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Flocking and turning: a new model for self-organized collective motion
Andrea Cavagna, Lorenzo Del Castello, Irene Giardina, Tomas Grigera, Asja Jelic, Stefania Melillo, Thierry Mora, Leonardo Parisi, Edmondo Silvestri, Massimiliano Viale, Aleksandra M. Walczak
TL;DR
Collective turning in polarized flocks is not captured by standard overdamped alignment dynamics because rotational information and behavioral inertia are missing. The paper introduces a phase–spin model based on rotational symmetries and conservation laws, derives its velocity dynamics, and finds that it supports experimentally consistent propagating turns while recovering Vicsek and long-scale hydrodynamic limits.
Problem
The Vicsek model does not reproduce the rapid, density-independent propagation of directional and curvature information observed during collective turns in natural flocks.
Method
The paper constructs microscopic flocking equations from generalized phase and spin coordinates, rotational symmetry, spin conservation, and spin inertia, then derives the corresponding velocity dynamics.
Results
The inertial spin model accurately describes turns initiated by one bird and propagating through the flock, whereas neglecting spin conservation prevents collective turns.
Takeaways & Limitations
Retaining rotational momentum enables undamped phase propagation while preserving the model’s agreement with Vicsek dynamics in the overdamped limit and flocking hydrodynamics at large scales.
Takeaways & Limitations
At asymptotically large length scales with rotational dissipation, the model reduces to the overdamped Vicsek limit and standard flocking hydrodynamics.
Abstract
from arXiv · showhide
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.
I. SETTING UP THE STAGE
The paper introduces a microscopic flocking model by moving from constrained particle variables to effective rotational coordinates, then using symmetry and conservation laws to organize the dynamics. The approach is motivated by collective turns that require dynamics beyond hydrodynamic descriptions.
- I. SETTING UP THE STAGE: The model targets collective turning and information propagation, which occur on finite spatial and temporal scales beyond hydrodynamics.The authors present the microscopic equations as a proposed model whose validity is checked against numerical simulations.
- I. SETTING UP THE STAGE: The formalism replaces complicated constraint-enforcing forces with effective canonical variables, while treating real flock motion as dissipative at the individual level.The paper argues that the relevant constraint can be handled through an appropriate choice of variables and Hamiltonian structure.
- I. SETTING UP THE STAGE: A constrained point is more conveniently described by a polar angle and its conjugate angular momentum than by standard position and momentum variables.The angular coordinate automatically enforces confinement to a circle, while the conjugate momentum generates rotations.
- I. SETTING UP THE STAGE: Rotational symmetry conserves the generator of rotation, whereas dissipation would eventually stop the rotation without external angular-momentum injection.This example supplies the mechanical template later generalized to velocity rotations in flocking.
B. Circular motion in the internal space of velocity
For constant-speed birds, the velocity direction is treated as an internal rotation described by phase and spin. Rotational symmetry conserves total spin, providing the mechanism for coordinated turning in the many-particle system.
- B. Circular motion in the internal space of velocity: Constant-speed motion requires a phase variable that parametrizes velocity orientation, analogous to the angular coordinate for constrained circular motion.The phase describes rotation in the internal velocity space rather than rotation of position in external space.
- B. Circular motion in the internal space of velocity: The spin is the generator conjugate to the velocity phase and measures resistance to changing the bird’s spin through the generalized inertia χ.In the force-free idealization, conserved spin produces constant angular velocity of the velocity direction.
- B. Circular motion in the internal space of velocity: Dissipation causes spin and angular velocity to decay in the absence of external spin injection.The idealized conserved-spin motion therefore becomes transient once dissipative effects are included.
- B. Circular motion in the internal space of velocity: For interacting particles, rotationally symmetric alignment interactions conserve total spin even though individual spins change.In the continuum limit, this conservation law becomes a spin-field continuity equation associated with propagation of undamped turns.
- B. Circular motion in the internal space of velocity: Phase rotations act on velocities and produce equal-radius trajectories, matching the observed turning behavior of cohesive flocks.This correspondence supports phase and spin as the appropriate canonical variables for flock turning.
II. DETERMINISTIC EQUATIONS FOR THE VELOCITY AND THE SPIN
The paper derives velocity equations from the phase–spin Hamiltonian structure and extends them to three dimensions. The resulting pseudo-Hamiltonian dynamics conserve speed while making spin mediate changes in trajectory curvature.
