Interactions among neighboring birds in a flock cause an alignment of their flight directions. We show that the minimally structured (maximum entropy) model consistent with these local correlations correctly predicts the propagation of order throughout entire flocks of starlings, with no free parameters. These models are mathematically equivalent to the Heisenberg model of magnetism, and define an "energy" for each configuration of flight directions in the flock. Comparing flocks of different densities, the range of interactions that contribute to the energy involves a fixed number of (topological) neighbors, rather than a fixed (metric) spatial range. Comparing flocks of different sizes, the model correctly accounts for the observed scale invariance of long ranged correlations among the fluctuations in flight direction.
@article{arxiv.1107.0604,
title = {Statistical mechanics for natural flocks of birds},
author = {William Bialek and Andrea Cavagna and Irene Giardina and Thierry Mora and Edmondo Silvestri and Massimiliano Viale and Aleksandra M Walczak},
journal= {arXiv preprint arXiv:1107.0604},
year = {2015}
}