PHYSICAL REVIEW E VOLUME 58, NUMBER 4 OCTOBER 1998
Flocks, herds, and schools: A quantitative theory of flocking
John Toner Institute of Theoretical Science, Materials Science Institute, and Department of Physics, University of Oregon, Eugene, Oregon 97403-5203
Yuhai Tu IBM Thomas J. Watson Research Center, P.O. Box 218, Yorktown Heights, New York 10598 ͑Received 16 April 1998͒ 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 ͗vជ ͘Þ0. This coherent motion of the flock is an example of spontaneously broken sym- metry: 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 nϭd component isotropic ferromagnet. In spatial dimensions dϾ4, 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 dϽ4, 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 dϭ2, 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 dϭ2 ͑the ‘‘Mermin-Wagner’’ theorem͒. 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., horizontal͒ 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 dϭ3. ͓S1063-651X͑98͒08410-4͔
PACS number͑s͒: 87.10.ϩe, 64.60.Cn, 05.60.ϩw
I. INTRODUCTION ͑2͒ The interactions are purely short ranged: each ‘‘boid’’ only responds to its neighbors, defined as those ‘‘boids’’ A wide variety of nonequilibrium dynamical systems with within some fixed, finite distance R0 , which is assumed to be many degrees of freedom have recently been studied using much less than L, the size of the ‘‘flock.’’ powerful techniques developed for equilibrium condensed ͑3͒ The ‘‘following’’ is not perfect: the ‘‘boids’’ make matter physics ͑e.g., scaling, the renormalization group, etc.͒. errors at all times, which are modeled as a stochastic noise. One of the most familiar examples of a many-degree-of- This noise is assumed to have only short-ranged spatiotem- freedom, nonequilibrium dynamical system is a large flock poral correlations. of birds. Myriad other examples of the collective, coherent ͑4͒ The underlying model has complete rotational symme- motion of large numbers of self-propelled organisms occur try: the flock is equally likely, a priori, to move in any di- in biology: schools of fish, swarms of insects, slime molds, rection. herds of wildebeest, etc. The development of a nonzero mean center-of-mass ve- Recently, a number of simulations of this phenomenon locity ͗vជ ͘ for the flock as a whole therefore requires sponta- have been performed ͓1–3͔. Following Reynolds ͓3͔, we will neous breaking of a continuous symmetry ͑namely, rota- use the term ‘‘boid’’ and bird interchangeably for the par- tional͒. ticles in these simulations. All of these simulations have sev- In an earlier paper ͓4͔, we formulated a continuum model eral essential features in common: for such dynamics of flocking, and obtained some exact re- ͑1͒ A large number ͑a ‘‘flock’’͒ of point particles sults for that model in spatial dimensions dϭ2 ͑appropriate ͑‘‘boids’’͒ each move over time through a space of dimen- for the description of the motion of land animals on the sion d (ϭ2,3, ...),attempting at all times to ‘‘follow’’ ͑i.e., Earth’s surface͒. Our most surprising result ͓4͔ was that two- move in the same direction as͒ its neighbors. dimensional moving herds with strictly short-ranged interac-
1063-651X/98/58͑4͒/4828͑31͒/$15.00PRE 58 4828 © 1998 The American Physical Society PRE 58 FLOCKS, HERDS, AND SCHOOLS: A QUANTITATIVE . . . 4829 tions appear to violate the Mermin-Wagner theorem ͓5͔,in greater than the mean interbird distance͒. Our description is that they can acquire long-ranged order, by picking out a then valid for distances large compared to l 0 , and for times consistent direction of motion across an arbitrarily large t much greater than some microscopic time t0 , presumably herd, despite the fact that this involves spontaneously break- of order l 0 /vT , where vT is a typical speed of a bird. Col- ing a continuous ͑rotational͒ symmetry. lective motion of the flock as a whole then requires that Of course, this result does not, in fact, violate the ͗vជ (rជ,t)͘Þ0; where the averaging can be considered an en- Mermin-Wagner theorem, since flocks are a nonequilibrium semble average, a time average, or a spatial average. Equiva- dynamical system. What is fascinating ͑at least to us͒ about lently, long-ranged order must develop for the flock as a our result is that the nonequilibrium aspects of the flock dy- whole to move; i.e., the equal-time velocity autocorrelation namics that make the long-distance, long-time behavior of function: the flock different from that of otherwise analogous equilib- rium systems are fundamentally nonlinear, strong-fluctuation C͑Rជ ͒ϵ͗vជ ͑Rជ ϩrជ,t͒•vជ ͑rជ,t͒͘ ͑1.1͒ effects. Indeed, a ‘‘breakdown of linearized hydrodynam- ics,’’ analogous to that long known to occur in equilibrium must approach a nonzero constant as the separation ͉Rជ ͉ ϱ; specifically, fluids ͓6͔ in spatial dimensions dϭ2, occurs in flocks for all → dϽ4. This breakdown of linearized hydrodynamics is essen- C͑Rជ ϱ͒ ͉͗vជ ͉͘2. ͑1.2͒ tial to the very existence of the ordered state in dϭ2. Fur- → → thermore, it has dramatic consequences even for dϾ2. Thus the average velocity ͗vជ ͘ of the flock is precisely The physics of this breakdown is very simple: above d analogous to the order parameter sជ in a ferromagnetic sys- ϭ4, where the breakdown does not occur, information about ͗ ͘ what is going on in one part of the flock can be transmitted to tem, where sជ is a local spin. another part of the flock only by being passed sequentially Our most dramatic result is that an intrinsically nonequi- through the intervening neighbors via the assumed short- librium and nonlinear feature of our model, namely, convec- ranged interactions. Below dϭ4, where the breakdown oc- tion, suppresses fluctuations of the velocity vជ at long wave- curs, this slow, diffusive transport of information is replaced lengths, making them much smaller than the analogous ជs by direct, convective transport: fluctuations in the local ve- fluctuations found in ferromagnets, for all spatial dimensions locity of the flock became so large, in these lower dimen- of the flock dϽ4. Specifically, the connected piece C (Rជ )of sions, that the motion of one part of the flock relative to C ជ another becomes the principal means of information trans- the correlation function C(R), defined as port, because it becomes faster than diffusion. There is a sort ជ ជ ជ of ‘‘negative feedback,’’ in that this improved transport ac- CC͑R͒ϭC͑R͒Ϫ lim C͑RЈ͒, ͑1.3͒ ͉Rជ Ј͉ ϱ tually suppresses the very fluctuations that give rise to it, → leading to long-ranged order in dϭ2. The purpose of the which is a measure of the fluctuations, decays to zero much present paper is to study the properties of the ‘‘ordered more rapidly, as ͉Rជ ͉ ϱ, than the analogous correlation state’’ of the flock, i.e., the state in which all members of the function in magnets. Quantitatively,→ for points whose sepa- flock are moving in the same average direction. Specifically, ជ ជ we will do the following: ration RϵRЌ lies perpendicular to the mean direction of mo- ͑1͒ We will give the details of the derivation of the results tion of the flock, of Ref. ͓4͔, and give detailed predictions for numerous cor- ជ 2 relation functions that can be measured in both experiments CC͑R͒ϰRЌ , ͑1.4͒ and simulations. In particular, we will show that two propa- gating sound modes exist in flocks, with unusually aniso- where the universal ‘‘roughness exponent’’ tropic speeds, whose detailed dependence on the direction of ϭϪ 1 ͑1.5͒ propagation we predict, making possible extremely stringent 5 quantitative tests of our theory. We also calculate their at- exactly,indϭ2, and is Ͻ1Ϫd/2, its value in magnetic sys- tenuations, which show highly anomalous, and strongly an- tems, for all dϽ4. For dϾ4, ϭ1Ϫd/2 for flocks as well as isotropic, scaling. for magnets. ͑2͒ We will formulate and study the most complete gen- The physical mechanism for this suppression of fluctua- eralization of the model of Ref. ͓4͔ for spatial dimensions tions is easy to understand: increased fluctuations in the di- dϾ2. rection of motion of different parts of the flock actually en- ͑3͒ We will include the effect of spatial anisotropy ͑e.g., hance the exchange of information between those different the fact that birds prefer to fly horizontally rather than verti- parts. This exchange, in turn, suppresses those very fluctua- cally͒ on flock motion. tions, since the interactions between birds tend to make them We describe the flock with coarse-grained density and all move in the same direction. velocity fields (rជ,t) and vជ (rជ,t), respectively, giving the These nonequilibrium effects also lead to a spatial anisot- average number density and velocity of the birds at time t ropy of scaling between the direction along (ʈ) and those within some coarse-graining distance l 0 of a given position orthogonal to (Ќ) the mean velocity ͗vជ ͘. The physical origin rជ in space. The coarse-graining distance l 0 is chosen to be of the anisotropy is also simple: if birds make small errors as small as possible, consistent with being large enough that ␦ in their direction of motion, their random motion perpen- the averaging can be done sensibly ͑in particular, l 0 must be dicular to the mean direction of motion ͗vជ ͘ is much larger 4830 JOHN TONER AND YUHAI TU PRE 58
than that along ͗vជ ͘; the former is ϰ␦, while the later is proportional to 1Ϫcos ␦ϳ␦2. As a result, any equal-time correlation function in the system of any combination of
fields crosses over from dependence purely on ͉RជЌ͉ to de- pendence purely on Rʈ when Rʈ ͉RជЌ͉ Ϸ , ͑1.6͒ l ͩ l ͪ 0 0
where l 0 is the bird interaction range. The universal anisotropy exponent
3 ϭ 5 , ͑1.7͒ exactly in dϭ2, and is Ͻ1 for all dϽ4. In particular, the connected, equal-time, velocity autocor-
relation function CC(Rជ ) obeys the scaling law
͑Rʈ /l 0͒ C ͑Rជ ͒ϭ Rជ 2 f , ͑1.8͒ C ͉ Ќ͉ v ͩ ͉͑RជЌ͉/l ͒ ͪ 0 FIG. 1. A snapshot of a simulated flock that has reached a sta- where f (x)isauniversal scaling function. We have been v tistically steady state. Note the enormous fluctuations in the density. unable to calculate this scaling function, even in dϭ2 where Quantitatively, the statistics of the spatial Fourier transform C (qជ ) we know the exponents exactly. However, the scaling form obtained from this picture agree with our quantitative prediction ͑1.8͒ immediately implies that equation ͑1.10͒. ជ 2/ ជ CC͑R͒ϰRʈ , when Rʈ /l 0ӷ͑RЌ /l 0͒ . ͑1.9͒ birds in a flock become infinitely bigger than those in a fluid or an ideal gas. This fact is obvious to the eye in a picture of So far, our discussion has focused on velocity fluctua- a flock ͑see Fig. 1͒. Quantitatively, we predict that the spa- tions. The density (Rជ ,t) shows huge fluctuations as well: tially Fourier transformed, equal-time density-density corre- 2 indeed, at long wavelengths, the fluctuations of the density of lation function C(qជ )ϵ͉͗(qជ ,t)͉ ͘ obeys the scaling law
q1ϪdϪϪ2 , q Ӷq 3ϪdϪϪ2 Ќ ʈ Ќ qЌ qʈl 0 Ϫ2 3ϪdϪϪ2 C ͑qជ ͒ϭ f Y͑ ជ ͒ϰ q qЌ , ͑l 0qЌ͒ ӷl 0q ӷqЌl 0 ͑1.10͒ 2 q ͉͉ ʈ q ͩ ͑qЌl ͒ ͪ 0 Ϫ3ϩ ͑1ϪdϪ2/͒ 2 ͭ q qЌ , ͑qЌl 0͒ Ӷqʈl 0 , ʈ
where Y(qជ ) is a finite, nonvanishing, O(1) function of the angle qជ between the wave vector qជ and the direction of mean
flock motion, q͉͉ and qជЌ are the wave vectors parallel and perpendicular to the broken symmetry direction, and qЌϭ͉qជЌ͉. 3 1 In dϭ2, ϭ 5 and ϭϪ 5 ,so
Ϫ ͑6/5͒ q , q ӶqЌ ͑4/5͒ Ќ ʈ qЌ qʈl 0 Ϫ2 ͑4/5͒ ͑3/5͒ C͑qជ͒ϭ f Y͑ជ͒ϰ q q , ͑l 0qЌ͒ ӷl 0q ӷqЌl 0 ͑1.11͒ 2 ͑3/5͒ q ͉͉ Ќ ʈ q ͩ͑qЌl ͒ ͪ 0 Ϫ4 2 ͑3/5͒ ͭq qЌ , ͑qЌl 0͒ Ӷq l 0 . ʈ ʈ
͑1͒ Calculate the complex numbers The most important thing to note about C(qជ ) is that it diverges as ͉qជ͉ 0, unlike C(qជ ) for, say, a simple fluid or gas, or, indeed,→ for any equilibrium condensed matter sys- ជ ជ tem, which goes to a finite constant ͑the compressibility͒ as ͑qជ ,t͒ϭ͚ eiq•ri͑t͒ ͑1.12͒ ͉qជ͉ 0. i →This correlation function should be extremely easy to measure in simulations, and in experiments on real herds or flocks, in which, say, video tape allows one to measure the for a variety of qជ ’s. ͑2͒ Average the squared magnitude of this number over positions rជi(t) of all the birds ͑labeled by i) in the flock at a variety of times t. The recipe is simple: time. The result is C(qជ ). PRE 58 FLOCKS, HERDS, AND SCHOOLS: A QUANTITATIVE . . . 4831
FIG. 2. Polar plot of the direction-dependent sound speeds cϮ(qជ ), with the horizontal axis along the direction of mean flock FIG. 3. Plot of the damping Im vs qxϵ͉qជЌ͉ where qជЌ is the motion. projection of wave vector qជ perpendicular to the direction of the mean flock velocity ͗vជ ͘ for fixed projection q of qជ parallel to ͗vជ ͘. ជ y Time-dependent correlation functions of and v in flocks Note that, for small ͉qជ͉, the crossover between Im ϰqz and also show interesting anomalous scaling behavior. However, x Im ϰqz/ occurs only for directions of propagation qˆ very nearly it is not so simple to summarize as the equal-time correlation y parallel to the mean flock velocity ͗vជ ͘, since Ͻ1. functions. Indeed, time-dependent correlation functions ͑or, equivalently, their spatiotemporal Fourier transforms͒ do not Note that this last result implies that the lifetime of the have a simple scaling form. This is because the collective wave is ϰqϪz/ for q l ӷ(q l ) and ͉qជ͉ 0; this only normal modes of the flock consist of propagating, damped ʈ ʈ 0 Ќ 0 → longitudinal ‘‘sound’’ modes ͑i.e., density waves͒, as well as, happens for directions of propagation very nearly parallel to ជ 6 3 in dϾ2, shear modes. The sound modes exhibit two different ͗v͘. Note also that in dϭ2 where zϭ 5 and ϭ 5 , z/ϭ2 types of scaling: the period T of a wave is proportional to its and the damping is conventional for sound modes propagat- ing parallel to the mean motion of the flock. For all other wavelength ͑the constant of proportionality being the in- directions of propagation, however, it is unconventional, and verse sound speed͒; while the lifetime of the mode is pro- characterized by zϭ 6 . portional to z, with z being another universal exponent. In 5 This behavior of the damping i.e., Im ), is summarized most systems ͑e.g., fluids, crystals͒ exhibiting sound modes, ͑ zϭ2, corresponding to conventional diffusive or viscous in Fig. 3. For dϾ2, the ‘‘hydrodynamic’’ mode structure damping ͓6͔. In flocks, however, we find also includes dϪ2 ‘‘hyperdiffusive’’ shear modes, with identical dispersion relations
