (31–33) represents a full description of the system’s dynamics in terms of the variables \((\vec {v}_i, \vec {s}_i)\). Being a first order system it is relatively easy to implement numerically, and we shall discuss results of simulations later in the paper. For the time being, we note that there are four important parameters entering Eqs. (31–33): the alignment strength \(J\), the moment of inertia \(\chi \) of the spin, the viscous coefficient \(\eta \), and the temperature \(T\). The spin inertia, \(\chi \), is the new ingredient in the dynamics.

### 4.2 A Closed Equation for the Velocity

*not*the conventional inertia (i.e. the mass of the particle) but measures the resistance of a particle to change the curvature of its trajectory. The second term at the l.h.s. is a contribution arising from the constraint on the speed, the analogue of what would be the centripetal acceleration for circular motion in \(\vec {r}\) space. Its presence is a signature of the underlying Hamiltonian structure of the equations and is therefore essential to recover the appropriate deterministic limit when \(\eta \rightarrow 0\). The last term in the l.h.s. is the dissipative term. In the r.h.s. we find the effect of the social force and the noise.

As compared to other second order equations that have been considered in the literature [38, 39, 40, 41, 42], in Eq. (37) the connection between inertial terms and symmetries is automatically implemented. We stress that this is in fact a crucial point, since—as we showed in the previous sections—the way information propagates is strongly influenced by this connection.

### 4.3 Overdamped Limit: the Vicsek Model

### 4.4 Steady State Distribution of Velocity and Spin

If we now look at the explicit form of the Hamiltonian in (42), we see that the velocities are described by a Heisenberg model on a random euclidean lattice (the one realized by the particles positions). In the ordered phase, where the rotational symmetry is spontaneously broken, this distribution generates soft Goldstone modes and long range correlations corresponding to fluctuations of the individual velocities perpendicular to the global velocity of the flock [45]. Long-range correlations in velocity fluctuations have been experimentally observed in natural flocks of birds [26]. Besides, in [25] using inference techniques on empirical data, we showed that the equal time properties of the velocity fluctuations can be fully described by a Boltzmann distribution with local pairwise alignment interactions. The model we have discussed so far is therefore fully consistent with all experimentally known dynamic *and* static features of real flocks of birds.

## 5 Information Propagation in the Ordered Phase

Equations (31–33) give rise at low noise to polarized flocks, much as in the Vicsek model. In Sect. 3 we have seen that for a deterministic flock the rotational symmetry and the concomitant conservation law have strong implications on the properties of the ordered phase. Now we would like to understand whether this scenario holds in presence of noise and dissipation.

*two*relevant phases are left. By using (44–46) into Eq. (37) and expanding at the first order (spin wave expansion), we obtain for each one of the two phases, \(\varphi _z\) and \(\varphi _y\), the same equation,

In general, solving equation (47) is not an easy task, because the dynamics of the \(\vec {\varphi }\)’s is coupled in a non-trivial way with the movement of the particles in space, i.e. the network \(n_{ij}\) changes with time. Hydrodynamic theories of flocking address this problem by coarse-graining the microscopic dynamics and looking at the large-scale long-time behavior. Here, however, we are interested in addressing the dynamics of the system on shorter timescales. We can therefore assume that \(n_{ij}\) does not change significantly, as indeed happens in flocks during collective turns [24]. Later in Sect. 6 we perform numerical simulations of model (1) where network rearrangements on the scale of turns are fully taken into account.

*attenuated*propagating waves, as the frequency has both a real and an imaginary part. For \(k < k_0\) we have

*evanescent*waves with exponential decay and we recover the overdamped Vicsek-like behavior.

