An alternative explanation for scale-free speed correlations in starling flocks: coarse-graining in time

Abstract: In a celebrated series of experimental observations, starling flocks have been shown to be characterized by scale-free, long-ranged spatial correlations in their velocity fluctuations. While this is expected for velocity orientation correlations on the basis of simple symmetry-breaking arguments, the same scaling-free behaviour for speed (i.e. the absolute value of birds’ velocity) correlations cannot be explained by the same symmetry-based argument. Possible explanations so far put forward required the implicit or explicit fine-tuning of a speed control parameter. In this work we explore a different possibility, investigating the effects of the experimental discrete temporal sampling of individual bird trajectories. We argue that observed velocity may well be a time coarse-grained observable, that is, the sum over many faster course corrections taken by the bird. A simple argument shows such a time coarse-grained speed to be linked with the squared fluctuations of (soft modes) transversal velocities, which may thus acquire a long-range correlation. Our idea is numerically tested by measuring spatial correlations between coarse-grained speeds in the on-lattice equilibrium XY model and the off-lattice out-of-equilibrium Vicsek model in two dimensions. Saturation of the speed correlation length is found in the equilibrium XY model, while in the non-equilibrium Vicsek model ordered symmetry-broken phase shows scale-free behaviour with a correlation length ξ is found to be proportional to system size L. We conclude that in non-equilibrium flocking models, the temporal coarse-graining procedure is able to reproduce scale-free behaviour at system sizes which are relevant to the experimental observations. We believe that this mechanism might find applications beyond the case of starling flocks and perhaps be relevant for other experimental observations of collective motion.