6 zϭ 5 , dϭ2, ͑1.13͒ qʈl 0 ϭ␥q Ϫiqz f . ͑1.16͒ s ʈ Ќ s ͩ͑qЌl ͒ ͪ and zϽ2 for all dϽ4, for sound modes propagating orthogo- 0 nal to the mean direction of flock motion. That is, sound The dispersion relations for Ϯ and s can be directly modes are much more heavily damped at long wavelengths probed by measuring the spatiotemporally Fourier trans- in flocks than in most ͓7͔ equilibrium condensed matter sys- formed density-density and velocity-velocity autocorrelation tems. functions The full dispersion relation for the sound modes is 2 C͑qជ ,͒ϵ͉͗͑qជ ,͉͒ ͘,
z qʈl 0 ϭc ͑ ជ ͒qϪiq f , ͑1.14͒ Ϯ Ϯ q Ќ Ϯ Cij͑qជ,͒ϵ͗vi͑qជ,͒vj͑Ϫqជ,Ϫ͒͘, ͑1.17͒ ͩ͑qЌl ͒ ͪ 0
respectively. Experimentally, or in simulations, C(qជ ,) can where qជ is the angle between qជ and ͗vជ ͘, and the direction- be calculated by temporally Fourier transforming the spa- dependent sound speeds cϮ(qជ ) are given by Eq. ͑4.11͒ of tially Fourier transformed density ͑1.12͒: Sec. IV, with ␥ and 1 flock-dependent parameters and 0 nϩ1 the mean number density of ‘‘birds’’ in the flock. A polar ͑ ͒ Ϫit n͑qជ ,͒ϭ n͑qជ ,t͒e , nϭ0,1,2, . . . plot of these sound speeds is given in Fig. 2. The exponent t͚ϭn is the universal anisotropy exponent described earlier, and ͑1.18͒ f Ϯ(x) are universal scaling functions that we have been un- able to calculate. However, we do know some of their limits: over a set of long ‘‘bins’’ of time intervals of length ӷt0 ͑the ‘‘microscopic’’ time step͒, and then averaging the f ͑x 0͒ constϾ0; f ͑x ϱ͒ϰxz/. ͑1.15͒ squared magnitude ͉(qជ ,)͉2 over bins: Ϯ → → Ϯ → 4832 JOHN TONER AND YUHAI TU PRE 58
FIG. 5. Geometry of the anisotropic model. Birds prefer to fly in the ‘‘easy’’ x-y plane. We take their ͑spontaneously chosen͒ direc- tion of motion within that plane to be y. The in-plane direction perpendicular to that is x. In general d, there are dϪ2 ‘‘hard’’ FIG. 4. Plot of the spatiotemporally Fourier-transformed density directions ជrH perpendicular to this easy plane. The anisotropy of correlation function C(qជ ,)vsfor fixed qជ . It shows two sharp scaling is between x and the other dϪ1 directions y,ជrH . asymmetrical peaks at ϭcϮ(qជ )q associated with the sound modes of the flock, where cϮ(qជ ) are the sound mode speeds. The change the exponents , z, and . Since does not renor- widths of those peaks are the second mode dampings Im Ϯ(qជ ) 2 z ϰqЌ f Ϯ͓qʈl 0/(qЌl 0) ͔. malize at one-loop order, all we can say at this point is that there are three possibilities: nmax 2 1 At higher order, renormalizes to zero. If this is the ͉͑qជ ,͉͒ ͑ ͒ 2 ជ case, we can show that C͑q,͒ϵ ͚ . ͑1.19͒ nϭ0 nmax 2͑dϩ1͒ dϩ1 3Ϫ2d Closed form expressions for these correlation functions in zϭ , ϭ , ϭ , ͑1.21͒ 5 5 5 terms of the scaling functions f L and f s are given in Sec. V. Although these expressions look quite complicated, the behavior they predict is really quite simple, as illustrated exactly, for all d in the range 2рdр4. Note that these re- sults linearly interpolate between the equilibrium results z in Fig. 4, where C is plotted as a function of for ϭ2, ϭ1, and ϭ1Ϫd/2 in dϭ4, and our 2d results z fixed q. As shown there, C has two sharp peaks at z ϭ 6 , ϭ 3 , and ϭϪ 1 in dϭ2. ϭcϮ(qជ )q, of width ϰqЌ f L͓qʈl 0/(qЌl 0) ͔ and height 5 5 5 Ϫ(2ϩzϩ3ϩdϪ3) ͑2͒ At higher order, grows upon renormalization and ϰq g͓q l /(q l ) ͔. Thus, c ( ជ ) can be 2 Ќ ʈ 0 Ќ 0 Ϯ q * simply extracted from the position of the peaks, while the reaches a nonzero fixed point value 2 at some new fixed exponents , z, and can be determined by comparing their point that differs from the 2ϭ0 fixed point we have studied previously, at which Eq. ͑1.21͒ holds. The exponents , z, widths and heights for different qជ ’s. and would still be universal ͑i.e., depend only on the di- The scaling properties of the flock are completely summa- mension of space d) for all flocks in this case, but those rized by the universal exponents z, , and .Indϭ2, our universal values would be different from Eq. ͑1.21͒. predictions for these exponents are ͑3͒2 is unrenormalized to all orders. Should this happen, 6 3 1 would parametrize a fixed line, with continuously varying zϭ 5 , ϭ 5 , ϭϪ 5 . ͑1.20͒ 2 values of the exponents z, , and . These results are exact and universal for all flocks with the We reiterate: we do not know which of the above possi- simple symmetries we discussed at the outset. bilities holds for dϾ2. However, whichever holds is univer- For dÞ2, the situation is less clear. We have performed a sal; that is, only one of the three possibilities above applies one-loop, 4Ϫ⑀ expansion to attempt to calculate these expo- to all flocks. We do not, however, know which one that is. nents, and find that, to this order, the model appears to have We also study an anisotropic model for flocking, which a fixed line with continuously varying exponents z, , and . incorporates the possibility that birds are averse to flying in Whether this is an artifact of our one-loop calculation, or certain directions ͑e.g., straight up or straight down͒. In par- actually happens, is unclear. A two-loop calculation might ticular, we consider the case in which, for arbitrary spatial clarify matters, but would be extremely long and tedious. dimensions dу2, there is an easy plane for motion ͑i.e., a ͑One loop was hard enough.͒ deϭ2 dimensional subspace of the full d-dimensional The origin of this complication is an additional convective space͒. In this case, the relevant pieces of the 2 vertex be- nonlinearity ͓8͔ not discussed in Ref. ͓4͔. This new term come a total derivative, and can be absorbed into the nonlin- ͑whose coefficient is a parameter we call 2) is unrenormal- ear term considered in Ref. ͓4͔, for all spatial dimensions d, ized at one loop order, leading to the apparent fixed line at not just dϭ2 as in the isotropic model. Hence, we are able to that order. In two dimensions, this extra term can be written obtain exact exponents for this problem for all spatial dimen- as a total derivative, and can be absorbed into the nonlinear sions d, not just dϭ2. term considered in Ref. ͓4͔. Hence, in dϭ2, the results of ͓4͔ We again find anisotropic, anomalous scaling for dϽ4. are sound. In dϾ2, however, this new term has a different The anisotropy of scaling is between the direction in the easy structure, and could, if it does not renormalize to zero, plane ͑call it x) perpendicular to the mean direction of mo- PRE 58 FLOCKS, HERDS, AND SCHOOLS: A QUANTITATIVE . . . 4833 tion ͑call it y; which, of course, also lies in the easy plane͒, while in dϭ2, where the model becomes identical to the and all dϪ1 other directions, including y ͑see Fig. 5͒. That isotropic model ͑the easy plane of motion being the entire is, the equal time, velocity-velocity autocorrelation function space in that case͒, we again recover the isotropic results, 3 1 obeys the scaling law ϭ 5 , and ϭϪ 5 . For the physical case dϭ3, we have
y/ ͒ rជ / 2 ͑ l 0 ͉ H͉ l 0 1 3 Cv͑Rជ ͒ϭx f v , , ͑1.22͒ ͑dϭ3͒ϭϪ 2 , ͑dϭ3͒ϭ 4 . ͑1.25͒ ͩ ͑x/l ͒ ͑x/l ͒ ͪ 0 0 The Fourier-transformed, equal-time density (Ϫ) correla- where rជ denotes the dϪ2 components of rជ in the ‘‘hard’’ H ជ directions orthogonal to the easy plane, with the scaling ex- tion function C(q) also obeys a scaling law ponents and given by C ͑qជ ͒ϭq1Ϫ2Ϫ͑dϪ1͒͑q2ϩq2͒Ϫ1 1Ϫd x x y ϭ , ͑1.23͒ 7Ϫd qyl 0 ͉qជ H͉l 0 ϫ f A , Y ͑ ͒, ͑1.26͒ a xy ͩ ͑qxl 0͒ ͑q l ͒ ͪ 3 x 0 ϭ ͑1.24͒ 7Ϫd where Y a(xy) is a finite, nonzero, O(1) function of the Ϫ1 A exactly, for all spatial dimensions d in the range 2рdр4. angle xyϭtan (qx /qy), and the scaling function f fol- For dϾ4, ϭ1Ϫd/2 and ϭ1, as in the isotropic case, lows
2 2 2 const, l 0͑qyϩ͉qជ H͉ ͒Ӷ͑l 0qx͒ qyl 0 ͉qជ H͉l 0 ϩ ϩ Ϫ f A , ϰ 2 ͓1 2 ͑d 1͔͒/2 ͑1.27͒ qx 2 2 ͩ ͑qxl 0͒ ͑q l ͒ ͪ , l ͑q ϩ͉qជ ͉2͒ӷ͑l q ͒, x 0 2 2 0 y H 0 x ͭ ͩ q ϩ͉qជ H͉ ͪ y
where l 0 is a ‘‘microscopic’’ length ͑of the order the inter- specialize this model to the ‘‘broken symmetry’’ state, in bird distance͒ and a dimensionless nonuniversal constant which the flock is moving with a nonzero mean speed ͗vជ ͘.In of order unity. Sec. IV, we linearize the broken symmetry state model, and Finally, the hydrodynamic mode structure of this aniso- calculate the correlation functions and scaling laws in this tropic flock consists of a pair of propagating longitudinal linear approximation. In Sec. V, we study the anharmonic sound modes, with dispersion relation given, in the coordi- corrections in the broken symmetry state, show that they nate system of Fig. ͑5͒,by diverge in spatial dimensions dϽ4, derive the new scaling laws that result in that case, calculate the exact exponents in ជ dϭ2, and discuss the difficulties that prevent us from obtain- z qyl 0 ͉qH͉l 0 ϭc ͑ជ,ជ͒qϪiq f , , Ϯ q q x A ing these exponents for 2ϽdϽ4. In Sec. VI, we repeat all of ͩ͑qxl 0͒ ͑q l ͒ ͪ x 0 the above for the anisotropic model. In Sec. VII, we describe 1.28 ͑ ͒ in some detail how our predictions might be tested experi- mentally, both by observations of real flocks of living organ- where f is a universal scaling function, c ( ជ , ជ ) is given A Ϯ q q isms, and in simulations. And, finally, in Sec. VIII, we dis- by Eq. ͑6.22͒ of Sec. 6 and the dynamical exponent cuss some of the open questions remaining in this problem, and suggest some possible directions for future research. 6 3 zϭ2ϭ ϭ , ͑1.29͒ 7Ϫd 2 II. THE ISOTROPIC MODEL where the last equality holds in dϭ3. Note that this value of In this section, we formulate our model for isotropic z again reduces to that of the isotropic model in dϭ2, and flocks. As discussed in the Introduction, the system we wish ϭ d 4. to model is any collection of a large number N of organisms The spatiotemporally Fourier-transformed density-density ͑hereafter referred to as ‘‘birds’’͒ in a d-dimensional space, 2 correlation function ͉͗(qជ ,)͉ ͘ has the same structure as with each organism seeking to move in the same direction as that illustrated for the isotropic problem in Fig. 4, with the its immediate neighbors. modification that qЌ is replaced by qx , and the scaling func- We further assume that each organism has no ‘‘com- tion f L is replaced by f A . The detailed expression for pass;’’ i.e., no intrinsically preferred direction in which it ͉͗(qជ ,)͉2͘ is given by Eq. ͑6.31͒ of Sec. VI. wishes to move. Rather, it is equally happy to move in any The remainder of this paper is organized as follows: In direction picked by its neighbors. However, the navigation of Sec. II, we formulate the isotropic model. In Sec. III, we each organism is not perfect; it makes some errors in at- 4834 JOHN TONER AND YUHAI TU PRE 58 tempting to follow its neighbors. We consider the case in same long-distance behavior. This belief can be justified by which these errors have zero mean; e.g., in two dimensions, our renormalization-group treatment of the continuum a given bird is no more likely to err to the right than to the model. left of the direction picked by its neighbors. We also assume So, given this lengthy preamble, what are the symmetries that these errors have no long temporal correlations; e.g., a and conservation laws of flocks? bird that has erred to the right at time t is equally likely to err The only symmetry of the model is rotation invariance: either left or right at a time tЈ much later than t. since the ‘‘birds’’ lack a compass, all directions of space are Although the continuum model we propose here will de- equivalent. Thus, the ‘‘hydrodynamic’’ equations of motion scribe the long-distance behavior of any flock satisfying the we write down cannot have built into them any special di- symmetry conditions we shall specify in a moment, it is in- rection picked ‘‘a priori’’; all directions must be spontane- structive to first consider an explicit example: the automaton ously picked out by the motion and spatial structure of the studied by Vicsek et al. ͓1͔. In this discrete time model, a flock. As we shall see, this symmetry severely restricts the number of boids labeled by i in a two-dimensional plane allowed terms in the equation of motion. ជ with positions ͕ri(t)͖ at integer time t, each chooses the Note that the model does not have Galilean invariance: direction it will move on the next time step ͑taken to be of changing the velocities of all the birds by some constant duration ⌬tϭ1) by averaging the directions of motion of all boost vជ b does not leave the model