Natural flocks of birds appear to be precisely in this underdamped, quasi-deterministic regime [24]. Experiments on large flocks of starlings performing collective turns show that, once the turn (i.e. a direction change) is initiated by some individuals at the edge of the flock, the turning front propagates through the group with a linear dispersion law with very limited attenuation. The relationship (50) between the experimentally measured speed of information propagation \(c_s\) and the alignment strength \(J\), predicted by the model in the deterministic limit, is also remarkably verified [24]. Thus, far from being an idealized construction, the deterministic model introduced previously truly captures some important mechanisms occurring in real biological groups.

### 5.1 Transition to the Hydrodynamic Regime

The natural flocks investigated in [24] live in the quasi-deterministic regime \(k>k_0\). Other systems, however, (or even very large flocks) might well explore larger length-scales. This is certainly the case for a variety of fluids or large assemblies of active individuals. As mentioned in the previous section, hydrodynamic theories of flocking offer a sophisticated and detailed description of the \(k\rightarrow 0\) regime, which has been investigated in numerical simulations and experiments on granular active matter. While we do not want to give a complete account of this description (we refer the reader to [20]) we briefly comment on what happens as these large length (and time) scales are approached. On the one hand, as we noted in the previous section, when \(k\rightarrow 0\) the frequency Eq. (51) becomes purely imaginary indicating that there are not any more purely directional propagating modes. The physical reason is that for times scales \(t\gg \tau =2\chi /\eta \) the global spin relaxes due to dissipation and does not act as a quasi-conserved variable, thus the conservation law that determined the propagating nature of the modes is lost. In other terms, the second order terms of the microscopic dynamics become irrelevant, as well as the coupling between direction and curvature. On the other hand, on such large time scales the movement of individuals becomes relevant, and crucial: orientational fluctuations (which would be evanescent on a fixed network) couple with fluctuations in density. As a consequence, there are propagating hybridized longitudinal sound modes transporting both directional and density disturbances. The speed of propagation of such modes usually depends on the average density (as in standard sound), contrary to the purely directional propagating modes of the underdamped regime, where the propagation speed \(c_s\) does not depend of the flock density and is determined by the global alignment strength \(J\) (i.e. the degree of polar order). Finally, if the interaction between the moving particles and the surrounding medium becomes important the hydrodynamic theory also predicts shear-bend modes [20, 46] that propagate directional changes. These modes are transverse (i.e. they do not exist in \(d=2\)) and have limited speed in a co-moving reference frame, and are therefore well distinct from the turning modes observed in flocks, which are longitudinal and have very fast propagation speeds [24].

Summarizing, in the quasi-deterministic regime \(k>k_0\), where the spin is a quasi-conserved variable, there are propagating modes thanks to the coupling between velocity and spin and the consequent conservation law; these turning modes transport changes of direction *and* curvature and describe collective turns in flocks. In the hydrodynamic regime, due to the movement of the network, changes in direction couple with changes in local density generating propagating sound modes; this regime is universal, i.e. independent of the details of the microscopic dynamics, and does not describe transport of curvature.

## 6 Numerical Simulations

In the previous sections we have used analytical arguments to derive the predictions of the inertial spin model on how orientational information propagates in the ordered phase. We did that, however, working under two assumptions: we disregarded effects due to the reshuffling of the interaction network, and we considered the continuous space limit. While these assumptions are reasonable in certain cases, we want to test the behavior of the model in full generality. We performed numerical simulations for a wide range of the relevant parameters \(\chi \), \(\eta \), and \(J\) and explored how the values of these parameters determine the collective dynamics of the system.

We are interested in understanding how, for different values of \(\eta \), \(\chi \) and \(J\), directional information propagates in the system, and whether the simulated flock is able to sustain coherent collective turns as natural flocks do. In real flocks collective turns start locally: an individual bird, the initiator, starts to turn and then the information about turning propagates through the system with a linear dispersion law [24]. To mimic this situation, we consider the system at low enough values of noise as to generate an ordered flock with large polarization (\(\Phi > 0.95\) as in natural flocks), and we initialize all the \(\vec {s}_i=0\) so that the flock as a whole performs a straight motion. Then, we chose at random a particle inside the flock and make it artificially turn changing its flight direction with a ramp-like time dependence (red line in Fig. 3d,e,f). The change in direction of the initiator affects—through Eq. (55)—the dynamics of the spins of nearby particles, which might or might not cause a collective movement. We then look at the behavior of the whole flock in time to assess whether the signal has propagated, and what kind of dispersion law is obeyed.

