Boundary Information Inflow Enhances Correlation in Flocking

Abstract

The most conspicuous trait of collective animal behaviour is the emergence of highly ordered structures. Less obvious to the eye, but perhaps more profound a signature of self-organization, is the presence of long-range spatial correlations. Experimental data on starling flocks in 3d show that the exponent ruling the decay of the velocity correlation function, C(r) ∼ 1/r γ , is extremely small, γ ≪ 1. This result can neither be explained by equilibrium field theory, nor by off-equilibrium theories and simulations of active systems. Here, by means of numerical simulations and theoretical calculations, we show that a dynamical field applied to the boundary of a set of Heisemberg spins on a 3d lattice, gives rise to a vanishing exponent γ, as in starling flocks. The effect of the dynamical field is to create an information inflow from border to bulk that triggers long range spin wave modes, thus giving rise to an anomalously long-ranged correlation. The biological origin of this phenomenon can be either exogenous -information produced by environmental perturbations is transferred from boundary to bulk of the flock -or endogenous -the flock keeps itself in a constant state of dynamical excitation that is beneficial to correlation and collective response.PACS numbers: 05.65.+b, 75.10.Hk, 05.50.+q Flocking, the collective motion displayed by large groups of birds, is one of the most spectacular examples of emergent collective behavior in nature, and it has fascinated inquiring minds since a long time [1]. Statistical physicists have tackled the problem via minimal models of self propelled particles (SPP) [2,3] and hydrodynamic continuum theories [4][5][6]. Such studies showed that flocking can be interpreted as a spontaneous symmetry breaking phenomenon occurring in a "moving ferromagnetic spin system", a sort of non-equilibrium counterpart of the well known Heisenberg model [7]. The basic ingredients of this description -self propulsion, lack of Galileian invariance and momentum conservation, local ferromagnetic interactions -define an extremely rich universality class, able to describe systems as diverse as vertebrate herds [8] Flocking, however, remains a prominent instance of collective animal motion for two reasons. First, it involves large numbers of individuals, hence justifying a statistico-mechanical approach to the problem. Second, unlike for most 3d systems, for flocks of starlings (Sturnus vulgaris) we have experimental data [14,15], against which theories and models can be tested. The statistical analysis of individual positions and velocities has revealed several unexpected physical features that need to be explained. In particular, it was found in [16] where C ij = δv i ·δv j and r ij is the distance between birds i and j. In systems of finite size L, due to the global constraint i δv i = 0, the function C(r) has a zero, which can be used as a finite-size definition of the correlation length ξ, C(r = ξ) = 0. In starling flocks it was found that ξ ∼ L, namely the correlation function is scale-free [16...

Scale-Free Correlations in Flocking Systems with Position-Based Interactions

“…In the current case, our results suggest that the scale-free dynamics observed in starling flocks may not require a subtle explanation like the development of critical dynamics, but could instead be simply due to the presence of position-based interactions that had been completely neglected in some of the previous models. We note that more recent publications based on the same data provide further arguments for the critical dynamics of starling flocks [4,5,13], but that they all neglect position-based interactions and could thus also overlook the effects discussed here. In particular, it is shown in [4] (published after the initial submission of this paper) that a Hamiltonian with only velocity-based interactions fits the data correctly when its control parameter is poised near a critical point and the boundary conditions of the flock are matched to the experiments.…”

Milling and meandering: Flocking dynamics of stochastically interacting agents with a field of view

“…In this regard, the existence of the wriggling pattern, which has a rougher boundary, is due to the fact that the stochasticity at the edge is reduced for larger values of σ. These results are intriguing in light of recent observations that the boundary of a flock plays an important role in its emergent dynamical properties [28]. Additionally, we note that as the alignment probability in our model is dependent on θ max , there is an inherent spatial anisotropy in the stochastic interactions.…”

Density regulation in strictly metric-free swarms

“…Those in the bulk are the safest from predation, so are least likely to try to change their relative position or the properties of the swarm as a whole [33]. Recent studies on large starling flocks have also noted that there is a local rise in bird density toward the edge of flocks, which would tend to support some distinction between edge and bulk, such as we introduce here [19,34].…”