invariant. Indeed, such a of those birds within a circle of radius R0 ͑in the most con- venient units of length R ϭ1) on the previous time step boost is impossible in a model that strictly obeys Vicsek’s 0 rules, since the speeds of all the birds will not remain equal ͑updating is simultaneous͒. The distance R0 is assumed to be ӶL, the size of the flock. The direction the bird actually to v0 after the boost. One could image relaxing this con- moves on the next time step differs from the above described straint on the speed, and allowing birds to occasionally speed up or slow down, while tending on average to move at speed direction by a random angle i(t), with zero mean and stan- dard deviation ⌬. The distribution of i(t) is identical for all v0 . Then the boost just described would be possible, but birds, time independent, and uncorrelated between different clearly would change the subsequent evolution of the flock. birds and different time steps. Each bird then, on the next Another way to say this is that birds move through a resistive medium, which provides a special Galilean refer- time step, moves in the direction so chosen a distance v0⌬t, ence frame, in which the dynamics are particularly simple, where the speed v0 is the same for all birds. To summarize, the rule for bird motion is and different from those in other reference frames. Since real organisms in flocks always move through such a medium i͑tϩ1͒ϭ͗ j͑t͒͘ϩi͑t͒, ͑2.1͒ ͑birds through the air, fish through the sea, wildebeest through the arid dust of the Serengeti͒, this is a very realistic feature of the model. rជi͑tϩ1͒ϭrជi͑t͒ϩv0͓cos ͑tϩ1͒,sin ͑tϩ1͔͒, ͑2.2͒ As we shall see shortly, this lack of Galilean invariance allows terms in the hydrodynamic equations of birds that are ͗i͑t͒͘ϭ0, ͑2.3͒ not present in, e.g., the Navier-Stokes equations for a simple ͗ ͑t͒ ͑tЈ͒͘ϭ⌬␦ ␦ , ͑2.4͒ fluid, which must be Galilean invariant, due to the absence of i j ij ttЈ a luminiferous ether. where the average in Eq. ͑2.1͒ is over all birds j satisfying The sole conservation law for flocks is conservation of birds: we do not allow birds to be born or die ‘‘on the ͉rជ ͑t͒Ϫrជ ͑t͉͒ϽR ͑2.5͒ wing.’’ j i 0 In contrast to the Navier-Stokes equation, there is no con- servation of momentum. This is, ultimately, a consequence and i(t) is the angle of the direction of motion of the ith bird ͑relative to some fixed reference axis͒ on the time step of the absence of Galilean invariance. that ends at t. The flock evolves through the iteration of this Having established the symmetries and conservation laws rule. Note that the ‘‘neighbors’’ of a given bird may change constraining our model, we need now to identify the hydro- on each time step, since birds do not, in general, move in dynamic variables. They are the coarse-grained bird velocity exactly the same direction as their neighbors. field vជ (rជ,t) and the coarse-grained bird density (rជ,t). The This model, though simple to simulate, is quite difficult to field vជ (rជ,t), which is defined for all rជ, is a suitable weighted treat analytically. Our goal in our previous work ͓4͔ and this average of the velocities of the individual birds in some vol- paper is to capture the essential physics of this model in a ume centered on rជ. This volume is big enough to contain continuum, ‘‘hydrodynamic’’ description of the flock. enough birds to make the average well behaved, but should Clearly, some short-ranged details must be lost in such a have a spatial linear extent of no more than a few ‘‘micro- description. However, as in hydrodynamic descriptions of scopic’’ lengths ͑i.e., the interbird distance, or by a few times equilibrium systems ͓6͔, as well as many recent treatments the interaction range R ). By suitable weighting, we seek to ͓9͔ of nonequilibrium systems, our hope is that our con- 0 ជ ជ tinuum approach can correctly reproduce the long-distance, make v(r,t) fairly smoothly varying in space. long-time properties of the class of systems we wish to The density (rជ,t) is similarly defined, being just the study. This hope is justified by the notion of universality: all number of particles in a coarse-graining volume, divided by ‘‘microscopic models’’ ͑in our case, different specifications that volume. for the exact laws of motion for an individual bird͒ that have The exact prescription for the coarse graining should be the same symmetries and conservation laws should have the unimportant, as long as (rជ,t) is normalized so as to obey PRE 58 FLOCKS, HERDS, AND SCHOOLS: A QUANTITATIVE . . . 4835 the ‘‘sum rule’’ that its integral over any macroscopic vol- make the local vជ have a nonzero magnitude (ϭͱ␣/) in the ume ͑i.e., any volume large compared with the aforemen- ordered phase, where ␣Ͼ0. DL,1,2 are the diffusion constants tioned microscopic lengths͒ be the total number of birds in ͑or viscosities͒ reflecting the tendency of a localized fluctua- that volume. Indeed, the coarse-graining description just out- tion in the velocities to spread out because of the coupling lined is the way that one imagines, in principle, going over between neighboring ‘‘birds.’’ The ជf term is a random driv- from a description of a simple fluid in terms of equations of ing force representing the noise. We assume it is Gaussian motion for the individual constituent molecules to the con- with white noise correlations: tinuum description of the Navier-Stokes equation. We will also follow the historical precedent of the Navier- ͗ f ͑rជ,t͒f ͑rជЈ,tЈ͒͘ϭ⌬␦ ␦d͑rជϪrជЈ͒␦͑tϪtЈ͒, ͑2.9͒ Stokes ͓6͔ equation by deriving our continuum, long- i j ij wavelength description of the flock not by explicitly coarse where ⌬ is a constant, and i, j denote Cartesian components. graining the microscopic dynamics ͑a very difficult proce- Finally, P is the pressure, which tends to maintain the local dure in practice͒, but, rather, by writing down the most gen- number density (rជ) at its mean value 0 , and ␦ϭ eral continuum equations of motion for vជ and consistent Ϫ0 . with the symmetries and conservation laws of the problem. The final equation ͑2.8͒ is just conservation of bird num- This approach allows us to bury our ignorance in a few phe- ber ͑we do not allow our birds to reproduce or die ‘‘on the nomenological parameters ͑e.g., the viscosity in the Navier- wing’’͒. Stokes equation͒ whose numerical values will depend on the Symmetry allows any of the phenomenological coeffi- detailed microscopic rules of individual bird motion. What cients , ␣, , , D in Eqs. ͑2.6͒ and ͑2.7͒ to be func- terms can be present in the EOM’s, however, should depend i n i tions of the squared magnitude vជ 2 of the velocity, and of only on symmetries and conservation laws, and not on the ͉ ͉ the density as well. microscopic rules. To reduce the complexity of our equations of motion still further, we will perform a spatial-temporal gradient expan- III. THE BROKEN SYMMETRY STATE sion, and keep only the lowest-order terms in gradients and We are mainly interested in the symmetry broken phase, time derivatives of vជ and . This is motivated and justified specifically in whether fluctuations around the symmetry by our desire to consider only the long-distance, long-time broken ground state destroy it ͑as in the analogous phase of properties of the flock. Higher-order terms in the gradient the 2D XY model͒. For ␣Ͼ0, we can write the velocity field expansion are ‘‘irrelevant’’: they can lead to finite ‘‘renor- as vជ ϭv xˆ ϩជ␦v, where v xˆ ϭ͗vជ ͘ is the spontaneous aver- malization’’ of the phenomenological parameters of the 0 ʈ 0 ʈ age value of vជ in the ordered phase. We will choose v long-wavelength theory, but cannot change the type of scal- 0 ing of the allowed terms. ϭͱ␣/ ͑which should be thought of as an implicit condition 2 With this lengthy preamble in mind, we now write down on v0 , since ␣ and  can, in general, depend on ͉vជ͉ ); with the equations of motion: this choice, the equation of motion for the fluctuation ␦vʈ of vʈ is ជ ជ ជ ជ ជ ជ ជ ជ ជ 2 ͒ tvϩ1͑v•ٌ͒vϩ2ٌ͑•v͒vϩ3ٌ͉͑v͉ץ ʈ␦Ϫ2␣␦vʈϩirrelevant terms. ͑3.1͒ץt␦vʈϭϪ1ץ ͒ ϭ␣vជ Ϫ͉vជ͉2vជ Ϫٌជ PϩD ٌជ ٌ͑ជ vជ B • Note now that if we are interested in ‘‘hydrodynamic’’ 2 ជ ជ ជ 2 ជ ជ modes, by which we mean modes for which frequency ϩDTٌ vϩD2͑v•ٌ͒ vϩ f , ͑2.6͒ 0 as wave vector q 0, we can, in the hydrodynamic → → .t␦vʈ relative to ␣␦vʈ in Eq. ͑3.1͒ץ ϱ (,q 0) limit, neglect n → PϭP͑͒ϭ n͑Ϫ0͒ , ͑2.7͒ The resultant equation can trivially be solved for ␦vʈ : n͚ϭ1
ʈ␦, ͑3.2͒ץvʈϭϪD␦ ץ ϩٌ ͑vជ͒ϭ0, ͑2.8͒ where we have defined another diffusion constant D t • ץ ϵ1 /2␣. Inserting Eq. ͑3.2͒ in the equations of motion for vជ and , we obtain, neglecting ‘‘irrelevant’’ terms: where , DB , D2 , and DT are all positive, and ␣Ͻ0 in the Ќ ␦ disordered phase and ␣Ͼ0 in the ordered state ͑in mean- ជ ជ ជ ជ ជ ជ ជ ជ ʈvЌϩ1͑vЌ•ٌЌ͒vЌϩ2ٌ͑Ќ•vЌ͒vЌץ␥tvЌϩץ :field theory͒. The origin of the various terms is as follows the terms on the left-hand side of Eq. ͑2.6͒ are the analogs ជ ជ ជ ជ 2 ជ 2ជ ជ , ʈ vЌϩfЌץof the usual convective derivative of the coarse-grained ve- ϭϪٌЌPϩDBٌЌٌ͑Ќ•vЌ͒ϩDTٌЌvЌϩDʈ locity field vជ in the Navier-Stokes equation. Here the ab- ͑3.3͒ sence of Galilean invariance allows all three combinations of ␦ץ one spatial gradient and two velocities that transform like ជ ជ ជ ជ 2 ,␦͉͉ץʈ␦ϭDץϩoٌЌ vЌϩٌЌ ͑vЌ␦͒ϩv0 • • tץ vectors; if Galilean invariance did hold, it would force 2 ϭ3ϭ0 and 1ϭ1. However, Galilean invariance does not ͑3.4͒ hold, and so all three coefficients are nonzero phenomeno- 2 logical parameters whose nonuniversal values are deter- where D , DB , DT , and DʈϵDTϩD2v0 are the diffusion mined by the microscopic rules. The ␣ and  terms simply constants, and we have defined 4836 JOHN TONER AND YUHAI TU PRE 58
␥ϵ v . ͑3.5͒ 1 0 ͓Ϫi͑Ϫ␥qʈ͒ϩ⌫T͑qជ ͔͒vជ T͑qជ ,͒ϭជf T͑qជ ,͒, ͑4.1͒ The pressure P continues to be given, as it always will, by ជ ជ Eq. ͑2.7͒. ͓Ϫi͑Ϫ␥qʈ͒ϩ⌫L͑q͔͒vLϩi1qЌ␦ϭ f L͑q,͒, From this point forward, we will treat the phenomenologi- ͑4.2͒ cal parameters , ␥, and D appearing in Eqs. ͑3.3͒ and i i ជ ͑3.4͒ as constants, since they depend, in our original model ͓Ϫi͑Ϫv0qʈ͒ϩ⌫͑q͔͒␦ϩi0qЌvLϭ0, ͑4.3͒ ͑2.6͒, only on the scalar quantities ͉vជ͉2 and (rជ), whose where fluctuations in the broken symmetry state away from their mean values v2 and are small. Furthermore, these fluctua- 0 0 ជ ជ ជ tions lead only to ‘‘irrelevant’’ terms in the equations of qЌ•vЌ͑q,͒ vL͑qជ ,͒ϵ ͑4.4͒ motion. qЌ It should be emphasized here that, once nonlinear fluctua- and tion effects are included, the v0 in Eq. ͑3.4͒ will not be given by the ‘‘mean’’ velocity of the birds, in the sense of qជЌvL vជ T͑qជ ,͒ϭvជЌ͑qជ ,͒Ϫ ͑4.5͒ ͉͚ivជ i͉ q ͗v͘ϵ , ͑3.6͒ Ќ N are the longitudinal and transverse ͑to qជЌ) pieces of the ve- where N is the number of birds. This is because, in our con- locity, ជf (qជ ,) and f (qជ ,) are the analogous pieces of the tinuum language, T L Fourier-transformed random force ជf (qជ ,), and we have de- d fined wave-vector-dependent transverse, longitudinal, and ͉͗͵͑rជ,t͒vជ ͑rជ,t͒d r͉͘ ͉͗vជ ͉͘ ͗v͘ϭ ϭ ͑3.7͒ dampings ⌫L,T, : ͗͵͑rជ,t͒ddr͘ ͗͘ ជ 2 2 ⌫L͑q͒ϵDLqЌϩDʈqʈ , ͑4.6͒ while v0 in Eq. ͑3.5͒ is 2 2 ⌫T͑qជ ͒ϭDTq ϩD q , ͑4.7͒ v0ϭ͉͗vជ ͑rជ,t͉͒͘. ͑3.8͒ Ќ ʈ ʈ
Once fluctuates, so that ϭ ϩ␦, the ‘‘mean’’ velocity ជ 2 ͗ ͘ ⌫͑q͒ϭDqʈ , ͑4.8͒ of the birds where we have defined D ϵD ϩD , q ϭ͉qជ ͉. ជ ជ ជ L T B Ќ Ќ ͗v͘ ͗͗͘v͘ ͗␦v͘ ជ ͗v͘ϭ ϭ ϩ , ͑3.9͒ Note that in dϭ2, the transverse velocity vT does not ͯ ͗͘ ͯ ͯ ͗͘ ͗͘ ͯ exist: no vector can be perpendicular to both the xʈ axis and qជ in two dimensions. This leads to many important simpli- which only ϭ ជ if the correlation function ជ Ќ v0ϵ͉͗v͉͘ ͗␦v͘ fications in dϭ2, as we will see later; these simplifications ϭ0, which it will not, in general. For instance, one could make it ͑barely͒ possible to get exact exponents in dϭ2 for easily imagine that denser regions of the flock might move the full, nonlinear problem. faster, in which case ͗␦vជ ͘ would be positive along ͗vជ ͘. The normal modes of these equations are dϪ2 purely Thus, ͗vជ ͘ measured in a simulation by simply averaging the diffusive transverse modes associated with vជ T , all of which speed of all birds, as in Eq. ͑3.6͒, will not be equal to v0 in have the same eigenfrequency Eq. ͑3.5͒. Indeed, we can think of no simple way to measure v , and so chose instead to think of it as an additional phe- 0 ϭ q Ϫi qជ ϭ q Ϫi D q2 ϩD q2 , 4.9 nomenological parameter in the broken symmetry state equa- T ␥ ʈ ⌫T͑ ͒ ␥ ʈ ͑ T Ќ ʈ ʈ ͒ ͑ ͒ tions of motion ͑3.3͒. It should, in simulations and experi- ments, be determined by fitting the correlation functions we and a pair of damped, propagating sound modes with com- will calculate in the next section. One should not expect it to plex ͑in both senses of the word͒ eigenfrequencies be given by ͗v͘ as defined in Eq. ͑3.7͒. Similar considerations apply to ␥: it should also be vϮ͑qជ ͒ vϯ͑qជ ͒ ϮϭcϮ͑qជ ͒qϪi⌫L Ϫi⌫ thought of as an independent, phenomenological parameter, ͫ 2c ͑ ជ ͒ͬ ͫ 2c ͑ ជ ͒ͬ 2 q 2 q not necessarily determined by the mean velocity and nonlin- ear parameter through Eq. 3.5 . 1 ͑ ͒ 2 2 vϮ͑qជ ͒ ϭcϮ͑qជ ͒qϪi͑DLqʈ ϩDЌqЌ͒ ͫ 2c ͑ ជ ͒ͬ 2 q IV. LINEARIZED THEORY OF THE BROKEN SYMMETRY STATE 2 vϯ͑qជ͒ ϪiDqʈ , ͑4.10͒ As a first step towards understanding the implications of ͫ2c ͑ជ͒ͬ 2 q these equations of motion, we linearize them in vជЌ and ␦ ϵϪ0 . Doing this, and Fourier transforming in space and where qជ is the angle between qជ and the direction of flock time, we obtain the linear equations motion ͑i.e., the xʈ axis͒, PRE 58 FLOCKS, HERDS, AND SCHOOLS: A QUANTITATIVE . . . 4837