- (1)
**Overdamped regime:**\(\eta ^2/\chi > n_c J (a/L)^2 \). In this regime, given a system of size \(L\), some attenuated propagation occurs up to certain spatial scales (\(k \ge k_0\), with \(k_0=1/(2 a \sqrt{n_cJ\chi })>1/L\)). On larger scales (\(k<k_0\)) however, dissipation takes over leading to an exponential decay of the signal. The extreme case occurs for \(\eta ^2/\chi \rightarrow \infty \) (or \(\chi /\eta ^2\rightarrow 0\)), corresponding to the Vicsek model, when propagation of orientational perturbations does not occur (unless we reach hydrodynamic lenght-scales where density and orientational modes couple, see [12, 20, 28] and Sect. 5.1). - (2)
**Underdamped regime:**\(\eta ^2/\chi \ll n_c J (a/L)^2\). In this regime there is linear propagation of the signal throughout the whole system with negligible attenuation. The speed of information propagation is determined solely by the ratio of alignment to inertia, \(c_s=a \sqrt{n_c J/\chi }\).

We performed simulations for values of the parameters corresponding to these two different regimes. As can be seen from Fig. 3, what we observe is fully consistent with what is described above. The top panels in this figure display the trajectories of the individual particles in the flock for three decreasing values of \(\eta ^2/\chi \), the trajectory of the initiator being represented by the thick black line. The intermediate panels show, for the same events, the cosine of the velocity of each particle with the original flight direction of the flock. Here the initiator is represented as a red line, and one can see the ramp-like dependence of the perturbation. The lower panels reproduce the individual acceleration profiles, and are particularly useful to determine if and how particles turn, and how the turning signal is attenuated during time.

The first two cases in Fig. 3 (panels a,b) belong to the overdamped regime. For values of \(\eta ^2/\chi \) much larger than \(n_c J (a/L)^2\) (which is of order 1 for our simulations) the turn of the initiator does not trigger any collective turn, since the signal does not propagate and is quickly damped (panel a). We can see from the cosine curves (panel d) that some of the neighbors feel the disturbance caused by the initiator, but this effect is very small: no particle follows the initiator, no collective turn is triggered and the flock ultimately keeps its original flight direction (see also attached Video 1). For smaller values of \(\eta ^2/\chi \) (panel b) there is some propagation, but it is progressively attenuated as the signal travels through the group and eventually decays (panels e,h). Some particles are reached by the directional perturbation before the signal is damped (corresponding to scales \(k>k_0\)), and follow the initiator changing their flight direction and performing a turn. Most of the particles, however, are not significantly affected by the perturbation, consistently with a strong damping of the signal on large enough scales. This effect is clearly seen by looking at the acceleration profiles of the particles (Fig. 3h). Some particles display a profile similar to the one of the initiator with a clear peak marking the occurrence of a turn. The height of the peak, however, decreases as the turning information is transmitted (i.e. particles start turning one after the other) and at some point it abruptly drops to zero (see inset). In this case, the flock breaks into two subgroups: the largest component keeps the original flight direction, while a smaller fragment is dragged by the initiator (see attached Video 2). In this respect we note that, due to active nature of the particles, the effect of overdamping is even stronger than discussed in Sect. 5: not only the turning information fails to percolate through the group, but the flock breaks up and looses cohesion.