- II. DETERMINISTIC EQUATIONS FOR THE VELOCITY AND THE SPIN: The velocity is a noncanonical observable, so its evolution is obtained from Poisson brackets with the Hamiltonian rather than by directly using canonical equations.The derivation also includes the kinematic relation between velocity and particle position.
- II. DETERMINISTIC EQUATIONS FOR THE VELOCITY AND THE SPIN: In three dimensions, three phases parametrize rotations about the Cartesian axes, and the spin becomes a three-dimensional vector.For example, the z-phase rotates the velocity projection in the xy plane while leaving its z component unchanged.
- II. DETERMINISTIC EQUATIONS FOR THE VELOCITY AND THE SPIN: The derived inertial spin equations retain a pseudo-Hamiltonian structure and conserve individual speed, v_i·s_i, and the Hamiltonian H.The velocity and spin are not canonical pairs, but vector products preserve these quantities in the deterministic dynamics.
- II. DETERMINISTIC EQUATIONS FOR THE VELOCITY AND THE SPIN: Alignment forces act through neighboring velocities, with interaction strength J and connectivity n_ij defining the social contribution.The force is obtained from the Hamiltonian expressed in velocity variables.
- II. DETERMINISTIC EQUATIONS FOR THE VELOCITY AND THE SPIN: Zero spin yields straight trajectories, while constant nonzero spin produces circular motion with curvature radius R ∼ v0χ/|s_i|.Forces change spin and therefore curvature rather than changing velocity direction abruptly; χ measures resistance to changing curvature.
B. Spin conservation and information propagation
The model links global and local spin conservation to undamped, linearly propagating phase disturbances that carry turning information through polarized flocks. A fixed interaction network permits the continuum derivation, while moving-neighbor networks violate this approximation.
- Spin conservation: Global spin conservation follows from the continuous rotational symmetry and implies that local curvature excitations must be carried away rather than dissipated locally.A strong misalignment excites local spin; conservation of the global spin produces propagating modes.
- Continuum description: In an ordered flock, phase variations can be represented by a continuum phase ϕ(r,t) and spin field s(r,t) when they are smooth over distances larger than the neighbor spacing.The interaction potential is approximated using spatial gradients and the average number of interacting neighbors.
- Continuum description: The spin equation becomes a continuity equation with current j = n_ca^2J∇ϕ, and the phase consequently obeys d’Alembert’s equation.The local conservation law is the continuum counterpart of global spin conservation.
- Information propagation: The resulting deterministic flock has undamped propagating phase modes with linear dispersion and speed c_s determined by alignment strength and generalized inertia.These modes transmit turning information: a local turn changes nearby particles’ spin and causes a collective turn.
- Information propagation: The continuum result requires a time-independent interaction matrix, an assumption generally violated by off-lattice moving particles.Without fixed connectivity, the continuous Laplacian and d’Alembert equation cannot be obtained in this form.
C. Dynamical universality class
The paper places flock turning in a dynamical class where a non-conserved direction field co-evolves with a conserved spin generated by rotational symmetry. This structure produces undamped linear propagation, unlike systems whose conserved field is not a symmetry generator.
- Symmetry and conservation: Rotational symmetry connects the equations of motion to a conserved quantity and to linear propagation of local disturbances in flight-direction dynamics.The paper identifies this symmetry–conservation–propagation connection as novel in the context of flocking particles.
- Universality class: Flock equations share the formal structure and dynamical universality class of planar ferromagnets and superfluid helium, classified as model F.In these systems, a non-conserved order parameter co-evolves with a conserved quantity.
- Universality class: Linear spin propagation requires the conserved quantity to generate a spontaneously broken continuous symmetry, not merely to be conserved.When the conserved field is instead the order parameter, as in the isotropic ferromagnet, phase propagation becomes dissipative and diffusive.
III. A NEW MODEL OF COLLECTIVE MOTION
The inertial spin model adds dissipation and noise to a spin-conserving deterministic structure while preserving fixed particle speed. Its spin inertia, alignment, friction, and temperature govern the dynamics, with friction damping curvature and turning.
- Motivation for dissipation: The deterministic inertial spin model conserves spin too effectively: without forces, angular velocity remains constant forever.The model therefore requires dissipation to prevent undamped turning over infinite distances.