␥ϩv0 The linear equations ͑4.1͒–͑4.3͒ are easily solved for the cϮ͑qជ ͒ϭ cos͑qជ ͒Ϯc2͑qជ ͒, ͑4.11͒ 2 fields ␦, vជ T , and vL in terms of the random forces:
␥Ϫv0 ជ ជ ជ ជ ជ v ͑ ជ ͒ϭϮ cos͑ ជ ͒ϩc ͑ ជ ͒, ͑4.12͒ vT͑q,͒ϭGTT͑q,͒fT͑q,͒, ͑4.14͒ Ϯ q 2 q 2 q
1 v ͑qជ ,͒ϭG ͑qជ,͒f ͑qជ,͒ϩG ͑qជ,͒f ͑qជ,͒, 2 2 2 2 L LL L L c ͑ ជ ͒ϵ ͑␥Ϫv ͒ cos ͑ ជ ͒ϩc sin ͑ ជ ͒, 2 q ͱ4 0 q 0 q ͑4.15͒ ͑4.13͒
␦͑qជ ,͒ϭGL͑qជ ,͒f L͑qជ ,͒ϩG͑qជ,͒f͑qជ,͒, and c0ϵͱ10. A polar plot of this highly anisotropic sound speed is given in Fig. 2. We remind the reader that ͑4.16͒ here and hereafter, we only keep the leading-order terms in the long-wavelength limit, i.e., for small qʈ and qЌ . where the propagators are
1 GTTϭ , ͑4.17͒ Ϫi͑Ϫ␥qʈ͒ϩ⌫T͑qជ͒
i͑Ϫv0qʈ͒Ϫ⌫͑qជ͒ GLLϭ , ͑4.18͒ ͓Ϫcϩ͑qជ͒q͔͓ϪcϪ͑qជ͒q͔ϩi͓⌫L͑qជ͒ϩ⌫͑qជ͔͒Ϫiqʈ͓v0⌫L͑qជ͒ϩ␥⌫͑qជ͔͒
i1qЌ GLϭ , ͑4.19͒ ͓Ϫcϩ͑qជ ͒q͔͓ϪcϪ͑qជ͒q͔ϩi͓⌫L͑qជ͒ϩ⌫͑qជ͔͒Ϫiqʈ͓v0⌫L͑qជ͒ϩ␥⌫͑qជ͔͒
i0qЌ GLϭ , ͑4.20͒ ͓Ϫcϩ͑qជ ͒q͔͓ϪcϪ͑qជ͒q͔ϩi͓⌫L͑qជ͒ϩ⌫͑qជ͔͒Ϫiqʈ͓v0⌫L͑qជ͒ϩ␥⌫͑qជ͔͒
i͑Ϫ␥qʈ͒Ϫ⌫L͑qជ͒ Gϭ . ͑4.21͒ ͓Ϫcϩ͑qជ͒q͔͓ϪcϪ͑qជ͒q͔ϩi͓⌫L͑qជ͒ϩ⌫͑qជ͔͒Ϫiqʈ͓v0⌫L͑qជ͒ϩ␥⌫͑qជ͔͒
In writing the definitions of the propagators ͑4.14͒–͑4.16͒, we have introduced a fictitious force f in the equation of motion ͑4.3͒. Of course, this force is, in fact, zero; but the propagators G and GL nonetheless prove useful in the perturbative treatment of the nonlinear corrections to this linear theory, so we have included f here. Given the expressions ͑4.14͒–͑4.21͒ for the velocity and density in terms of the random force ជf , and the autocorrelation ͑2.9͒ of that random force, it is straightforward to calculate the correlations of the densities and velocities. We find
Ќ Ќ Cij͑qជ,͒ϵ͗vi ͑Ϫqជ,Ϫ͒vj ͑qជ,͒͘ Ќ Ќ qi qj ϭG ͑qជ,͒G ͑Ϫqជ,Ϫ͒ f ͑qជ,͒f ͑Ϫqជ,Ϫ͒ ϩG ͑qជ,͒G ͑Ϫqជ,Ϫ͒ f ͑qជ,͒f ͑Ϫqជ,Ϫ͒ TT TT ͗ Ti Tj ͘ LL LL 2 ͗ L L ͘ qЌ
Ќ Ќ ϵCTT͑qជ,͒Pij͑qជ͒ϩCLL͑qជ,͒Lij͑qជ͒, ͑4.22͒ where
Ќ Ќ Ќ qi qj Lij͑qជ͒ϵ 2 , ͑4.23͒ qЌ
Ќ Ќ Ќ Pij͑qជ͒ϵ␦ijϪLij͑qជ͒ ͑4.24͒ are longitudinal and transverse projection operators that project any vector perpendicular to both the flock motion and qជЌ ,
⌬ C ͑qជ,͒ϭ ͑4.25͒ TT 2 2 ͑Ϫ␥qʈ͒ ϩ⌫T͑qជ͒ 4838 JOHN TONER AND YUHAI TU PRE 58 and
2 ⌬͑Ϫv0qʈ͒ C ͑qជ,͒ϭ . ͑4.26͒ LL 2 2 2 ͓Ϫcϩ͑qជ͒q͔ ͓ϪcϪ͑qជ͒q͔ ϩ͕͓⌫L͑qជ͒ϩ⌫͑qជ͔͒Ϫqʈ͓v0⌫L͑qជ͒ϩ␥⌫͑qជ͔͖͒
The transverse and longitudinal correlation functions Eqs. ͑4.25͒ and ͑4.26͒ are plotted as functions of for fixed qជ in Fig. 6. Note that they have weight in entirely different regions of frequency: CTT is peaked at ϭ␥qʈ , while CLL has two peaks, 2 at ϭcϮ(qជ )q. Since all three peaks have widths of order q , there is little overlap between the transverse and the longitu- dinal peaks as ͉qជ͉ 0. The density-density→ correlation function
2 2 ⌬0qЌ C ϭ ͑4.27͒ 2 2 2 ͓Ϫcϩ͑qជ͒q͔ ͓ϪcϪ͑qជ͒q͔ ϩ͕͓⌫L͑qជ͒ϩ⌫͑qជ͔͒Ϫqʈ͓v0⌫L͑qជ͒ϩ␥⌫͑qជ͔͖͒
looks almost identical to C , especially when one notes 2 2 LL ␥Ϫv0 q F qជ ; , 2 ʈ ϩ1, 4.30 that near the frequencies ϭcϮ(qជ )q where both peak, the ͑ ␥͒ϵͱ ͑ ͒ ͩ 2v ͪ ͩ qЌ ͪ 2 0 numerator of Eq. ͑4.26͒ ͕ϭ͓cϮ(qជ )qϪv0qʈ͔ ͖, differs from ជ that of Eq. ͑4.27͒ only by a ͉q͉-independent factor of ␥Ϫv q 2 2 0 ʈ ͓c ( ជ )qϪv q ͔ / q . A ͑qជ ;,␥͒ϵϮF͑qជ ;,␥͒ϩ , ͑4.31͒ Ϯ q 0 ʈ 0 1 Ќ Ϯ 2v q Given these Fourier-transformed correlation functions, it ͩ 0 ͪ Ќ is straightforward, and instructive, to Fourier transform back and to real time. In particular, it is simple to calculate the spa- tially Fourier-transformed equal-time velocity correlation v0 function: ϵ . ͑4.32͒ ͱ10 ϱ d Ќ The second equality in Eq. ͑4.29͒ is obtained from the first ͗vi͑qជ ,t͒v j͑Ϫqជ ,t͒͘ϭPij͑qជ͒ CTT͑qជ,͒ ͵Ϫϱ2 2 simply by canceling common factors of 10qЌ out of the ϱ numerator and denominator of various terms. Ќ d ϩLij͑qជ͒ CLL͑qជ,͒ Note, and this will prove to be crucial later, that (qˆ ) ͵Ϫϱ2 depends only on qˆ , diffusion constants, and the dimension- Ќ Ќ ⌬ P ͑qជ͒ L ͑qជ͒ 1 less ratios and ␥/v0 . This last fact is essential for our ij ˆ ij ϭ ϩ͑q͒ ϰ 2 , renormalization-group scaling analysis, as we will show 2ͫ⌫ ͑qជ͒ ⌫ ͑qជ͒ͬ q T L later. ͑4.28͒ The 1/q2 divergence of Eq. ͑4.28͒ as ͉qជ͉ 0 reflects the enormous long-wavelength fluctuations in this→ system. where the second integral over frequency has been evaluated These fluctuations predicted by the linearized theory are 2 strong enough to destroy long-ranged order in dр2. To see in the limit of ͉qជ͉ 0, so that c(qជ )qӷ⌫Lϰq , and the fac- → this, calculate the mean-squared fluctuations in vជ (rជ,t)ata tor (qˆ ) depends only on the direction qˆ of qជ , not its mag- Ќ nitude, and is given by the sadly complicated expression given point ជr, and time t. This is simply the integral of the trace of Eq. ͑4.28͒ over all qជ : 2 1 ͓c ͑ ជ ͒qϪv q ͔ d ˆ ϩ q 0 ʈ d q ͑q͒ϵ ͉͗vជ ͑rជ,t͉͒2͘ϭ ͗v ͑qជ ,t͒v ͑Ϫqជ ,Ϫt͒͘ c ͑ ជ ͒qͫ c ͑ ជ ͒qϪv q ϩ͓c ͑ ជ ͒qϪ␥q ͔⌫ /⌫ Ќ d i i 2 q ϩ q 0 ʈ ϩ q ʈ L ͵ ͑2͒
2 d ͓cϪ͑qជ ͒qϪv0qʈ͔ ⌬ d q ͑dϪ2͒ ϩ ϭ ͵ d 2 2 c ͑ ជ ͒qϪv q ϩ͓c ͑ ជ ͒qϪ␥q ͔⌫ /⌫ ͬ 2 ͑2͒ DTq ϩD q Ϫ q 0 ʈ Ϫ q ʈ L ͫ ͩ Ќ ʈ ʈ 2 ជ 1 Aϩ͑q,,␥͒ ͑qˆ ͒ ϵ ϩ . ͑4.33͒ F͑qជ ,,␥͒ͫ A ͑qជ ,,␥͒ϪA ͑qជ ,,␥͒⌫ ͑qជ ͒/⌫ ͑qជ ͒ 2 2 ϩ Ϫ L DLq ϩD q ͪ Ќ ͬ ʈ ʈ A2 ͑qជ ,,␥͒ Ϫ The last integral clearly diverges in the infrared (͉qជ͉ 0) for ϩ , ͑4.29͒ → A ͑qជ ,,␥͒ϪA ͑qជ ,,␥͒⌫ ͑qជ ͒/⌫ ͑qជ ͒ͬ ជ Ϫ ϩ L dр2. The divergence in the ultraviolet (͉q͉ ϱ) for dу2is not a concern, since we do not expect our theory→ to apply for where we have defined ͉qជ͉ larger than the inverse of a microscopic length ͑such as PRE 58 FLOCKS, HERDS, AND SCHOOLS: A QUANTITATIVE . . . 4839
In the next section, we will show that this prediction is invalidated by nonlinear effects, and, in fact, much of the scaling of correlation functions and propagators is changed from that predicted by the linearized theory in spatial dimen- sions dр4.
V. NONLINEAR EFFECTS AND BREAKDOWN OF LINEAR HYDRODYNAMICS IN THE BROKEN SYMMETRY STATE A. Scaling analysis In this section we analyze the effect of the nonlinearities in Eqs. ͑3.3͒ and ͑3.4͒ on the long length and time behavior of the system, for spatial dimensions dϽ4. We will rescale
lengths, time, and the fields vជЌ and ␦ according to
FIG. 6. Plot of CLL(qជ,) and CTT(qជ,)vsfor identical fixed z xជЌ bxជЌ , x b x , t b t, qជ . Note the smallness of the overlap between the transverse and → ʈ→ ʈ → longitudinal peaks. ͑5.1͒ vជ bvជ , ␦ b␦, Ќ→ Ќ → the interaction range l 0). Presumably, at larger wave num- choosing the scaling exponents to keep the diffusion con- bers, the correlation function falls off fast enough that the stants DB,T,,ʈ , and the strength ⌬ of the noise fixed. The wave-vector integral in Eq. ͑4.33͒ converges in the ultravio- reason for choosing to keep these particular parameters fixed let. rather than, e.g., 1 , is that these parameters completely de- Indeed, we will in subsequent calculations mimic the ef- termine the size of the equal time fluctuations in the linear- fect of this putative more rapid decay of correlations as ͉qជ͉ ized theory, as can be seen from Eq. ͑4.33͒. Under the res- ϱ with a sharp ultraviolet cutoff. We will restrict integrals calings ͑5.1͒, the diffusion constants rescale according to over→ wave vectors to hypercylindrical shell with long ͑very D bzϪ2D and D bzϪ2D ; hence, to keep them B,T→ B,T ,ʈ→ ,ʈ long͒ axis along the direction of flock motion xʈ : fixed, we must choose zϭ2 and ϭ1. The rescaling of the random force ជf can then be obtained from the form of the f -f ͉qជЌ͉Ͻ⌳, Ϫϱрqʈрϱ ͑4.34͒ correlations Eq. ͑2.9͒ and is, for this choice of z and ,
with the ultraviolet cutoff ⌳ of order the inverse of a micro- ជf bϪ1Ϫd/2ជf . ͑5.2͒ → scopic length ͑e. g., l 0). Obviously, this is quite an arbitrary choice of ultraviolet To maintain the balance between ជf and the linear terms in vជЌ cutoff, and any result that depends on the precise form of this in Eq. ͑3.3͒, we must rescale the velocity field according to cutoff will not be accurately calculated by this prescription. However, universal, long-wavelength properties of the flock vជЌ b vជЌ ͑5.3͒ should be unaffected by the precise choice of cutoff, and it is → on those properties that we will focus our attention. with The infrared divergence in Eq. ͑4.33͒ for dр2 cannot be dismissed so easily, since our hydrodynamic theory should ϭ1Ϫd/2, ͑5.4͒ get better as ͉qជ͉ 0. Indeed, in the absence of nonlinear ef- which is the roughness exponent for the linearized model. fects, this divergence→ is real, and signifies the destruction of That is, we expect vជЌ fluctuations on length scale L to scale long-ranged order in the linearized model by fluctuations, like L. Therefore, the linearized hydrodynamic equations, even for arbitrarily small noise ⌬, in spatial dimensions d neglecting the nonlinear convective term and the nonlineari- р2, and in particular in dϭ2, where the integral in Eq. ជ ͑4.28͒ diverges logarithmically in the infrared. This is so ties in the pressure, imply that vЌ fluctuations grow without bound ͑like L)asL ϱfor dр2, where the above expres- ជ 2 since, if ͉͗vЌ͉ ͘ is arbitrarily large even for arbitrarily small sion for becomes positive.→ Thus, this linearized theory pre- ⌬, our original assumption that vជ can be written as a mean dicts the loss of long-range order in dр2, as we saw in Sec. value ͗vជ ͘ plus a small fluctuation vជЌ is clearly mistaken; IV by explicitly evaluating the real space fluctuations. indeed, the divergence of vជЌ suggests that the velocity can Making the rescalings as described in Eqs. ͑5.1͒, the equa- swing through all possible directions, implying that ͗vជ ͘ϭ0 tion of motion ͑3.3͒ becomes for dр2. ជ ␥v ជ ␥ ជ ជ ជ ជ ជ ជ ʈvЌϩb ͓1͑vЌ•ٌЌ͒vЌϩ2ٌ͑Ќ•vЌ͒vЌ͔ץ␥ tvЌϩbץ In dϭ2, this result is very reminiscent of the familiar Mermin-Wagner-Hohenberg ͑MWH͒ theorem ͓5͔, which ϱ states that in equilibrium, a spontaneously broken continuous ជ ␥n n ជ ជ ជ ϭϪٌЌ ͚ b n͑␦͒ ϩDBٌЌٌ͑Ќ•vЌ͒ symmetry is impossible in dϭ2 spatial dimensions, pre- ͩnϭ1 ͪ cisely because of the type of logarithmic divergence of fluc- 2 ជ 2 ជ ជ vЌϩfЌ ͑5.5͉͉͒ץtuations that we have just found here. ϩDTٌЌvЌϩD͉͉ 4840 JOHN TONER AND YUHAI TU PRE 58 with d d qd ជ ជ i͑q•rϪt͒ vជЌϾ͑rជ,t͒ϭ d vជЌ͑qជ,͒e , ͑5.9͒ ͵Ͼ ͑2͒ ␥ϭϩ1ϭ2Ϫd/2, ͑5.6͒ d d qd ជ ជ ជ ជ ជ ជ i͑q•rϪt͒ vЌϽ͑r,t͒ϭ ͵ d vЌ͑q,͒e , ͑5.10͒ ␥vϭzϪϭ1, ͑5.7͒ Ͻ ͑2͒
where denotes a wave-vector integral restricted to a hy- and ͐Ͼ Ϫ1 percylindrical shell b ⌳Ͻ͉qជЌ͉Ͻ⌳, where ⌳ is an ultravio- let cutoff, and ͐ likewise denotes an integral over the inte- d Ͻ rior of this shell: qជ ϽbϪ1 . and ជf are likewise ␥nϭzϪϩnϪ1ϭnϩ͑1Ϫn͒ . ͑5.8͒ ͉ Ќ͉ ⌳ ␦ Ќ 2 separated. ͑2͒ Average the EOM over the short wavelength fields
The scaling exponent for ␦ is given by ϭ, since the vជЌϾ , ␦Ͼ , and ជf Ͼ to get new, effective EOM for the long- density fluctuations ␦ are comparable in magnitude to the wavelength fields vជЌϽ and ␦Ͻ , with ‘‘intermediate’’ renor- I vជЌ fluctuations. To see this, note that the eigenmode of the malized parameters DЌ , etc. This average is performed per- linearized equations of motion that involves ␦ is a sound turbatively in the nonlinearities in the EOM. The mode, with dispersion relation ϭcϮ(qជ )q. Inserting this perturbation theory can be represented graphically; the inter- into the Fourier transform of the continuity equation ͑3.4͒, ested reader is referred to the previously mentioned ͓6͔ for ជ ជ ជ further details on the mechanics of this. we see that ␦ϳqЌ•vЌ /qЌ . The magnitude of qЌ drops out of the right-hand side of this expression; hence ␦ scales like ͑3͒ We now rescale the time, space, and the fields in the EOM according to Eq. ͑5.1͒ in order to restore the original ͉vជЌ͉ at long distances. Therefore, we will choose ϭϭ 1Ϫd/2. ultraviolet cutoff ⌳ of the problem. We will choose rescaling The first two of these scaling exponents for the nonlin- exponents z, , and to produce fixed points. earities to become positive as the spatial dimension d is de- Of course, the exponents are, in fact, completely arbitrary. creased are ␥ and ␥ , which both become positive for d We need not choose them to produce fixed points. However, 2 it is very convenient to do so, since, as we will show in more ជ ជ ជ ជ ជ ជ Ͻ4, indicating that the 1(vЌ•ٌ)vЌ ,2(ٌЌ•vЌ)vЌ , and detail later, the values of z, , and that do produce fixed 2 ,2ٌជ Ќ(␦ ) nonlinearities are all relevant perturbations for points are exactly the values of the physical observable time dϽ4. So, for dϽ4, the linearized hydrodynamics will break anisotropy, and roughness exponents that characterize the down. scaling properties of various correlation functions. What can we say about the behavior of Eqs. ͑3.3͒ and Performing this RG procedure, we find the following re- ͑3.4͒ for dϽ4, when the linearized hydrodynamics no longer cursion relations: holds? The standard approach for such problems is the dy- namical renormalization group. In most cases, this approach dDB,T ϭ͓zϪ2ϩGD ͑g͔͒D , ͑5.11͒ is only practical near the upper critical dimension ͑in our dl B,T B,T case dcϭ4), and yields the anomalous exponents in an ex- pansion in ⑀ϭd Ϫd. This approach will obviously not be of c dD much use in our problem in dϭ2, where the ostensibly small ʈ, ϭ͓zϪ2ϩGʈ,͕͑gi͖͔͒Dʈ, , ͑5.12͒ parameter in this expansion ⑀ϭ2. We will nonetheless un- dl dertake this approach in Sec. V B, and show that, for unfor- tunate technical reasons, we learn little even near dϭ4. For- dn ϭ͓zϩ͑nϪ1͒Ϫ1ϩG͕͑g ͖͔͒ , ͑5.13͒ tunately, as we show in Sec. V C, because of the various dl n i n symmetries in Eq. ͑3.3͒, we can obtain the exact scaling exponents in dϭ2. d0 ϭ͑zϪ1͒ , ͑5.14͒ dl 0 B. Renormalization-group analysis, d<4
