No abstract

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 v = 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 1 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 GinsburgLandau theory for an n = d component isotropic ferromagnet. In spatial dimensions d > 4, 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 d < 4, non-linear 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, longrange 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.

We review the past decadeÕs theoretical and experimental studies of flocking: the collective,

We construct the equations of motion for the coupled dynamics of order parameter and concentration for the nematic phase of driven particles on a solid surface, and show that they imply (i) giant number fluctuations, with a standard deviation proportional to the mean and (ii) long-time tails ∼ t −d/2 in the autocorrelation of the particle velocities in d dimensions despite the absence of a hydrodynamic velocity field. Our predictions can be tested in experiments on aggregates of amoeboid cells as well as on layers of agitated granular matter.

Fluids of exciton-polaritons, excitations of two dimensional quantum wells in optical cavities, show collective phenomena akin to Bose condensation. However, a fundamental difference from standard condensates stems from the finite life-time of these excitations, which necessitate continuous driving to maintain a steady state. A basic question is whether a two dimensional condensate with long range algebraic correlations can exist under these non-equilibrium conditions. Here we show that such driven two-dimensional Bose systems cannot exhibit algebraic superfluid order except in low-symmetry, strongly anisotropic systems. Our result implies, in particular, that recent apparent evidence for Bose condensation of exciton-polaritons must be an intermediate scale crossover phenomenon, while the true long distance correlations fall off exponentially. We obtain these results through a mapping of the long-wavelength condensate dynamics onto the anisotropic Kardar-Parisi-Zhang equation.arXiv:1311.0876v2 [cond-mat.stat-mech]