Statistical mechanics for natural flocks of birds

Abstract

Flocking is a typical example of emergent collective behavior, where interactions between individuals produce collective patterns on the large scale. Here we show how a quantitative microscopic theory for directional ordering in a flock can be derived directly from field data. We construct the minimally structured (maximum entropy) model consistent with experimental correlations in large flocks of starlings. The maximum entropy model shows that local, pairwise interactions between birds are sufficient to correctly predict the propagation of order throughout entire flocks of starlings, with no free parameters. We also find that the number of interacting neighbors is independent of flock density, confirming that interactions are ruled by topological rather than metric distance. Finally, by comparing flocks of different sizes, the model correctly accounts for the observed scale invariance of long-range correlations among the fluctuations in flight direction.

Fig. 1.

The raw data. (A) One snapshot from flocking event 28 - 10, N = 1,246 birds (see SI Appendix, Table S1). (B) Instantaneous vector velocities of all the individuals in this snapshot, normalized as .

Fig. 2.

Setting the strength and range of interactions. (A) The predicted strength of correlation, C_int, as a function of the interaction strength J, with n_c = 11, for the snapshot in Fig. 1. Matching the experimental value of C_int = 0.99592 determines J = 45.73. (Inset) Zoom of the crossing point; error bars are obtained from the model’s predictions of fluctuations of C_int(J,n_c). (B) The log-likelihood of the data per bird () as a function of n_c with J optimized for each value of n_c; same snapshot as in A. There is a clear maximum at n_c = 11. (Inset) the log-likelihood per bird for other snapshots of the same flocking event. (C) The inferred value of J for all observed flocks, shown as a function of the flock’s size. Each point corresponds to an average over all the snapshots of the same flock. Error bars are standard deviations across multiple snapshots. (D) As in C but for the inferred values of n_c. Averaging over all flocks we find n_c = 21.2 ± 1.7 (black line). (E) The inferred value of the topological range as a function of the mean interbird distance in the flock, for all flocks. Error bars are standard deviations across multiple snapshots of the same flock. (F) As in E but for the metric range r_c. If interactions extend over some fixed metric distance r₀, then we expect and r_c = constant; we find the opposite pattern, which is a signature of interactions with a fixed number of topological neighbors (14).

Fig. 3.

Correlation functions predicted by the maximum entropy model vs. experiment. The full pair correlation function can be written in terms of a longitudinal and a perpendicular component, i.e., . Because the two components have different amplitudes, it is convenient to look at them separately. (A) Perpendicular component of the correlation, , as a function of the distance; the average is performed over all pairs ij separated by distance r. Blue diamonds refer to experimental data (for the snapshot in Fig. 1), red circles to the prediction of the model in Eq. 4. The dashed line marks the maximum r that contributes to C_int, which is the only input to the model. The correlation function is well fitted over all length scales. In particular, the correlation length ξ, defined as the distance where the correlation crosses zero, is well reproduced by the model. (Inset) ξ vs. size of the flock, for all the flocking events; error bars are standard deviations across multiple snapshots of the same flocking event. (B) Four-point correlation function , where the pairs ij and kl are as shown in the Inset (see also SI Appendix). The figure shows the behavior of C₄(r₁; r₂) as a function of r₂, with r₁ = 0.5. (C) Longitudinal component of the correlation , as a function of distance. Note that in the spin wave approximation, C^L(r) = 1 - C₄(0; r) - S². (D) Similarity between the predicted mean value of flight direction, , and real data, for all individual birds in the interior of the flock. The similarity can be quantified through the local overlap , which is plotted as a function of the distance of the individual from the border. Maximal similarity corresponds to q_i = 1. (Inset) Full distribution P(q) for all the interior birds.

Fig. 4.

Maximum entropy analysis for a model of self-propelled particles. (A) Inferred value of the parameter J vs. microscopic strength of alignment forces used in the simulation. (B) Inferred value of n_c vs. the true number of interacting neighbors in the simualtion. Slopes of the lines are 2.2 and 2.7, respectively. Error bars are standard deviations across 45 snapshots of the same simulation.