In this subsection, we analyze the effect of the relevant d1,2, ϭ͓Ϫ1ϩzϩG1,2,͕͑gi͖͔͒1,2, , ͑5.15͒ nonlinearities 1 ,2 , and 2 on the broken symmetry state dl in spatial dimensions dϽ4. Our tool is the dynamical renormalization group ͑for de- d⌬ tails, see, e.g., the excellent description in Forster, Nelson, ϭ͓zϪϪ2ϩ1ϪdϩG ͕͑g ͖͔͒⌬, ͑5.16͒ dl ⌬ i and Stephen ͓6͔͒. We will summarize the essential features of this procedure here; readers interested in details are re- dv0 ferred to ͓6͔. ϭ͑zϪ͒v , ͑5.17͒ We proceed through the iteration of the following 3 steps: dl 0
͑1͒ We separate the fields vជЌ and ␦, and the random forces ជf into short and long wavelength components, accord- d␥ ϭ͑zϪ͒␥, ͑5.18͒ ing to dl PRE 58 FLOCKS, HERDS, AND SCHOOLS: A QUANTITATIVE . . . 4841 where we have taken bϭ1ϩdl , dl Ӷ1, to obtain differen- This particular correlation function is important because it tial recursion relations, the G’s represent graphical ͑i.e., per- gives us our best measure of the size of the velocity fluctua- turbative͒ corrections, and the ͕gi͖’s are a set of dimension- tions, and will ultimately determine whether or not these less coupling constants involving ratios of powers of the fluctuations destroy the long-ranged orientational order of dynamical parameters. We have also dropped ‘‘irrelevant’’ the flock ͑thereby driving its mean velocity to zero͒. terms in these recursion relations. The coupling constant Cij(qជ) is, of course, a function of the flock dynamical is the coefficient of the ជ (vជ ) nonlinearity in the -Ќ• Ќ␦ parameters DB,T , Dʈ, , ⌬, etc., as well as of qជ . Furtherٌ equation of motion. This coupling constant is equal to 1, and more, at small qជ , it is difficult to calculate in spatial dimen- must, up to trivial rescaling corrections, remain equal to 1 sion dϽ4 due to the nonlinear terms, for the reasons dis- upon renormalization. This is a simple consequence of the cussed above. So let us follow this renormalization-group fact that mass conservation is exact; that is, the equation of matching procedure to relate C (qជ; B0 ) where B0 de- motion for must remain the simple continuity equation ij ͕ i ͖ ͕ i ͖ notes the set of dynamical parameters D0 , D0 , ⌬ , etc. in ϩٌជ (vជ )ϭ0, except for the trivial changes introduced B,T ʈ 0 ץ t • the unrenormalized model, to the same correlation function by rescaling. This implies that G( g )ϭ0, exactly, for all ͕ i͖ in the renormalized model, a renormalization group time l ͕gi͖. We will later use this fact to obtain exact values for the later: scaling exponents , z, and in dϭ2. It is worth noting that ␥ is treated as an independent variable here, only its bare value is related to the bare values of 1 and v0 through Eq. 0 C ͑qជЌ ,q ;͕B ͖͒ ͑3.5͒. ij ʈ i Although there is no symmetry argument forbidding ͑2ϩϩdϪ1͒l l l ϭe Cij͓e qជЌ ,e qʈ ;͕Bi͑l ͖͔͒, renormalization of v0 and ␥, simple power counting shows that there are no relevant graphical corrections to them; this ͑5.20͒ is why no graphical corrections appear in Eqs. ͑5.17͒, ͑5.18͒. As mentioned earlier, the rescaling exponents , z, and where the ͕B (l )͖ denote the renormalized parameters. are arbitrary. We chose them to produce fixed points only for i In the discussion that follows, we will first consider the computational convenience. However, there is no choice of case (q /⌳)Ӷ(q /⌳). At the conclusion of the discussion rescaling exponents that will keep all the parameters fixed. ʈ Ќ of this special case, we will briefly indicate how the general For instance, to keep D and D fixed, we will have to B,T ʈ, case can be treated to obtain the scaling laws quoted in the choose zϾ1. However, with this choice of z, the relations Introduction. For the case (q /⌳)Ӷ(q /⌳), we will ͑5.13͒ and ͑5.14͒ show that and flow to infinity. ʈ Ќ 1 0 choose ϭ (qជ )ϭln(⌳/q ), where ⌳ is the ultraviolet Which of the parameters, then, should we choose to keep l l * Ќ Ќ fixed? That is, which is most convenient to keep fixed? The cutoff, on the right-hand side, and obtain answer to this question is provided by the renormalization group matching formalism. This approach enables one to use ជ 0 the renormalization group to relate correlation functions in Cij͑qЌ ,qʈ ;͕Bi ͖͒ the original, unrenormalized model at long distances and 2ϩϩdϪ1 large times to the same correlation functions in the renormal- ⌳ qʈ ϭ Cij ⌳, ;͕Bi͓l ͑qជЌ͔͖͒ . ͩqЌͪ * ized system at shorter distances and times. The advantage of ͩ ͑qЌ /⌳͒ ͪ this approach is that long distance, large time correlation ͑5.21͒ functions are hard to calculate in dϽ4, since, as we showed from our earlier scaling arguments ͓and can also verify from the renormalization-group recursion relations ͑5.11͒–͑5.12͔͒, Now, if the original qЌ was small (Ӷ⌳) and we have chosen these are not accurately calculable from the harmonic theory the rescaling exponents , , and z so that the nonlinearities developed in Sec. IV, since the nonlinearities , , and 1,2 1(l ), 2(l ), (l ), and 2(l ) on the right-hand side of 2 have very large effects at long distances. By mapping Eq. ͑5.21͒ flow, as l ϱ,toO(1) fixed point values these correlation functions onto those at short distances in (* ,*), then ( →), ( ), ( ), and ( ) 1,2, 2 1 l * 2 l * l * 2 l * the renormalized equations of motion, we circumvent this can be replaced by those fixed point values, since will be l * problem. Clearly, there is a caveat here: even at short dis- large. Because those fixed point values are, by assumption, tances, the correlation functions in the renormalized model O(1), then, up to O(1) correction factors coming from these can only be calculated accurately in the harmonic theory if nonlinearities, the right-hand side of Eq. ͑5.21͒ can be evalu- the nonlinear couplings in that renormalized model are not ated in the harmonic approximation Eqs. ͑4.22͒, ͑4.25͒, and too big. This suggests that the convenient choice of the res- ͑4.26͒. ͓The correction factors are only of O(1)—i.e., not caling exponents , z, and is that which keeps the nonlin- divergent—because the right-hand side of Eq. ͑5.21͒ is earities , , and fixed. 1,2 2 ជ ជ Let us illustrate these considerations explicitly for one evaluated at large q(͉qЌ͉ϭ⌳), where the infrared diver- very important correlation function: the equal time spatially gences associated with the strong relevance of the nonlineari- Fourier-transformed velocity-velocity autocorrelation func- ties do not matter. It is precisely because of those infrared tion: divergences that we could not evaluate the left-hand side of ͑5.21͒ directly, but rather were forced to go through this seemingly circuitous RG matching formalism.͔ Making that Cij͑qជ͒ϵ͗vi͑qជ,t͒vj͑Ϫqជ,t͒͘. ͑5.19͒ harmonic approximation on the right-hand side, we obtain 4842 JOHN TONER AND YUHAI TU PRE 58
on l :exp͓(zϪ)l ͔, as can be seen from their recursion qʈ * * C ⌳, ;͕⌫͑l ͖͒ relations , and the combination ij * … ͩ ͑qЌ /⌳͒ ͪ
⌬ qʈ ϭ * PЌ͑qˆ ͒ ͑l ͒ . ͑5.26͒ 2 2 ij Ќ * ͑q /⌳͒⌳ DT*⌳ ϩD*ʈ ͓qʈ /͑qЌ /⌳͒ ͔ Ќ
⌬ By combining the recursion relations ͑5.17͒, ͑5.13͒, and ϩ * 2 2 ͑5.14͒ into a recursion relation for : DL*⌳ ϩD*ʈ ͓qʈ /͑qЌ /⌳͒ ͔
d d ln v0 1 d qʈ Ќ ϫ ⌳, ;͕B ͑l ͖͒ L ͑qˆ ͒, ͑ln ͒ϭ Ϫ ͑ln 1ϩln 0͒ϭ1Ϫ i * ij Ќ dl dl 2 dl ͩ ͑qЌ /⌳͒ ͪ ͑5.27͒ ͑5.22͒ we find where we have used the fact that we have chosen the scaling exponents to make ⌬ and all the diffusion coefficients ͕Di͖ ͑1Ϫ͒l ͑l ͒ϭe 0 , ͑5.28͒ flow to fixed points ⌬*, ͕Di*͖. We wish to show that this expression depends on qជ only through the scaling ratio u which implies that ϵqʈ /(qЌ /⌳) . The first ͑transverse͒ term in Eq. ͑5.22͒ ex- plicitly has this property. The second ͑longitudinal term͒ ⌳ 1Ϫ would also, except for the factor, to which we now turn. ϭ el 1Ϫϭ . ͑5.29͒ ͑l *͒ 0͑ *͒ 0 From Eq. ͑4.29͒ for , we see that to calculate this factor we ͩ qЌ ͪ must calculate Using this in Eq. ͑5.26͒, we see that the combination
qʈ F ⌳, ;͕Bi͑l ͖͒ * qʈ qʈ ͩ ͑qЌ /⌳͒ ͪ ͑l ͒ ϭ ͑5.30͒ * 0 ⌳͑qЌ /⌳͒ qЌ 2 2 ␥͑l ͒Ϫv0͑l ͒ qʈ ϭ * * 2͑l ͒ ϩ1, ͱͩ 2v ͑l ͒ ͪ * takes on precisely the value it would take on using the un- 0 * ͩ ⌳͑qЌ /⌳͒ ͪ renormalized parameters and the unrescaled wave vector qជ . ͑5.23͒ Hence, the same is true of F, AϮ , and BϮ . And, therefore, the same is true of (qˆ ). qʈ A ⌳, ;͕B ͑l ͖͒ Thus, we can replace „⌳,qʈ /(qЌ /⌳) ;͕Bi(l )͖… in Eq. Ϯ i * * ͩ ͑qЌ /⌳͒ ͪ ˆ 0 ͑5.22͒ with (q;͕Bi ͖), its unrenormalized value straight from the linearized theory. Doing so, and recalling that the qʈ ϭϮF ⌳, ;͕Bi͑l ͖͒ unrenormalized (qˆ ) was O(1) for all directions qˆ of qជ ,we ͩ ͑qЌ/⌳͒ * ͪ see from Eq. ͑5.22͒ that the correlation function Cij is largest when ͓␥͑l ͒Ϫv0͑l ͔͒ qʈ Ϫ * * ͑l ͒ * 2v0͑l ͒ ͩ⌳͑qЌ/⌳͒ ͪ * qʈ qЌ ϳ . ͑5.31͒ ͑5.24͒ ⌳ ͩ ⌳ ͪ and For ͉qជ͉Ӷ⌳, where our theory applies, Eq. ͑5.31͒ implies that q ӷq ͑since Ͻ1). In that limit, (qˆ ) 1; using this in qʈ ʈ Ќ B ⌳, ;͕B ͑l ͖͒ the expression 5.22 for C in the renormalized→ system, and Ϯ i * ͑ ͒ ij ͩ ͑qЌ /⌳͒ ͪ using Eq. ͑5.22͒ in turn in our expression ͑5.21͒ for Cij in the original model, we obtain, for qʈӷqЌ , the scaling law qʈ ϭϮF ⌳, ;͕B ͑l ͖͒ i * ͩ ͑qЌ /⌳͒ ͪ ͑qʈ /⌳͒ C ͑qជ͒ϭqϪ͑2ϩdϩϪ1͒f . ͑5.32͒ ij Ќ ij ͓␥͑l ͒Ϫv ͑l ͔͒ q ͩ͑qЌ /⌳͒ ͪ 0 ʈ ϩ * * ͑l ͒ , 2v ͑l ͒ * 0 * ͩ⌳͑qЌ /⌳͒ ͪ Note that the range q͉͉ӷqЌ for which this scaling law ͑5.25͒ holds includes those qជ ’s which dominate the fluctations, all of which are clearly dependent only on the fixed point namely, those with q͉͉ /⌳տ(qЌ /⌳) . ជ ជ value of the ratio ␥( )/v ( ) which is just a number of Integrating Cij(q) over all q gives the equal-time, root- l * 0 l * „ O(1) since ␥( ) and v ( ) have the same dependence mean-squared real-space fluctuation of vជ : l * 0 l * Ќ PRE 58 FLOCKS, HERDS, AND SCHOOLS: A QUANTITATIVE . . . 4843
ddq scaling law ͑1.8͒. This directly and simply relates the RG to 2 ͉͗vជЌ͑rជ,t͉͒ ͘ϭ Cii͑qជ͒ physically observable correlation functions, so this is the ͵ ͑2͒d choice we will make. dϪ1 A scaling law similar to Eq. ͑5.32͒ can be derived, by d qЌ ͑qʈ /⌳͒ ϭ qϪ͑2ϩdϩϪ1͒ dq f precisely the same type of arguments, for the equal-time ͵ d Ќ ͵ ʈ ii ͑2͒ ͩ͑qЌ /⌳͒ ͪ density-density correlation function: dϪ1 d qЌ ϭA , ͑5.33͒ q3ϪdϪϪ2 ͵ 2ϩdϪ1 Ќ qʈl 0 qЌ C ͑qជ ͒ϭ f Y͑ ជ ͒, ͑5.38͒ 2 q q ͩ ͑qЌl ͒ ͪ 0 where the final proportionality was obtained by scaling qЌ out of the qʈ integral via the change of variables qʈ where, in writing this relation, we have used the fact that the ϵ⌳Qʈ(qЌ /⌳) and we have defined the qЌ-independent ‘‘roughness’’ exponent for , ϭ, the ‘‘roughness’’ ex- 2ϩdϩ constant Aϵ͐dQʈ fij(Qʈ)⌳ . The final integral in Eq. ponent for vជЌ . ͑5.33͒ clearly converges in the infrared (͉qជ ͉ 0) limit if Ќ → The alert reader will have noticed that neither of the scal- and only if Ͻ0. Furthermore, if is Ͼ0, and we impose an ing laws ͑5.32͒ and ͑5.38͒ derived so far involves the time Ϫ1 infrared cutoff ͉qជЌ͉ϾLЌ in Eq. ͑5.33͒, where LЌ is the rescaling exponent z. This is unsurprising, since we have lateral ͑i.e., Ќ direction͒ spatial extent of the system, we considered only equal-time correlation functions up to now. easily obtain To fully study the dynamics of the model, we need to consider correlations between different times, as well as po- 2 2 ͉͗vជЌ͑vជ ,t͉͒ ͘ϭCЈLЌ . ͑5.34͒ sitions. These different time correlation functions will in- volve z. Indeed, the connected real-space, equal-time, velocity au- It is easiest to work with the space and time Fourier trans- tocorrelation function discussed in the Introduction is given form C (qជ,), defined by by ij
ជ ជ ជ ជ ជ ជ ជ ជ 2 ͗v ͑qជ ,͒v ͑Ϫqជ ,Ј͒͘ϵ␦͑ϩЈ͒C ͑qជ,͒. ͑5.39͒ CC͑R͒ϵ͗v͑rϩR,t͒•v͑r,t͒͘Ϫ͉͗v͑r,t͉͒͘ i j ij ជ ជ ជ ជ ជ ϭ͗vЌ͑rϩR,t͒•vЌ͑r,t͒͘ We will find, as we have asserted many times already in
d this paper, that Cij(qជ,) does not have a simple scaling d q ជ ជ ϭ C ͑qជ͒eiq•R form, unlike the equal-time correlations. Nonetheless we can ͵ d ii ͑2͒ derive an expression for it in terms of functions of qជ that do show simple scaling behavior, namely, effective wave- ddq ͑q /⌳͒ ϭ qϪ͑2ϩdϩϪ1͒f ʈ vector-dependent diffusion constants that diverge as ͉qជ͉, ͵ d Ќ ii ͑2͒ ͩ͑qЌ /⌳͒ ͪ 0. → i͑qជ Rជ ϩq R ͒ We begin this derivation by separating Cij into its trans- ϫe Ќ• Ќ ʈ ʈ . ͑5.35͒ verse and longitudinal parts: Making the changes of variable Ќ Ќ Cij͑qជ,͒ϵLijCL͑qជ,͒ϩPijCT͑qជ,͒, ͑5.40͒ Qជ Ќ Qʈ qជ ϵ , q ϵ , ͑5.36͒ Ќ Ќ Ќ Ќ Ќ ʈ Ќ where L ϵq q /q2 and P ϭ␦ ϪL Ϫ␦ ␦ are, re- ͉RជЌ͉ ͉RជЌ͉ ij i j Ќ ij ij ij i,͉͉ j,͉͉ spectively, the longitudinal and transverse projection opera- tors defined in Sec. IV. Both the transverse and longitudinal we obtain the scaling law ͑1.8͒ for Cc(Rជ ) quoted in the In- troduction, with pieces CL,T(qជ ,) obey the same renormalization-group transformation
Qʈ f ͑u͒ϵ ddϪ1Q dQ f v Ќ ʈ ii ជ 0 ͵ ͩQ ͪ CL,T͑qЌ ,qʈ ,;͕Bi ͖͒ Ќ
ជ ˆ ͑2ϩzϩϩdϪ1͒l l l zl i͑QЌ•RЌϩQʈu͒ Ϫ͑2ϩdϩϪ1͒ ជ ϫe QЌ . ͑5.37͒ ϭe CL,T͑e qЌ ,e qʈ ,e ;͕Bi͑l ͖͒͒, ͑5.41͒ This shows that the we obtain from the renormalization group by the prescription we have chosen—namely, making where we have been careful to take into account the rescaling the specific set of parameters DB,T,ʈ , ⌬, 1,2, , and 2 flow to fixed points—is precisely the physical roughness exponent of the delta function in Eq. ͑5.39͒ in deriving the argument of defined by the velocity fluctuations. the exponential in Eq. ͑5.41͒. To summarize: if we chose the rescaling exponents , z, As for the equal-time correlation function, it is convenient l l and so as to make the particular subset of the dynamical here to choose the rescaling factor e such that e qជЌ is right l ជ parameters DB,T,ʈ, , ⌬, 1,2, , and 2 flow to nonzero fixed on the Brillouin zone boundary; i.e., e ͉qЌ͉ϭ⌳. Making this points, then those and are the ones that appear in the choice, and taking ⌳ϭ1, we obtain 4844 JOHN TONER AND YUHAI TU PRE 58
