Low-temperature marginal ferromagnetism explains anomalous scale-free correlations in natural flocks

Abstract

We introduce a new ferromagnetic model capable of reproducing one of the most intriguing properties of collective behaviour in starling flocks, namely the fact that strong collective order coexists with scale-free correlations of the modulus of the microscopic degrees of freedom, that is the birds' speeds. The key idea of the new theory is that the single-particle potential needed to bound the modulus of the microscopic degrees of freedom around a finite value, is marginal, that is, it has zero curvature. We study the model by using mean-field approximation and Monte Carlo simulations in three dimensions, complemented by finite-size scaling analysis. While at the standard critical temperature, T c , the properties of the marginal model are exactly the same as a normal ferromagnet with continuous symmetry-breaking, our results show that a novel zero-temperature critical point emerges, so that in its deeply ordered phase the marginal model develops divergent susceptibility and correlation length of the modulus of the microscopic degrees of freedom, in complete analogy with experimental data on natural flocks of starlings.

Renormalization group study of marginal ferromagnetism

“…A different approach, still based on ferromagnetism, was proposed in [16], and successfully tested against experimental data in [15]. The idea of the new approach is to have zero curvature of the bare potential from the outset, without the need to decrease the overall amplitude of the bounding potential.…”

Comment on "Velocity and Speed Correlations in Hamiltonian Flocks" by M. Casiulis et al. [arXiv:1911.06042]

“…Scale-free velocity correlations have also been observed in real flocks of birds [2]. M. Casiulis et al analyze one flock in the dataset of [2] and claim that the data support the rigid-body rotation scenario also in real flocks, hence concluding that previous explanations of scale-free correlations in flocks, namely Goldstone modes in the velocity orientations [2][3][4] and a marginal (or nearcritical) mode in the speed [5,6], are unnecessary.…”

Velocity and Speed Correlations in Hamiltonian Flocks

“…On the experimental side, most quantitative data were obtained in artificial systems [17][18][19][20]. One noticeable exception is the large scale observational and data analysis effort conducted by the Starflag project [21][22][23][24][25][26][27]: the individual three-dimensional trajectories of a few thousand birds in compact flocks were obtained and analyzed. In particular, such starling flocks display correlations for the bird speeds and velocities that have been described as being long-ranged and scale-free [21,27].…”

“…[21], while the hydrodynamic theories of flocking are good candidates to explain the correlations of the displacement vectors, in particular the ones of their orientations, they do not explain the scale-free nature of the speed correlations. Explaining the latter within a critical framework requires either the application of an external dynamical field [22], or the introduction of a free energy with a marginal direction [27].…”

“…In the latter, the scaling and functional form of the correlations stem from rigid body rotations around the center of mass of the flock. Hence the question: in bird flocks, what is the part of the correlations due to, on the one hand, the Goldstone mode along the longitudinal direction of an effective potential (C d ) [21] and the marginality of its transverse direction (C sp ) [27] and, on the other hand, rotations? It is not obvious which effect is stronger.…”