Anomalous Behaviour of Mutual Information in Finite Flocks

The existing consensus is that flocks are poised at criticality, entailing long correlation lengths and a maximal value of Shannon mutual information in the large-system limit. We show, by contrast, that for finite flocks in the long observation time limit, mutual information may not only fail to peak at criticality—as observed for other critical systems—but also diverge as noise tends to zero. This result carries implications for other finite-size, out-of-equilibrium systems, where observation times may vary widely compared to time scales of internal system dynamics; thus it may not be assumed that mutual information locates the phase transition.