0 CL,T͑qជЌ ,qʈ ,;͕Bi ͖͒ with the sound frequencies Ϯ(qជ ;␥,0 ,͕Bi͖) obtained in the harmonic theory: q Ϫ͑2ϩzϩϩdϪ1͒ ˆ ʈ ϭqЌ CL,T qЌ , , ;͕Bi͑l ͖͒ , ͩ q qz * ͪ Ќ Ќ ͑qជ ;␥, ,͕B ͖͒ ͑5.42͒ Ϯ 0 i ␥ϩv0 ϭ q cos ជ where, on the right-hand side, we have defined 2 q
2 1 Ϫv q cos ជ ͑␥ 0͒ q 2 2 l ϭln . ͑5.43͒ Ϯ ϩ q sin ជ * ជ ͱͩ 2 ͪ 1 0 q ͩ ͉qЌ͉ͪ 2 ͑␥ϩv0͒qʈ ͑␥Ϫv0͒qʈ For the moment, let us focus on the longitudinal piece ϭ Ϯ ϩ q2 , 2 ͱͩ 2 ͪ 1 0 Ќ CL(qជ ,). As we argued for the equal-time correlation func- tion, here too we can evaluate the right-hand side in the ͑5.47͒ harmonic approximation, Eq. ͑4.26͒. This gives
0 2 CL͑qជЌ ,q ,;͕B ͖͒ ʈ i qʈ ⌫L͑l ͒ϵDL*ϩD*ʈ , ͑5.48͒ ⌬ ͓Ϫv ͑l *͒q qzϪ͔2qϪ͑2ϩ3zϩϩdϪ1͒ * ͩ q ͪ 0 0 ʈ Ќ Ќ Ќ ϭ * , D ͑5.44͒ 2 qʈ ⌫͑l ͒ϵD* , ͑5.49͒ where * ͩ q ͪ Ќ
2 2 ϭ Ϫϩ͑l ͒ ϪϪ͑l ͒ z * z * and D* , D* , and ⌬ are the fixed point values of DL , D , D ͭͫ qЌ ͬ ͫ qЌ ͬ L ʈ * ʈ and ⌬, to which those parameters will have flown for l * large, as it will be for small qЌ . ϩ ͓⌫L͑l ͒ϩ⌫͑l ͔͒Ϫv0͑l ͒ The complication of this expression—that is, the fact that ͩ qz ͪ * * * ͫ Ќ stops it from having a simple scaling form—is that the pa- 2 rameters ␥(l ), (l ), and (l ) that appear implicitly ␥͑l ͒ q * 1 * 0 * ϫ ϩ * ʈ , in ͑5.44͒ do not flow to field point values for our ‘‘canoni- ⌫L͑l ͒ ⌫͑l ͒ ͩ * v0͑l ͒ * ͪͩ q ͪ cal’’ choice of , z, and , as discussed earlier. Physically, * Ќ ͬ ͮ this reflects the fact that the scaling of the sound speeds ( ͑5.45͒ ϰq) is different from that of their dampings ͑damping rate ϰq2 in harmonic theory; we will show damping rate is ϰqz q Ќ ˆ ʈ here, in a moment͒. Ϯ͑l ͒ϵϮ qЌ , ;␥͑l ͒,0͑l ͒,͕Bi͑l ͖͒ , * ͩ q * * * ͪ To proceed, it is first useful to reorganize Eq. ͑5.44͒ Ќ 4z ͑5.46͒ slightly; by multiplying numerator and denominator by qЌ ,
0 CL͑qជЌ ,qʈ ,;͕Bi ͖͒
zϪ 2 ͑zϪϪ2ϩ1Ϫd͒ ⌬ ͓Ϫv0͑l ͒qʈqЌ ͔ qЌ ϭ * * . Ϫqz 2 Ϫqz 2ϩ qz ⌫ ϩ⌫ Ϫ v ⌫ ϩ␥ ⌫ q q2zϪ 2 ͓ Ќ ϩ͑l *͔͒ ͓ Ќ Ϫ͑l *͔͒ ͓ Ќ͓ L͑l *͒ ͑l *͔͒ ͓ 0͑l *͒ L͑l *͒ ͑l *͒ ͑l *͔͒ ʈ Ќ ͔ ͑5.50͒
Next, we solve the recursion relations for ␥( ), ( ), and ( ): l * 1 l * 0 l * v ϭe͑zϪ͒l v ϭ0 ϭv 0 qϪz , ͑5.51͒ 0͑l *͒ * 0͑l ͒ 0͑ ͒ Ќ ␥ ϭe͑zϪ͒l ␥ ϭ0 ϭ␥ 0 qϪz , ͑5.52͒ ͑l *͒ * ͑l ͒ ͑ ͒ Ќ ϭe͑zϪ1͒l ϭ0 ϭ 0 q1Ϫz , ͑5.53͒ 1͑l *͒ * 1͑l ͒ 1͑ ͒ Ќ ϭe͑zϪ1͒l ϭ0 ϭ 0 q1Ϫz , ͑5.54͒ 0͑l *͒ * 0͑l ͒ 0͑ ͒ Ќ PRE 58 FLOCKS, HERDS, AND SCHOOLS: A QUANTITATIVE . . . 4845 where in the second equality in each equation we have used ϭln(1/ qជ ). Using these results and the expressions ͑5.46͒ l * ͉ Ќ͉ and ͑5.47͒ for ( ), we see that z and drop out of the combinations Ϯ l * 2 Ϫ z z 1 qʈ Ϫz ␥͑0͒ v0͑0͒ qʈ Ϫz 2͑1Ϫz͒ qЌϮ͑l ͒ϭqЌ ͓␥͑0͒ϩv0͑0͔͒ qЌ Ϯ qЌ ϩ1͑0͒0͑0͒qЌ * ͫ 2 q ͱͩ 2 q ͪ ͬ Ќ Ќ
2 1 ͓␥͑0͒Ϫv0͑0͔͒qʈ ϭ ͑␥ϩv ͒q Ϯ ϩ ͑0͒ ͑0͒q2 2 0 ʈ ͱͩ 2 ͪ 1 0 Ќ
ϭϮ͓qជЌ ,qʈ ;␥͑0͒,v0͑0͒,1͑0͒,0͑0͔͒ ͑5.55͒ and therefore the positions of the peaks in the full correlation function ͑5.50͒ are exactly those given by the harmonic theory using the bare parameters 1(0), v0(0), ␥(0), and 0(0), namely, wϮ͓qជ ;␥(0),v0(0),1(0),0(0)͔. This is a direct conse- quence of the fact that there are no ͑relevant͒ graphical renormalizations of the parameters ͑␥, 1 , and 0) that determine the sound speeds ͓see the recursion relations ͑5.14͒–͑5.17͔͒ and shows that the relevant nonlinearities below dϭ4donot alter the positions of the peaks in the spatiotemporally Fourier-transformed velocity-velocity autocorrelations. The same cannot, however, be said for their widths. Indeed, using the above result for the sound speeds and Eqs. ͑5.48͒ and
͑5.49͒, we see that CL(qជ ,) can be rewritten:
2 ͑zϪϪ2ϩ1Ϫd͒ ⌬ ͓Ϫv0͑0͒qʈ͔ qЌ C ͑qជ ,͒ϭ * , ͑5.56͒ L 2 2 R ជ R ជ R ជ R ជ 2 ͓Ϫcϩ͑qជ ͒q͔ ͓ϪcϪ͑qជ ͒q͔ ϩ͕͓⌫L͑q͒ϩ⌫ ͑q͔͒Ϫqʈ͓v0͑0͒⌫L͑q͒ϩ␥͑0͒⌫ ͑q͔͖͒
where the sound speeds are given by the harmonic result discussed earlier, cannot be determined without knowing the equation ͑4.11͒, and the renormalized dampings fixed-point values Di* of the diffusion constants. However, 2 regardless of their values, we still get a scaling law with the q q R ʈ z z ʈ same power of qЌ and the same scaling variable q͉͉ /qЌ as ⌫L͑qជ ͒ϭ D* ϩDL* qЌϵqЌ f L , ͑5.57͒ ʈ ͩ q ͪ ͩ q ͪ that found earlier in the opposite limit. ͫ Ќ ͬ Ќ In between these two limits we have to choose l to 2 * q q smoothly interpolate between the two limits. This choice will R qជ ϭD* ʈ qz qz f ʈ 5.58 clearly depend on the ratio q /q . Naively, one could imag- ⌫ ͑ ͒ Ќϵ Ќ ͑ ͒ ͉͉ Ќ ͩ q ͪ ͩ q ͪ 1/ Ќ Ќ ine simply choosing ϭln͓min(1/q ,O(1)/q )͔. A sub- l * Ќ ͉͉ tler choice would take into account the O(1) perturbative obey simple scaling laws. corrections we have neglected, and would presumably lead The exact form of the scaling laws that we have obtained to a smooth crossover of l between the two limits. here namely, e.g., f (q /q )ϭ͓D*(q /q )2ϩD*͔ , is not * „ L ʈ Ќ ʈ ʈ Ќ L … The moral of this discussion is threefold: correct, because our choice of l ϭln(1/qЌ) is only valid * when qЌӷq͉͉ . In the opposite limit qЌӶq͉͉ , the fluctuations ͑1͒ We always get scaling laws of the form ͑5.57͒ for the 2 become negligible in the renormalized problem once D͉͉q͉͉ dampings. 2 becomes ӷDB⌳ in the renormalized problem, because at ͑2͒ The renormalized damping functions and the noise this point the linearized approximation to the correlation strength are always of such a form that they depend only on functions is smaller than its largest value at the Brillouin q͉͉ for q͉͉ӷqЌ , and only on qЌ in the opposite limit. zone boundary. This means we can now stop the renormal- ͑3͒ We can only calculate the scaling function if we know ization at such that el q ϭ⌳ϫD*/D* , which implies l * * ͉͉ B ͉͉ the diffusion constants at the fixed point. that ϭln(1/q )/ϩO(1), where the O(1) factor is uni- l * ͉͉ versal, because it depends only on the fixed point values of This last point will stop us from calculating the crossover the diffusion constants. Performing the above calculations functions in dϭ2, even though, as we will see, we can cal- with this choice of , we now obtain culate the exponents there. l * We see from Eq. ͑5.57͒ that the physical significance of 2 qЌ qʈ the exponent z is that it gives the scaling of the peak widths ⌫R͑qជ ͒ϭ D*ϩD* qz/ϫO͑1͒ϵqz f , L ʈ L 2/ ʈ Ќ L ជ ͩ q ͪ ͩq ͪ ͑in )ofCL(q,) with qЌ , while the peak positions con- ͫ ͬ Ќ ʈ ͑5.59͒ tinue to obey the ‘‘zϭ1’’ scaling ϰq. Similar, but actually far simpler, arguments show that the where we have now defined f (q /q )ϵ͓D*(q /q )z/ L ʈ Ќ ʈ ͉͉ Ќ transverse correlation function CT(qជ ,) obeys 2Ϫz/ ϩDL*(qЌ/qʈ) ͔ϫO(1). Note that the precise form of this scaling function is different in this regime from that zϪϪ2ϩ1Ϫd f ⌬͑qʈ /qЌ͒qЌ found earlier for q ӷq . Furthermore, its exact form is un- C ͑qជ ,͒ϭ , ͑5.60͒ Ќ ͉͉ T 2 2 ជ certain, due to our uncertainty in the O(1) factor, which, as ͑Ϫ␥q͉͉͒ ϩ⌫T͑q͒ 4846 JOHN TONER AND YUHAI TU PRE 58
We hope the reader has not been too confused by the fact where ⌫T(qជ ) obeys the scaling law that we have restored the ultraviolet cutoff ⌳ϳ1/l 0 to the problem by going back to dimensionful units where ⌳Þ1. z qʈ ⌫T͑qជ ͒ϭqЌ f T , ͑5.61͒ This completes our discussion of how the renormalization ͩ q ͪ Ќ group, and, in particular, the exponents z, , and , relate to physically observable correlation functions and propagators. and f is a scaling function associated with ⌬. For general ⌬ Now, we turn to the problem of actually calculating those values of q and q , the same scaling function f should Ќ ͉͉ ⌬ exponents. also be present in all of the other correlation functions as well ͑wherever ⌬ appears͒, such as Eq. ͑5.56͒ for the longi- 2؍tudinal correlation function. C. Exponents in d Likewise, the propagators of the full nonlinear theory are To do this, we must calculate the graphical corrections in given, in dϽ4, by the harmonic expressions ͑4.18͒–͑4.21͒, Eqs. ͑5.11͒–͑5.16͒. The procedure for this, as discussed in except that ⌫L , ⌫ , and ⌫T in those expressions are replaced ͓6͔ involves the harmonic correlation functions and propaga- by the anharmonic scaling laws ͑5.57͒, ͑5.58͒, and ͑5.61͒. tors and vertices representing the nonlinearities 1,2, and To complete the specification of the scaling laws, we need 2 . Rather than actually calculating these corrections, we the asymptotic behavior of the scaling functions f ⌬,L,T,(u). will show that, when 2ϭ0, the structure of the theory is From Eqs. ͑5.57͒, ͑5.58͒, and the analogous result for Eq. such that we can determine the exponents , z, and ex- ͑5.61͒, and requiring that the second point of our tripartite actly. moral applies, we see that Consider first the 1 nonlinearity. Separating vជЌ into transverse and longitudinal components, const, u 0 f ͑u͒ϰ → ͑5.62͒ ⌬ ͭ u͑zϪϪ2ϩ1Ϫd͒/, u ϱ vជ ϵvជ ϩvជ ͑5.69͒ → Ќ T L and this can be written as const, u 0 ͑vជ ٌជ ͒vជ ϭ͑vជ ٌជ ͒vជ ϩ͑vជ ٌជ ͒vជ ϩ͑vជ ٌជ ͒vជ f ͑u͒ϰ → ͑5.63͒ Ќ• Ќ Ќ T• Ќ T L• Ќ L T• Ќ L L,T, ͭ uz/, u ϱ, → ជ ជ ជ ϩ͑vL•ٌЌ͒vT . ͑5.70͒ which implies that Now consider the graphs that can be constructed from vជ z L qЌ , qʈӶqЌ ជ ជ ϪvT , the cross terms in this expression. These will always ⌫L,T,͑q͒ϰ z/ ͑5.64͒ ͭ q , qʈӷqЌ . mix transverse and longitude propagators and correlation ʈ functions in the internal integrals over momentum and fre- The simplest summary of the scaling of all correlation func- quency. But, as noted earlier in our discussion of the har- tions and propagators is as follows: simply use the harmonic monic theory, the peaks in the longitudinal propagators and expressions for them, except that diffusion constants DT,B, correlation functions occur at different frequencies ͓ should be replaced by wave-vector-dependent quantities that ϭϮ(qជ )͔ than those in the transverse propagators and cor- diverge as qជ 0, according to the scaling law relation function, which occur at ϭ0. Furthermore, the → overlap between these peaks is negligible, since their widths z ͑qʈ /⌳͒ (ϰq with zϾ1) are much less than this offset in peak po- D ͑qជ ͒ϭqzϪ2 f , ͑5.65͒ Ќ T,B, Ќ T,B, sitions. This implies that the integral over wave vectors and ͩ ͑qЌ /⌳͒ ͪ frequencies of any graph that mixes transverse and longitu- the bare noise strength ⌬ should be replaced by dinal propagators and correlation functions will be much less ͑by powers of q) than any similar graph containing purely q zϪϪ2ϩ1Ϫd ͑q /⌳͒ transverse or purely longitudinal propagators and correlation ជ Ќ ʈ ⌬͑q͒ϭ⌬ f ⌬ ͑5.66͒ ជ ជ *ͩ ⌳ ͪ functions. Hence, the vLϪvT cross terms in Eq. ͑5.70͒ are ͩ ͑qЌ /⌳͒ ͪ irrelevant compared to the pure vជ L and vជ T terms. and the diffusion constant Dʈ should be replaced by Now let us consider those relevant pieces. The Fourier transform of the vជ T piece at wave vector qជ can be written in ͑qʈ /⌳͒ Fourier space: D ͑qជ ͒ϭqzϪ2 f ͑5.67͒ ʈ Ќ ʈ ͩ ͑qЌ /⌳͒ ͪ FT͓͑vជ ٌជ ͒v ͔ជϭi vជ ͑pជ͒͑qជ Ϫpជ ͒v ͑qជϪpជ͒ T• Ќ Ti q ͚ T Ќ Ќ Ti as can be seen by requiring that pជ
2 qʈ Ќ 2 z ϭiqj vT ͑pជ͒vT ͑qជϪpជ͒, ͑5.71͒ Dʈ͑qជ ͒q ϭD* qЌ ͑5.68͒ ͚ j i ʈ ͉͉ ͩq ͪ pជ Ќ where we have used the fact that vជ is transverse, so the right-hand side being the form of the qʈ-dependent term T ជ ជ ជ in Eq. ͑5.57͒. pЌ•vT(p)ϭ0. So this piece of the 1 vertex, which is a term PRE 58 FLOCKS, HERDS, AND SCHOOLS: A QUANTITATIVE . . . 4847
to two independent exact scaling relations between the three independent exponents , z, and . Requiring dDʈ, /dl ϭ0 implies
zϭ2, ͑5.74͒
while requiring d⌬/dl ϭ0 leads to FIG. 7. Feynmann graph renormalizing the noise correlations zϭϩ2ϩdϪ1. ͑5.75͒ when 2ϭ0. There is a factor of the external momentum ͉qជЌ͉ ϵqx associated with each vertex; hence this graph does not renor- malize ⌬, but, rather, only changes O(q2) pieces of the f Ϫ f cor- We emphasize that we have only shown that these relations relation function. ͑5.74͒ and ͑5.75͒ hold when 2ϭ0. We can obtain a third independent exact scaling relation between these three exponents, and thereby determine them tvi(qជ ,t), is proportional to the externalץ in the equation for exactly, when 2ϭ0, by considering the renormalization of momentum qជ . So is the purely longitudinal term, as can Ќ the nonlinearities 1 , , and 2 . easily be seen in real space. Since vជ L is longitudinal, we can We start deriving this third relation by noting that when write 2ϭ0 and 1ϭϵ, there can be no graphical renormal- ization of 1 . This is because for these parameter values the equations of motion ͑3.3͒ and ͑3.4͒ have an exact symmetry vជ ϭٌជ ͑5.72͒ L Ќ that we call pseudo-Galilean invariance: namely, they remain unchanged under the ‘‘boost’’ transformation: for some scalar field . Now the second term in Eq. ͑5.70͒ can be rewritten in terms of : rជ rជ Ϫvជ t, ͑5.76͒ Ќ→ Ќ b ,͒ ץ ץ͑ ץ ͒ϭ 1 ץ ץ͒͑ ץ͒ϭ͑ ץ ץ͒͑ ץvជ ٌជ ͒v ϭ͑͑ L• Ќ Li j j i j i j 2 i j j vជЌ͑rជ,t͒ vជЌ͑rជ,t͒ϩvជ b , ͑5.77͒ ͑5.73͒ →