- Model construction: Dissipation is coupled to spin and accompanied by noise, producing the stochastic inertial spin model.The construction parallels the need for damping in ordinary sound propagation while retaining spin-mediated propagation.
- Model construction: The final equations preserve the fixed-speed constraint |v_i| = v_0 while introducing generalized viscous damping and noise.The noise is represented by an independent vectorial term, and the velocity remains constrained to constant magnitude.
- The inertial spin model: The model is expressed in terms of velocities and spins, with alignment strength J, spin inertia χ, viscous coefficient η, and temperature T as its four important parameters.The spin inertia χ is the new ingredient in the dynamics.
- The inertial spin model: When social forces and noise vanish, nonzero friction makes spin decay exponentially, reducing trajectory curvature until motion becomes straight.The coefficient η acts as rotational dissipation rather than damping the fixed velocity magnitude.
B. A closed equation for the velocity
The paper rewrites the inertial spin dynamics as a closed generalized Langevin equation for velocity vectors. The equation combines curvature inertia, a speed-constraint term, dissipation, social forces, and noise, with symmetry determining the inertial structure.
- Closed velocity dynamics: Differentiating the velocity equation and using the spin equation yields a closed second-order equation for the velocities.This eliminates the spin variable from the explicit dynamical description.
- Closed velocity dynamics: The resulting equation is a generalized Langevin equation whose inertial coefficient χ measures resistance to changing trajectory curvature rather than particle mass.Its left-hand side also contains a constraint-induced centripetal term and dissipation, while the right-hand side contains social force and noise.
- Symmetry and propagation: Unlike other second-order flocking equations, the model automatically connects inertial terms to rotational symmetries and thereby shapes information propagation.The authors identify this symmetry–inertia connection as essential to the model’s deterministic limit.
C. Overdamped limit: the Vicsek model
The model’s overdamped limit is obtained when spin inertia is negligible relative to spin dissipation, yielding the continuous-time Vicsek model. The resulting dynamics preserves constant speed through the perpendicular component of the social force.
- Overdamped reduction: χ/η^2 → 0 gives the overdamped limit of the inertial spin model, recovering the continuous-time Vicsek equation.The limit follows from the full Langevin equation when spin inertia becomes negligible compared with dissipation.
- Overdamped reduction: Rescaling time by t′ = t/η removes η from the velocity equation and noise correlator, explaining the χ/η^2 scaling.
- Force interpretation: The double cross product projects the social force perpendicular to velocity, conserving each particle’s constant speed.Only the perpendicular force component changes the velocity direction; the parallel component cannot change its norm.
D. Steady state distribution of velocity and spin
The rotationally symmetric model admits a steady-state distribution for velocities and spins described through a Fokker–Planck equation. Its velocity sector has a Boltzmann form corresponding to a Heisenberg model on the particles’ interaction lattice.
- Steady-state distribution: The joint velocity–spin distribution obeys a Fokker–Planck equation with a steady-state solution.
- Steady-state distribution: The steady-state distribution factorizes into velocity and spin components, allowing velocities to be statistically described by expectation values.The ratio between noise and dissipation acts as an effective temperature in the Boltzmann distribution.
- Ordered phase: The velocity distribution is a Heisenberg model on a random Euclidean lattice and generates soft Goldstone modes with long-range transverse velocity correlations in the ordered phase.
IV. INFORMATION PROPAGATION IN THE ORDERED PHASE
In the ordered phase, coupling between velocity and spin produces directional modes whose propagation depends on the balance between inertia and dissipation. The model spans undamped propagation, attenuated waves, and Vicsek-like exponential damping across spatial scales.
- Linearized dynamics: The linearized phase dynamics is derived using two transverse phases around the collective velocity, assuming the interaction network remains fixed over turn timescales.The network is later allowed to rearrange in numerical simulations, while the fixed-network approximation enables analytic dispersion relations.
- Dispersion relation: The dispersion relation determines whether local flight-direction disturbances propagate through the flock or are damped, depending especially on inertia and dissipation.
- Limiting regimes: η → 0 yields undamped linear propagation, whereas χ/η^2 → 0 gives the Vicsek limit with purely imaginary diffusive dispersion and no propagating turns.
- Finite dissipation: For nonzero inertia and dissipation, k ≥ k0 supports attenuated propagating waves, while k < k0 gives evanescent waves with exponential decay.