In the underdamped regime (panel c), on the contrary, there is robust propagation of the signal through the whole group: all the particles follow the initiator changing their flight direction with negligible attenuation. Both the cosine curves and the acceleration profiles show that the particles turn in a highly coordinated way: the curves are very similar one to the other with a shift in time corresponding to the propagation of the turn from the initiator to the last particle (Fig. 3f,i). The flock performs a neat collective turn retaining shape and cohesion during motion (Fig. 3c, and Video 3).

In this regime, we can quantify the dispersion law using the method introduced in [24]. We rank all the particles according to their order of turning: the initiator has rank \(r=0\), then the particle that first starts turning after the initiator has rank \(r=1\) and so on. To determine the rank of a particle we follow two different procedures. We look at the cosine curve of the particle and determine its absolute turning delay from the initiator (and therefore its rank) as the time when the cosine reaches a threshold value (we choose 0.9—see inset in Fig. 3f). Alternatively, we proceed as in [24]: from the acceleration curves we compute the relative time delays between pairs of particles as the shift necessary to superimpose their curves. Once the relative delays are found for all particles we use a standard algorithm to rank them all. This method is more robust when it is not known a priori who is the initiator (as in experiments [24]). The two procedures give equivalent results.

The behavior displayed in Fig. 4 is analogous to what is observed in natural flocks of birds. The propagation curves look indeed very similar to the ones computed for real flocks and shown in [24]. Even more importantly, Fig. 4c is remarkably similar to what is found in real data, and the model correctly reproduces the experimentally found relationship between propagation speed and degree of order in the system. We can therefore conclude that Eq. (1) in the underdamped limit fully describe the correlated collective turning exhibited by real flocks. We remind that in this limit inertia and deterministic effects dominate over dissipation: it is the Hamiltonian structure of the dynamical equations, and the connected conservation law, that cause the linear dispersion law and the highly non-trivial relationship between speed of propagation and degree of order in the system.

## 7 Conclusions

The main point of our work is that momentum matters. More precisely, the powerful mathematical entanglement between symmetry, momentum and conservation, is the crucial ingredient giving rise to a dispersion relation, namely to a propagation law, identical to the one observed in real flocks of birds during global changes of direction. If momentum is disregarded, the dispersion relation changes dramatically and linear propagation collapses: the practical result is that no collective change of direction can be achieved by the group.

But what momentum is that? We hope we have convinced the reader that the momentum essential to our new model is *not* linear momentum, *nor* orbital angular momentum. Accordingly, the generalized inertia in our equation is not the mass, nor the moment of inertia. Linear momentum conservation is clearly violated in active matter systems: this is because the particles are, indeed, active. Momentum (and energy) is injected and dissipated continuously at the individual level. For this reason describing active matter systems in terms of Hamiltonian dynamics in the \((x,p)\) variables is impossible. But once we wrap all the complicated active mechanism into an effective constraint for the velocity, we can define an effective angular variable, the phase, which parametrizes the rotation of the velocity and satisfies the constraint. The relevant momentum then is the generator of this internal rotation, that is the spin. The resistance of the particle to changes in its spin is the turning inertia. This set of variables, \((\varphi , s, \chi )\), has nothing to do with the more familiar set, \((\theta , l, I)\), of polar angle, orbital angular momentum and moment of inertia.

This unusual definition of momentum is the precise reason why the propagation phenomenon described by the inertial spin model is so akin to superfluidity. One could object that linear propagation of information is in fact the most mundane thing of all, sound waves being its most obvious example. The point is that sound waves transport density fluctuations, the symmetry involved is translation, and the relevant momentum is \(p\). On the other hand, second sound waves, those associated to superfluidity, transport phase fluctuations, the symmetry is internal rotation, and momentum is \(s\), the spin. This transport of phase and spin *is* typical of superfluid systems that transport quantum phase. However weird this connection between flocks and quantum systems may seem, it is mathematically exact.