where the ‘‘boost’’ velocity vជ b is an arbitrary constant vector which is clearly a total derivative, whose Fourier transform in the Ќ plane. is proportional to qជЌ . This symmetry must be preserved upon renormalization Hence, the two relevant pieces of the 1 vertex are pro- with the same value of . Hence, there can be no graphical portional to the external momentum qជЌ . Clearly, the 2 renormalization of , when 1ϭ . That is, G1 ϭ0 when ϭ . Since Gϭ0 always, it is clear that, if Ͻ ini- term, being a total Ќ derivative, is also proportional to qជЌ in 1 1 tially, it will always remain so upon renormalization. Fourier space. Hence, when 2ϭ0, all the remaining relevant This implies that, for flocks that start with bare 1 less vertices are proportional to qជЌ . An immediate consequence of this is that ⌬ and D acquire no graphical renormalization. than the bare ͑which should be a finite fraction of all ʈ possible flocks , only two types of fixed points are possible: For ⌬, this can be seen by noting that any graph that renor- ͒ * * * malizes ⌬͑e.g., Fig. 7͒ must contain two external vertices, ͑I͒1ϭϭ0or͑II͒Þ0. The first type of fixed point ͑I͒, however, is readily seen each proportional to qЌ , and hence must be proportional to q2 . Therefore all renormalizations of ⌬ must be propor- to be unstable to , which must always be nonzero ͑and, in Ќ fact, ϭ1) initially. Hence, this fixed point is never reached, tional to q2 , and hence negligible, as ͉qជ͉ 0, relative to the Ќ → and we must flow to a fixed point of type II. To see that fixed bare ⌬. Likewise, any graph for the diffusion constants must points of type I are unstable, note that for such a fixed point, be proportional to Fig. 8, which must be proportional to at the only remaining relevant nonlinearity is 2 . But, by itself, ជ least one power of qЌ . Since Dʈ and D involve no powers 2 cannot renormalize any of the diffusion constants DB and of qជЌ , they cannot be renormalized graphically. Di . The reason for this is that any graph ͑e.g., Fig. 8͒ that Thus, when 2ϭ0, ⌬, D , and Dʈ get no graphical renor- renormalizes any diffusion constant must have an external malization. That is, Gʈ , G , and G⌬ in Eqs. ͑5.12͒ and velocity leg emerging from the right. However, using only ͑5.16͒ are, exactly, ϭ0. Thus, the requirement that ⌬, Dʈ , the 2 vertex, which only involves , we can only make and D flow to fixed points (dDʈ, /dl )ϭ0ϭd⌬/dl leads graphs with a leg emerging from the right. Therefore, at a fixed point of type I, GB,Tϭ0, exactly, in Eq. ͑5.11͒. Thus, to find a fixed point for DB,T , we must choose zϭ2. Combin- ing this with the previous exact scaling relations ͑5.74͒ and ͑5.75͒,wefindϭ1 and ϭ1Ϫd/2. But using these values ͑which are nothing but those that we found in the harmonic theory͒, we find that is a relevant perturbation at any fixed point of type I:
FIG. 8. Feynmann graph for diffusion constants. When 2ϭ0, ជ d d this graph is proportional to at least one power of ͉qЌ͉ϵqx , and, so, ϭ 2Ϫ ͑5.78͒ cannot renormalize D or D . dl ͩ 2ͪ ʈ 4848 JOHN TONER AND YUHAI TU PRE 58
for all spatial dimensions dϽ4. Note that ͑5.78͒ is exact at 1 so that the full vជЌ nonlinearity becomes 2 (1ϩ2) any fixed point of type I, since G ϭ0, exactly. 2 ˆ x(vx )x, which is just what we would get if we startedץϫ So fixed points of type I are unstable, and we will always with a ͑primed͒ model with Јϭ0 and Јϭ ϩ . This flow to a fixed point of type II. Using the recursion relation 2 1 1 2 later model, since it has Јϭ0, must have the ‘‘canonical’’ ͑5.15͒ for , and the fact that Gϭ0 always, we immedi- 2 exponents ͑5.80͒–͑5.81͒ and, hence, so must the ( , ) * 1 2 ately obtain that can be Þ0 if and only if a third exact model, which includes all possible dϭ2 models. So all mod- scaling relation is satisfied, namely, els in dϭ2 must have the canonical exponents ͑5.80͒– ͑5.82͒. ϭ1Ϫz. ͑5.79͒ Equivalently, we can derive this result by simply noting 1 The three relations ͑5.74͒, ͑5.75͒, and ͑5.79͒ that hold when that, in dϭ2, the full vЌϪvЌ vertex becomes 2 (1 v2), which is a total x derivative even when ) ץ( ϩ 2ϭ0 can be solved trivially, to find the exact scaling expo- 2 x x 2 Þ0. Furthermore, the dϭ2 model now has the pseudo- nents in all dϽ4 that describe flocks with 2ϭ0: Galilean invariance ͑5.76͒ and ͑5.77͒ when 1ϩ2ϭ . dϩ1 These two properties (vЌϪvЌ vertex total x derivative, and ϭ , ͑5.80͒ 5 pseudo-Galilean invariance at a special point͒ are all that we used to derive the ‘‘canonical’’ exponent ͑5.80͒–͑5.82͒;so 2͑dϩ1͒ those canonical exponents must hold in dϭ2. Setting dϭ2 zϭ , ͑5.81͒ in Eqs. ͑5.80͒–͑5.82͒, we obtain 5 ϭ 3 , ͑5.86͒ and 5 ϭ 6 3Ϫ2d z 5 , ͑5.87͒ ϭ . ͑5.82͒ 5 1 ϭϪ 5 . ͑5.88͒ Note that these match continuously, at the upper critical di- Note, in particular, that Ͻ0. This implies, as discussed ear- mension dϭ4, onto their harmonic values ϭ1, zϭ2, and lier, that the flock exhibits true long-ranged order. ϭ1Ϫd/2ϭϪ1, as they should. Bare Bare Using the exponents ͑5.86͒–͑5.88͒ in the general scaling If 1 Ͼ , then, besides the two cases that we dis- relations, such as Eqs. ͑5.31͒ and ͑5.33͒, we obtain all of the cussed above, there may be a third type of fixed point: ͑III͒ scaling results for correlation functions in dϭ2 quoted in the *ϭ0, 1*Þ0. Introduction. Note also that for this set of exponents zϪ For this type of fixed point to be stable, the exponents Ϫ2ϩ1Ϫdϭ0. Hence, from Eq. ͑5.66͒, we see that the and z have to satisfy noise strength ⌬ is a constant, independent of qជ , which ϩzϽ1. ͑5.83͒ makes sense since ⌬ is unrenormalized graphically. So, in the dϭ2 model, we can calculate all correlation functions The above inequality, together with Eq. ͑5.74͒ and Eq. from their harmonic expressions, except that we replace the ͑5.75͒, give Eqs. ͑5.80͒, ͑5.81͒, and ͑5.82͒ with the ϭ signs diffusion constants DB,T, with functions that diverge as qជ replaced by Ͻ. For simplicity, we will only discuss the cases 0 according to the scaling laws Bare Bare → with 1 Ͻ , where the exponents are given by Eqs. ͑5.80͒, ͑5.81͒, and ͑5.82͒. However, all the qualitative results Ϫ4/5 ͑qʈ /⌳͒ DB,T͑qជ ͒ϭqЌ fB,T 3/5 , ͑5.89͒ will be valid for case III, e.g., the spontaneous symmetry ͩ͑qЌ /⌳͒ ͪ broken phase will be more stable in case III if it is stable in 6 case II because of the inequality. The simplest possible sce- where we have used the exact dϭ2 exponents zϭ 5 and 3 nario is that there is only one stable fixed point, regardless of ϭ 5 in the general scaling law ͑5.65͒. Dʈ, , on the other Bare Bare whether Ͻ1 or not, and that it is type II, and has the hand, are, like ⌬, constants, since zϭ2 ͓see general equa- canonical exponents ͑5.80͒–͑5.82͒. We consider it highly tion ͑5.12͔͒, which also makes sense since Dʈ, are unrenor- probable that this is, in fact, the case. Even if it is not, Eqs. malized graphically. Hence, the only replacement needed to Bare turn the harmonic results into the correct results for the full, ͑5.80͒–͑5.82͒ do hold for some flocks ͑those with Bare nonlinear theory in dϭ2 is Eq. ͑5.89͒. Ͼ1 ). Our derivation of these results ͑5.80͒–͑5.82͒ depended only on the assumption that 2*ϭ0. Note, however, that in D. d>2 dϭ2, any flock is equivalent to a flock with 2ϭ0. This is Now we turn to dϾ2. Here the canonical exponents need because the 1 and 2 vertices become identical in dϭ2, not hold if 2*Þ0. The obvious thing to do, therefore, is to where vជ has only one component, which we will take to be Ќ determine whether a small 2 is a relevant or irrelevant per- x. That is, in dϭ2, turbation at the 2ϭ0 fixed point. We have attempted to do this to leading ͑one-loop͒ order in a 4Ϫ⑀ expansion. This v2͒xˆ , ͑5.84͒ ͑ ץ v ϭ 1 ץ ͑vជ ٌជ ͒vជ ϭ xˆ v 1 Ќ• Ќ Ќ 1 x x x 2 1 x x involves calculating the perturbative corrections G1,2 and G in the recursion relations ͑5.15͒–͑5.13͒ to one-loop or- 2 ជ vជ vជ ϭ xˆ v v ϭ 1 v2 xˆ 5.85 x͑ x ͒ ͑ ͒ der, and to linear order in 2 . Once this is done, we can findץx x͒ x 2 2ץ͑ 2ٌ͑Ќ• Ќ͒ Ќ 2 PRE 58 FLOCKS, HERDS, AND SCHOOLS: A QUANTITATIVE . . . 4849 a fixed point with 2ϭ0, and then calculate the linear We should emphasize that all flocks will be described by renormalization-group eigenvalue of 2 at this fixed point. only one of the three possibilities enumerated above ͑i.e., That is, we will expand the right hand side of the recursion one cannot have different possibilities realized in different relation ͑5.15͒ for 2 to linear order in 2 , obtaining flocks͒. Unfortunately, we have no idea which of the above possibilities is realized for 2ϽdϽ4. d2 ϭ␥22 ͑5.90͒ dl VI. ANISOTROPIC MODEL for 1ϭ1* , ϭ* , 2ϭ2* , and 2 small. If ␥2 is Ͻ0, Not all flocks, of course, are equally likely to move in any then the 2ϭ0 fixed point is ͑at least locally͒ stable, and the direction in the space they occupy. Flocks of birds, for in- canonical exponents ͑5.80͒–͑5.82͒ will hold for all d. Unfor- stance, although they occupy a dϭ3 dimensional volume tunately, an ͑extremely laborious͒ calculation ͑involving 14 ͑the air͒, are far more likely to move horizontally than ver- different Feynmann graphs͒ shows, after many seemingly tically. This is presumably because gravity breaks the rota- miraculous and unexpected cancellations between different tional symmetry between the horizontal plane and vertical graphs, that G , G , and G are exactly zero to one loop directions. 1 2 2 order. This implies that One can imagine a variety of ‘‘microscopic’’ rules, like the Vicsek rule described earlier, that would exhibit such ϩ1Ϫzϭ0͑⑀2͒ ͑5.91͒ anisotropy. For example, one could apply a ‘‘Vicsek’’ rule in three dimensions, selecting thereby a vector nˆ . Instead of and that, to this order at least, 1,2, and 2 can take on any moving along that vector, however, one could instead move value at the fixed point. That is, to this order, there appears to along a vector ‘‘compressed’’ along some ͑z͒ axis be a fixed ‘‘line’’ ͓actually, a fixed four-dimensional sub- space ( , , , )͔, instead of a single fixed point. This, 1 2 2 nជ Јϭsn zˆϩnជ , ͑6.1͒ unfortunately, eliminates all of our predictive power for the z Ќ exponents. For example, keeping Dʈ fixed leads to ˆ ˆ with sϽ1 and nជЌϭnϪnzz. This will tend to promote mo- tion in the x-y plane at the expense of motion in the z direc- zϪ2ϭϪGʈ͑1 ,2 , ,2͒. ͑5.92͒ tion. Alternatively, one could project all velocities into the Our earlier arguments show that Gʈ vanishes if 2 does; x-y plane, apply a Vicsek rule to them ͑while still sampling however, to this order 2* can be anything; hence, so can Gʈ , neighbors in three dimensions͒, and then add to this xy move and so we get no information about z and from this relation a random decorrelated step in the z direction ͓10͔. at all. Likewise the recursion relation for ⌬ leads to For technical reasons that will, we hope, become obvious, we will focus our attention on systems that, whatever their zϪϪ2ϩ1ϪdϭϪG⌬͑2 ,1 , ,2͒ ͑5.93͒ spatial dimension d, have an easy plane of motion; i.e., two components of velocity that are intrinsically favored over the with the right-hand side again vanishing if 2ϭ0, but taking other dϪ2. We will also assume perfect isotropy within this on any value you like if 2 can be anything, as it can, to this plane and within the dϪ2 dimensional ‘‘hard’’ subspace. order. The case of birds flying horizontally corresponds to dϭ3. So what actually happens for 2ϽdϽ4? Unfortunately, A natural extension of our fully isotropic model ͑EOM͒ to from our one-loop calculation, we cannot say, but can only this case is enumerate the possibilities: Possibility I: At higher order, proves to be irrelevant, 2 ͒ vជ ϩ ͑vជ ٌជ ͒vជ ϩ ٌ͑ជ vជ ͒vជ ϩ ٌជ ͉͑vជ͉ ץ 2 and flows to zero at the fixed point. In this case, the canoni- t 1 • 2 • 3 cal exponents ͑5.80͒–͑5.82͒ will hold, for all flocks, in all d ជ ជ ជ 2ជ ជ ជ ជ ជ ϭϪٌP͑͒ϩ␣vϪ͉v͉ vϪ␦␣vHϩDBٌٌ͑•v͒ in the range 2ϽdϽ4. e 2ជ H 2 ជ ជ ជ 2ជ ជ Possibility II: d2 /dl ϭ0 to all orders ͑i.e., exactly͒.In ϩDTٌevϩDT ٌHvϩD2͑v•ٌ͒ vϩf. ͑6.2͒ this case, there is a fixed line ͑or, more generally, D- dimensional subspace with Dу1) with exponents that vary Mass conservation, of course, still applies: as continuous functions of 2 , which can take on any value. Hence, the exponents , z, and will be continuously vari- ជ ជ tϩٌ•͑v͒ϭ0 ͑6.3͒ץ able functions of the parameters in the ordered phase. This behavior is somewhat reminiscent of that of the dϭ2 equi- and the pressure P() will still be given by the same expan- librium XϪY model, although here it is occurring for an sion in ␦ϭϪo entire range of spatial dimension 2ϽdϽ4, and, furthermore, is not associated in any way with the absence of true long- ϱ n ranged orientational order, since such order is actually P͑͒ϭ ͚ n͑␦͒ . ͑6.4͒ present in our model. nϭ1 Possibility III: d2 /dl Ͼ0 at higher order, and the or- dered phase is controlled by a new, *Þ0 fixed point. In In Eq. ͑6.2͒, vជ H denotes the dϪ2 ‘‘hard’’ components of vជ , 2 2 this case, the exponents will again be universal, but presum- i.e., those orthogonal to the dϭ2 easy plane. Likewise, ٌe 2 2 2 2 d 2 2 , xiץ/ ץxi and ͚iϭ3ץ/ ץably, different from the canonical ones ͑5.80͒–͑5.82͒. Unfor- and ٌH denote the operators ͚iϭ1 tunately, we have no idea what they will be. respectively, where iϭ1,2 are the ‘‘easy’’ Cartesian direc- 4850 JOHN TONER AND YUHAI TU PRE 58
