Flocks, herds, and schools: A quantitative theory of flocking

We present a quantitative continuum theory of ``flocking'': the collective coherent motion of large numbers of self-propelled organisms. In agreement with everyday experience, our model predicts the existence of an ``ordered phase'' of flocks, in which all members of even an arbitrarily large flock move together with the same mean velocity $〈\stackrel{\ensuremath{\rightarrow}}{v}〉\ensuremath{\ne}0.$ This coherent motion of the flock is an example of spontaneously broken symmetry: no preferred direction for the motion is picked out a priori in the model; rather, each flock is allowed to, and does, spontaneously pick out some completely arbitrary direction to move in. By analyzing our model we can make detailed, quantitative predictions for the long-distance, long-time behavior of this ``broken symmetry state.'' The ``Goldstone modes'' associated with this ``spontaneously broken rotational symmetry'' are fluctuations in the direction of motion of a large part of the flock away from the mean direction of motion of the flock as a whole. These ``Goldstone modes'' mix with modes associated with conservation of bird number to produce propagating sound modes. These sound modes lead to enormous fluctuations of the density of the flock, far larger, at long wavelengths, than those in, e.g., an equilibrium gas. Our model is similar in many ways to the Navier-Stokes equations for a simple compressible fluid; in other ways, it resembles a relaxational time-dependent Ginsburg-Landau theory for an $n=d$ component isotropic ferromagnet. In spatial dimensions $dg4,$ the long-distance behavior is correctly described by a linearized theory, and is equivalent to that of an unusual but nonetheless equilibrium model for spin systems. For $dl4,$ nonlinear fluctuation effects radically alter the long distance behavior, making it different from that of any known equilibrium model. In particular, we find that in $d=2,$ where we can calculate the scaling exponents exactly, flocks exhibit a true, long-range ordered, spontaneously broken symmetry state, in contrast to equilibrium systems, which cannot spontaneously break a continuous symmetry in $d=2$ (the ``Mermin-Wagner'' theorem). We make detailed predictions for various correlation functions that could be measured either in simulations, or by quantitative imaging of real flocks. We also consider an anisotropic model, in which the birds move preferentially in an ``easy'' (e.g., horizontal) plane, and make analogous, but quantitatively different, predictions for that model as well. For this anisotropic model, we obtain exact scaling exponents for all spatial dimensions, including the physically relevant case $d=3.$