Local equilibrium in bird flocks

Abstract

The correlated motion of flocks is an instance of global order emerging from local interactions. An essential difference with analogous ferromagnetic systems is that flocks are active: animals move relative to each other, dynamically rearranging their interaction network. The effect of this off-equilibrium element is well studied theoretically, but its impact on actual biological groups deserves more experimental attention. Here, we introduce a novel dynamical inference technique, based on the principle of maximum entropy, which accodomates network rearrangements and overcomes the problem of slow experimental sampling rates. We use this method to infer the strength and range of alignment forces from data of starling flocks. We find that local bird alignment happens on a much faster timescale than neighbour rearrangement. Accordingly, equilibrium inference, which assumes a fixed interaction network, gives results consistent with dynamical inference. We conclude that bird orientations are in a state of local quasi-equilibrium over the interaction length scale, providing firm ground for the applicability of statistical physics in certain active systems.

Fig. 1

Performance of the inference methods on the predicted interaction range n_c. A. Inferred versus real n_c obtained by applying our new inference method to simulated data generated with Eq. 1 at various interaction ranges. The method performs well for different values of the sampling rate dt. B. Dependence of the inferred n_c on the sampling time dt. On simulated data with n_c = 10 (dashed line), the inference method based on exact integration (red points) performs well regardless of the sampling time dt. By contrast, the inference method based on Euler’s integration method (green points) overestimates the true interaction range at large dt. C. A similar trend is observed when we apply the two inference procedures to real flocking data, as illustrated here on one flocking event. Note that in this case the true value is not known. Error bars represent standard errors over time frames.

Fig. 2

Comparison between the two relevant time scales of active matter, as inferred in 14 natural flocks using our inference method based on exact integration. Histograms of the neighbour exchange time τ_network versus the local alignment time τ_relax = 1/Jn_c, show that the relaxation of orientations is much faster than the turnover of neighbours. Note that the experimental sampling time dt = 0.2 s (dashed line) is of the same order as the alignment time, justifying the use of exact integration. Inset: the scatter plot of τ_relax versus τ_network shows no correlation between the two quantities.

Fig. 3

Inference on natural flocks. For each of the 14 flocking events, the parameters of the model were inferred using either the dynamical inference method presented here, with dt = 0.2 s, or an equilibrium inference method as in [13]. A. Both methods agree well on the predicted value of the alignment range n_c. B. While the dynamical method infers the alignment strength J and the noise amplitude T separately, the equilibrium method only infers their ratio J/T, the value of which is consistent between the two methods. Error bars represent standard errors over time frames.