What is surprising is that the new phenomenology of linear propagation is obtained at the rather low price of introducing just one more parameter to the standard Vicsek model, that is the spin inertia \(\chi \). Moreover, because of the standard Hamiltonian decoupling between coordinates and momenta, the static (short time) properties of the inertial spin model are exactly the same as the Vicsek model, so that we do not have to worry about how to recover the rich physical description of static correlations previously achieved. In the overdamped limit momentum and inertia become irrelevant and ones recovers the original Vicsek dynamics. Finally, the large length-scale behavior is described by the standard hydrodynamics of flocking [20]. Hence, the inertial spin model introduced here achieves quite a lot in terms of new and experimentally accurate phenomenoogy, at a very little cost in terms of modeling complexity.

At the more concrete numerical level, we have clearly shown that the inertial spin model describes turns accurately. A change of direction can successfully be initiated by just *one* bird and the turn propagates very efficiently to the rest of the flock. The speed of propagation is related to the strength of the alignment interaction and to the turning inertia exactly in the way predicted by our mathematical equations and observed in real experiments [24]. On the contrary, if spin conservation is neglected (overdamped or Vicsek case) no collective turn is achieved. Interestingly, due to the active nature of flocking, the effect of overdamping is even more dramatic than predicted in the fixed-lattice case: instead of a diffusive and damped propagation of the signal, one observes no propagation at all, as the failure to transport the turning information ultimately produces the break up of the flock.

An interesting question is what happens when considering very large systems and time scales. This is the realm addressed by the hydrodynamic theories of flocking [12, 20, 22]. On these scales the motion of the network cannot be disregarded and a non-trivial hybridization occurs between density modes (caused by the relative movement of the particles) and orientational modes (due to alignment between velocities). As we already discussed in the paper, if some rotational dissipation is present in the system (i.e. \(\eta \ne 0\)), the asymptotic long wavelength behavior of our model is the same as the overdamped Vicsek limit, and we therefore expect the predictions of current hydrodynamic theories to describe it appropriately. One can however wonder whether a hydrodynamic description exists that also accounts for the underdamped behavior occurring at smaller scales, where many relevant phenomena like collective turns in flocks take place. To do so, one would need to modify the hydrodynamic equations to include a weakly damped spin field. This opens new interesting perspectives at the theoretical level.

What we have done here and in Reference [24] is to push one step further the ambitious program to tame the vast richness of biological phenomena using the powerful conceptual framework of theoretical physics. This program is of course not new. Kinetic approaches [47, 48] and hydrodynamic theories of active matter [20, 22, 23] used tools as kinetic theory, Navier-Stokes equations and dynamical renormalization group to determine the long distance and large time scaling properties of flocks and to explain why the Mermin–Wagner theorem is violated. At the static level, the maximum entropy approach [25, 49, 50] endeavors to define a thermodynamic description of biological collective systems based exclusively on the experimental data. In this approach, theoretical physics tools, as the Goldstone theorem and scaling relations, have powerful consequences [26].

Here, however, we do something different. We bring into the arena the abstract power of symmetries and conservation laws in linking static properties to dynamical predictions at the level of the microscopic description of the system. Symmetry is not necessarily a dynamical concept: it is a very general and profound condition that a system may or may not enjoy. However, the conservation laws predicted by symmetry through Noether’s theorem and Hamilton’s description have deep dynamical implications, which, as we have seen, can be experimentally verified in natural system and in numerical simulations alike. To do this the right set of *effective* variables must be found—a recurrent and quite natural theme in physics. We believe that this kind of approach may have some generality in biological systems other than flocks.

## Notes

### Acknowledgments

We thank William Bialek, Serena Bradde, Paul Chaikin and Dov Levine for discussions. Work in Rome was supported by Grants IIT–Seed Artswarm, ERC–StG n.257126 and US-AFOSR - FA95501010250 (through the University of Maryland). Work in Paris was supported by Grant ERC–StG n. 306312.

## Supplementary material

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