- Finite flock scales: In finite flocks, sufficiently weak dissipation leaves damping slower than the crossing time, so information propagates effectively with limited attenuation across the group.Natural starling flocks are reported to occupy this underdamped, quasi-deterministic regime.
A. Transition to the hydrodynamic regime
At very large length and time scales, dissipation relaxes the global spin and removes the propagating directional modes. The resulting behavior connects to hydrodynamic flocking theories, while shorter-scale quasi-deterministic modes describe collective turns.
- Transition to hydrodynamics: As k → 0, the frequency becomes purely imaginary because dissipation relaxes the global spin and removes the conservation law supporting propagating modes.
- Transition to hydrodynamics: In the quasi-deterministic regime, spin is quasi-conserved and velocity–spin coupling supports propagating changes of direction and curvature.
- Transition to hydrodynamics: At hydrodynamic scales, network movement couples directional changes to local-density changes, producing propagating sound modes.
V. NUMERICAL SIMULATIONS
Numerical simulations test how dissipation, spin inertia, and alignment determine whether directional information propagates through an ordered flock and produces a coherent collective turn. The underdamped regime reproduces linear propagation and coordinated turning, whereas overdamping attenuates the signal and can destroy cohesion.
- Simulation setup: Simulations vary η, χ, and J across overdamped and underdamped regimes to test information propagation and collective turning in ordered flocks.The simulations use topological interactions and low noise to generate strongly polarized flocks.
- Regimes: For η^2/χ > n_cJ(a/L)^2, dissipation causes orientational signals to decay on large scales, with the Vicsek limit eliminating propagation except at hydrodynamic scales.The overdamped regime permits only attenuated propagation below a characteristic wave-number scale.
- Regimes: For η^2/χ ≪ n_cJ(a/L)^2, signals propagate linearly through the system with negligible attenuation, and the propagation speed depends on J/χ.The underdamped regime supports the signal behavior required for collective turns.
- Simulation outcomes: Strong overdamping prevents a collective turn, while intermediate overdamping attenuates the signal enough to break flock cohesion and produce fragments.In the strongest overdamped case, particles retain the original flight direction; in the intermediate case, the flock loses coherence and cohesion.
- Simulation outcomes: In the underdamped regime, all particles follow the initiator with negligible attenuation, turning coherently while the flock retains its shape and cohesion.The individual velocity and acceleration profiles are coordinated with time shifts corresponding to turn propagation.
- Propagation law: The turning-front distance grows linearly with time, and simulations reproduce the predicted dependence of propagation speed on J/χ and the experimentally observed relation with flock order.Ranks are inferred from cosine or acceleration profiles, and the two procedures give equivalent results.
VI. CONCLUSIONS
The paper argues that internal rotational momentum and its conservation are essential for propagating collective turns, and connects this mechanism to a broader symmetry-based description of active matter. The model adds spin inertia while retaining Vicsek static behavior and Toner–Tu hydrodynamics at large scales, though its underdamped finite-scale behavior is not captured by the standard hydrodynamic description.
- Conclusions: Conserved internal angular momentum and spin inertia generate the linear propagation law needed for collective turns, whereas removing momentum collapses propagation.The relevant momentum is the spin associated with internal rotation, not linear or orbital angular momentum.
- Conclusions: Spin transports phase fluctuations through internal rotational dynamics, making the model’s propagation mathematically analogous to second sound in superfluid systems.The analogy concerns the symmetry, transported quantity, and associated momentum.
- Conclusions: The model introduces spin inertia χ as one additional parameter while preserving the Vicsek model in the overdamped limit and matching its static short-time properties.Momentum and inertia become irrelevant in the overdamped limit.
- Conclusions: Numerically, the model allows a single bird to initiate a turn that propagates efficiently through the flock, with speed governed by alignment strength and turning inertia as predicted and observed experimentally.The conclusion emphasizes agreement between the model’s propagation law and real-flock observations.
- Scope and limitations: At very large systems and time scales, the model reduces to standard flocking hydrodynamics when rotational dissipation is present, while smaller-scale underdamped behavior would require a weakly damped spin field.Network motion hybridizes density and orientational modes in the asymptotic long-wavelength regime.
- Conclusions: The framework applies symmetry and conservation-law reasoning to microscopic active-matter dynamics rather than relying only on long-distance hydrodynamics.The authors present this as a broader theoretical-physics approach to biological collective motion.