We study a model of flocking for a very large system ͑N 320 000͒ numerically. We find that in the long wavelength, long time limit, the fluctuations of the velocity and density fields are carried by propagating sound modes, whose dispersion and damping agree quantitatively with the predictions of our previous work using a continuum equation. We find that the sound velocity is anisotropic and characterized by its speed c for propagation perpendicular to the mean velocity ͗ y͘, ͗ y͘ itself, and a third velocity l͗ y͘, arising explicitly from the lack of Galilean invariance in flocks. [S0031-9007 (98)06186-9] PACS numbers: 87.10. + e, 05.60. + w, 64.60.CnThe dynamics of "flocking" behavior of living things, such as birds, fish, wildebeest, slime molds, and bacteria has long attracted a great deal of attention among biologists, computer animators, and physicists [1][2][3]. It is crucial to correctly describe the interaction between members of the flock in order to understand and model the flocking behavior. As summarized in [2], a large flock does not have a global leader; instead, the impressive collective flocking phenomena is caused by individual members of the flock following the motion of their neighbors.In our earlier work [4], we studied the flocking dynamics by using continuum equations for the coarse-grained density field r͑x, t͒ and velocity field y͑ x, t͒, written as ≠ t y 1 l͑ y ? =͒ y a y 2 bj yj 2 y 2 =Pwhere b, D 1 , D 2 , and D L are all positive, and a , 0 in the disordered phase and a . 0 in the ordered state. The a and b terms simply make the local y have a nonzero magnitude ͑ p a͞b ͒ in the ordered phase. D L,1,2 are diffusion constants. The Gaussian random noise f has correlations:where D is a constant, and i, j denote Cartesian components. Finally, the pressure P P͑r͒ Pǹ 1 s n ͑r 2 r 0 ͒ n , where r 0 is the mean of the local number density r͑ r͒ and s n are coefficients in the pressure expansion. The final equation (2) reflects conservation of birds.In [4], we considered the special case of (1) with l 1. Just as the absence of the Galilean invariance for the flock motion allows a and b fi 0 in Eq. (1), likewise l need not be 1. In [5] and this paper, we consider the more generic case l fi 1, which leads to a different direction dependence of the sound speed than when l 1 [6].In the ordered phase where a . 0, the velocity field and the density field can be written as y y sxjj 1 dy, r r 0 1 dr, where r 0 and y sxjj are the space averaged density and spontaneous symmetry broken velocity, respectively. The spontaneous symmetry breaking of a vector field leads to large "Goldstone mode" fluctuations; in flocks, this mode is y Ќ , the projection of d y perpendicular tox jj [we will hereafter use "jj" ("Ќ") to denote the projection of any vector along (perpendicular to)x jj ]. Indeed, for equilibrium systems, such fluctuations are strong enough in two dimensions to destroy the long range order [7]. One of the remarkable predictions of our continuum model of flocking is that the ordered state is stable ...