2 y, ͑6.10͒ץtions, and iϭ3 d the ‘‘hard’’ ones. The term Ϫ␦␣͉vជ H͉ , ␦vyϭϪDy ␦␣Ͼ0 suppresses→ these components of velocity relative to those in the easy plane. vជ HϭϪDHٌជ H, ͑6.11͒ Equation ͑6.2͒ is not, of course, the most general aniso- tropic model we could write down. For instance, one could where we have defined the diffusion constants have anisotropy in the nonlinear terms: e.g., terms like vជ ٌជ )vជ could have different coefficients than (vជ 1) e• e H Dyϵ , ͑6.12͒ ជ ជ ជ 2␣ ’’,vH . However, because vH winds up being ‘‘massive(ٌ• in the sense of decaying to zero too rapidly ͑i.e., nonhydro- dynamically͒ at long wavelengths and times to nonlinearly 1 DHϵ ͑6.13͒ affect the hydrodynamic ͑long wavelength, long time͒ be- ␦␣ havior of the flock ͑in its low-temperature phase͒, any addi- and we have used the relation ͑6.4͒ for the pressure, and tional terms in Eq. 6.2 distinguishing ជ and ជ will have ͑ ͒ vH ve dropped all but the leading order linear terms in ␦, since no effect on the hydrodynamic behavior in the low- higher powers of ␦ in Eqs. ͑6.10͒ and ͑6.11͒ prove to be ‘‘temperature’’ phase. That is, Eq. ͑6.2͒ already contains irrelevant. enough anisotropy to generate all possible relevant, symme- Using the solutions ͑6.10͒ and ͑6.11͒, and taking, for the try allowed terms in the broken symmetry state. Hence, we reasons just discussed, will keep things simple and not generalize Eq. ͑6.2͒ further. As we did for the isotropic problem, we will now break vជ ͑rជ͒ϭ͓v ϩ␦v ͑rជ,t͔͒yˆ ϩv ͑rជ,t͒xˆ ϩvជ ͑rជ,t͒ ͑6.14͒ the symmetry of this model, i.e., look for solutions to the 0 y x H form we can write a closed system of equations for vx(rជ,t) and ជ vជ ͑rជ,t͒ϭ͗vជ ͘ϩ␦vជ ͑rជ,t͒. ͑6.5͒ ␦(r,t): ϩ ϩ ϭ D 2ϩD 2 , y HٌH͒␦ץx͑vx͒ ͑ yץ y␦ץt␦ v0ץ Now, however, the direction of the mean velocity ͗vˆ ͘ 6.15 ͓which we will choose as before, to be a static, spatially ͑ ͒ uniform solution of the noiseless (ជf ϭ0) version of ͑6.2͔͒ is 2ץ ͒2ϩ͑D␦͑ ץ ͒Ϫ␦͑ ץ v2͒ϭϪ͑ ץ v ϩ ץ␥v ϩ ץ ,not arbitrary, but must lie in the easy (1,2) plane. To see this t x y x 2 x x 1 x 2 x ʈ y let us, without loss of generality, write 2ϩD ٌ2 ͒v ϩf , ͑6.16͒ץ ϩD ˆ ˆ x x H H x x ͗vជ ͘ϭv0yyϩv0zz ͑6.6͒ where we have defined ϵ1ϩ2 , and ␥ϭ1v0 , and ˆ ˆ with v0y and v0z constants, y in the easy plane and z one of dropped irrelevant terms. the dϪ2 ‘‘hard’’ directions. To solve Eq. ͑6.2͒ with ជf ϭ0, Proceeding as we did in the isotropic model, we begin by these must obey linearizing these equations, Fourier transforming them, and determining their mode structure. 2 2 ␣v0yϪ͑v0yϩv0z͒v0yϭ0 ͑6.7͒ The result of the first two steps is the Fourier-transformed equations of motion and ͓Ϫi͑Ϫv0qy͒ϩ⌫͑qជ ͔͒␦͑qជ ,͒ϩiqx0vx͑qជ,͒ϭ0, 2 2 ͑␣Ϫ␦␣͒v0zϪ͑v0yϩv0z͒v0zϭ0. ͑6.8͒ ͑6.17͒
Subtracting v ϫ ͑6.8͒ from v ϫ ͑6.7͒ we obtain 0y 0z ͓Ϫi͑Ϫ␥qy͒ϩ⌫v͑qជ ͔͒vx͑qជ ,͒ϩi1qx␦͑qជ ,͒ϭ f x͑qជ ,͒, ␦␣v0yv0zϭ0, which implies that either v0y or v0z must be ͑6.18͒ zero. It is straightforward to show that the former solution is where we have defined unstable ͑with two linear eigenvalues ␣Ͼ0) to small vជ e fluc- tuations, while the latter is stable ͑with dϪ2 linear eigenval- 2 2 ⌫͑qជ ͒ϵDyqyϩDHqH , ͑6.19͒ ues Ϫ␦␣Ͻ0) to vជH fluctuations, so the solution with ͗vជ ͘ in the easy plane is the stable one. Furthermore, fluctuations in ជ 2 2 2 the ‘‘hard’’ directions are ‘‘massive,’’ in the sense of decay- ⌫v͑q͒ϵDʈqyϩDHqHϩDxqx . ͑6.20͒ ing rapidly to zero even at long wavelengths, and so can be Again as in the isotropic model, we first determine the neglected in the low-temperature phase ͑just like v fluctua- ʈ ជ tions in the isotropic case͒. Likewise, if we take eigenfrequencies (q) of these equations, finding
ˆ Ϯ͑qជ ͒ϭcϮ͑qជ ,qជ ͒qϪi⑀Ϯ͑qជ ͒, ͑6.21͒ ͗vជ ͘ϭv0y ͑6.9͒ where the sound speeds fluctuations in ␦vyϭvyϪv0 , will also be massive ͑with lin- ear eigenvalueϪ2␣). Eliminating the massive fields ␦vy and 1 cϮ͑qជ ,qជ ͒ϭ 2 ͑␥ϩv0͒cos qជ Ϯc2͑qជ ,qជ ͒ ͑6.22͒ vជ H in favor of the pressure, as we did for ␦vʈ in the isotropic case, gives with PRE 58 FLOCKS, HERDS, AND SCHOOLS: A QUANTITATIVE . . . 4851
1 2 2 2 2 ជ ជ ជ c2͑qជ ,qជ ͒ϵͱ 4 ͑␥Ϫv0͒ cos qជ ϩ10 sin qជ cos qជ , Cv͑q,͒ϭ͗␦͑q,͒vx͑Ϫq,Ϫ͒͘ ͑6.23͒ ⌬1qx͓Ϫv0qyϪi⌫͑qជ ͔͒ ϭ , ͑6.30͒ ជ 2 where qជ is the polar angle between q and the y axis, and qជ ͉Den͑qជ,͉͒ is the azimuthal angle, measured relative to the x axis, i.e., ជ the angle between the projection of q orthogonal to y, and and the x axis. A polar plot of this sound speed versus qជ for qជ ϭ0 ͓i.e., 2 2 qជ in the ‘‘easy’’ ͑i.e., x-y) plane͔ looks exactly like that for ⌬0qx C ͑qជ,͒ϭ , ͑6.31͒ 2 the isotropic model ͑Fig. 2͒. Indeed, any slice with fixed qជ ͉Den͑qជ,͉͒ looks qualitatively like that figure, although, as qជ /2 ជ ជ → ͑i.e., as qЌ , the projection of q orthogonal to y, approaches where we have defined orthogonality to the x axis͒, the sound velocity profile be- comes two circles with their centers on the y axis and both circles passing through the origin. Den͑qជ,͒ϭ͓Ϫcϩ͑qជ ,qជ͒q͔͓ϪcϪ͑qជ ,qជ͒q͔ The dampings ⑀ (qជ ) in Eq. ͑6.21͒ are 0(q2), and given Ϯ ϩi͕͓⌫ ͑qជ͒ϩ⌫ ͑qជ͔͒Ϫq ͓v ⌫ ͑qជ͒ by v y 0 v
ϩ␥⌫͑qជ͔͖͒, ͑6.32͒ cϮ͑qជ ,qជ͒ ⑀Ϯ͑qជ ͒ϭϮ ͓⌫v͑qជ͒ϩ⌫͑qជ͔͒ 2c2͑qជ ,qជ͒ which, of course, implies
v0cos͑qជ ͒ ␥ ϯ ⌫v͑qជ ͒ϩ ⌫ ͑qជ ͒ . ͑6.24͒ ជ 2 2 2 ͩ v ͪ ͉Den͑q,͉͒ ϭ͓Ϫcϩ͑qជ ,qជ͒q͔ ͓ϪcϪ͑qជ ,qជ͒q͔ 2c2͑qជ ,qជ ͒ 0 ជ ជ ជ Note that, unlike the isotropic problem in dϾ2, here there ϩ͕͓⌫͑q͒ϩ⌫v͑q͔͒Ϫqy͓v0⌫v͑q͒ are no transverse modes in any d: we always have just two 2 ϩ␥⌫͑qជ͔͖͒ . ͑6.33͒ longitudinal Goldstone modes associated with ␦ and vx . We can now again parallel our treatment of the isotropic model and calculate the correlation functions and propaga- These horrific expressions actually look quite simple when tors. The calculation is so similar that we will not repeat the plotted as a function of at fixed qជ ; indeed, such a plot of details, but merely quote the results: Cvv looks precisely like the solid line in Fig. 6: two asym- metrical peaks, centered at ϭcϮ(qជ ,qជ )q, with widths i͑Ϫv q ͒Ϫ⌫ ͑qជ͒ 2 0 y ⑀Ϯ(qជ )ϰq . Gvv͑qជ,͒ϭ , ͑6.25͒ Den͑qជ,͒ Note that, at this linear order, everything scales as it did in the isotropic problem: peak positions ϰq, widths ϰq2, and heights ϰ1/q4. i1qx Gv͑qជ ,͒ϭ , ͑6.26͒ Continuing to blindly follow the path we trod for the iso- Den͑qជ,͒ tropic problem, we can calculate the equal-time vxϪvx cor- relation function: i0qx Gv͑qជ ,͒ϭ , ͑6.27͒ Den͑qជ,͒ Cvv͑qជ͒ϵ͗vx͑qជ,t͒vx͑Ϫqជ,t͒͘ ˆ i͑Ϫ␥qy͒Ϫ⌫v͑qជ͒ ϱ d ⌬ ͑q͒ G ͑qជ,͒ϭ , ͑6.28͒ ϭ C ͑qជ,͒ϭ , ͑6.34͒ ͵ 2 vv 2 Den͑qជ,͒ Ϫϱ ⌫L͑qជ͒
⌬͓͑Ϫv q ͒2ϩ⌫2͑qជ͔͒ 0 y where (qˆ ) depends only on the direction qˆ of qជ , and is Cvv͑qជ,͒ϭ , ͑6.29͒ ͉Den͑qជ,͉͒2 given by
2 1 ͓cϩ͑qជ ,qជ ͒qϪv0qy͔ ͑qˆ ͒ϭ c ͑ ជ , ជ ͒q ͫ c ͑ ជ , ជ ͒qϪv q ϩ͓c ͑ ជ , ជ ͒qϪ v q ͔⌫ /⌫ 2 q q ϩ q q 0 y ϩ q q 1 0 y v
2 ͓cϪ͑qជ ,qជ ͒qϪv0qy͔ ϩ . ͑6.35͒ c ͑ ជ , ជ ͒qϪv q ϩ͓c ͑ ជ , ͒qϪ v q ͔⌫ /⌫ ͬ ϩ q q 0 y Ϫ q vecq 1 0 y v 4852 JOHN TONER AND YUHAI TU PRE 58
These fluctuations again diverge like 1/q2 as ͉qជ͉ 0, just as ϭ1Ϫz, ͑6.41͒ in the isotropic problem. → This completes our abbreviated discussion of the linear- whose solution is easily found in all dϽ4: ized theory of the anisotropic model. The most succinct sum- 3 mary of this linearized theory is that everything scales just as ϭ , ͑6.42͒ it did in the isotropic problem. This implies that the nonlin- 7Ϫd earities ͓i.e., the and 2 terms in the equations of motion ͑6.15͔͒ become relevant in and below the same upper critical 6 zϭ , ͑6.43͒ dimension ducϭ4 as in the isotropic problem. For dϽ4, 7Ϫd therefore, these nonlinearities will change the long-distance behavior of the anisotropic model. We will now treat these 1Ϫd ϭ . ͑6.44͒ nonlinearities using renormalization-group arguments similar 7Ϫd to those we used for the isotropic model in dϭ2. Now, how- ever, they will work for all d between 2 and 4. Note that these reduce to our isotropic results in dϭ2, as Notice that all of the nonlinearities in Eq. ͑6.15͒ are total they should, since the two models are identical there. They x derivatives, just as in the dϭ2 case for the isotropic prob- also reduce to the harmonic values zϭ2, ϭ1, and ϭϪ1, lem. Now, however, this is true in all spatial dimensions, not in dϭ4, as they should, since 4 is the upper critical dimen- just in dϭ2. ͑This, of course, is the reason we chose to sion. consider precisely two ‘‘soft’’ components.͒ Thus, we will In the physically interesting case of dϭ3, we obtain: now be able to derive exact exponents in this model for all 3 spatial dimensions. We will not go through the arguments in ϭ 4 , ͑6.45͒ detail, as they are virtually identical to those in the dϭ2 case 3 for the isotropic model, but will simply quote the conclu- zϭ 2 , ͑6.46͒ sions: 1 ͑1͒ There are no graphical corrections to any of the diffu- ϭϪ 2 . ͑6.47͒ sion constants in Eq. ͑6.15͒ except Dx . ͑2͒ The stable fixed point that controls the ordered phase As in the isotropic case, we can use scaling arguments here to show that the effect of the nonlinearities can be fully in- must have *Þ0 at least for (0)Ͻ(0), which is a finite fraction of all flocks, and corporated by simply replacing Dx everywhere it appears in the linearized expressions by the divergent, wave-vector- ͑3͒⌬and are not graphically renormalized. Point one suggests that, in constructing our dynamical dependent scaling form: renormalization group for Eq. ͑6.15͒, we should scale the x direction differently from both the y direction and the dϪ2 zϪ2 ͑qy /⌳͒ ͑qH /⌳͒ Dx͑qជ ͒ϭqx f , . ͑6.48͒ hard directions. Furthermore, since both the y direction and ͫ ͑q /⌳͒ ͑q /⌳͒ͬ x x the dϪ2 hard directions are alike in having their associated diffusion constants unrenormalized, we should scale these Doing this leads to all of the scaling laws for this anisotropic directions the same way. Therefore, in our renormalization problem quoted in the Introduction. group, we will rescale as follows: x bx,(y,xជ) → H b(y,xជ), t bzt. With these rescalings, the recursion re- VII. TESTING THE THEORY IN SIMULATIONS → H → lations for Di , iÞx, , ⌬, and become AND EXPERIMENTS In this section, we discuss how our theory can be tested in dD i simulations and direct observations of real flocks. The ϭ͑zϪ2͒Di ͑iÞx͒, ͑6.36͒ dl ‘‘real’’ flocks may include, e.g., mechanical, self-propelled ‘‘go carts’’ packed so densely that they align with their d⌬ neighbors ͓11͔, as well as aggregates of genuinely living ϭ͓zϪ2ϩ͑1Ϫd͒Ϫ1͔⌬, ͑6.37͒ dl organisms. We begin with a few suggestions about the best boundary conditions and parameter values for simulations or experi- d ϭ͑ϩzϪ1͒ . ͑6.38͒ ments, and then describe how the correlation functions and dl scaling exponents , z, and predicted by our theory can be measured. The most useful boundary conditions are ‘‘torus’’ All three relations are exact, since none of these parameters conditions; that is, reflecting walls in dϪ1 directions, and experiences any graphical renormalization. As in the isotro- periodic boundary conditions in the remaining direction, call pic case, we want all of these parameters to flow to fixed it y ͑see Fig. 9͒. The advantage of these conditions is that one points; this leads to three exact scaling relations between the knows a priori that, if the flock does spontaneously order, its three exponents , z, and : mean velocity will necessarily be in the periodic ͑y͒ direc- tion. zϭ2, ͑6.39͒ It might be objected that imposing such anisotropic boundary conditions breaks the rotation invariance our zϪ2ϩ͑1Ϫd͒ϭ1, ͑6.40͒ model requires, but this is not, in fact, the case. A ‘‘bird’’ PRE 58 FLOCKS, HERDS, AND SCHOOLS: A QUANTITATIVE . . . 4853
FIG. 10. More practical ‘‘track’’ geometry for experiments on real flocks. Data should only be taken from the cross-hatched region FIG. 9. Illustration of the optimal boundary conditions for simu- centered on the middle of the ‘‘straightaway.’’ lations and experiments to test our predictions. The top and bottom walls are reflecting, while periodic boundary conditions apply at the left and right walls ͑i.e., a bird that flies out to the right instantly mean motion through boundary conditions may also be reappears at the same height on the left͒. The mean direction of dreamed up by experimentalists more clever than we are. spontaneous flock motion, if any occurs, is clearly forced to be We strongly caution anyone attempting to test our results, horizontal by these boundary conditions. In spatial dimensions d however, that it is only through boundary conditions that one Ͼ2, one should choose reflecting boundary conditions in dϪ1 di- may prepick the direction of mean motion. Any approach that rections, and periodic in the remaining direction, thereby forcing prepicks this direction in the bulk of the flock, such as giving ͗vជ ͘ to point along that periodic direction. each bird a compass, letting them be blown by a wind, or run downhill, or follow a chemical scent, etc., will lead to a deep inside the box moves with no special direction picked model outside the universality class of our isotropic model, out a priori;itcanonly find out about the breaking of rota- since the starting model does not have any rotation invari- tion invariance on the boundary if the bulk of the flock spon- ance to be spontaneously broken ͑unless the anisotropy taneously develops long-range order. This is precisely analo- leaves an ‘‘easy plane’’ in which all directions are equiva- gous to the way one speaks of a ferromagnet as lent, in which case our anisotropic model of Sec. VI applies͒. spontaneously breaking a continuous symmetry even if it Indeed, such flocks of ‘‘birds with compasses’’ will be less orders in the presence of ordered boundary conditions. So, interesting than the models we have studied here, since the by imposing these boundary conditions, we know the direc- ‘‘compass’’ will introduce a ‘‘mass’’ that makes any fluctua- tion of the flock motion ͑the y direction in the simulation͒, tion away from the prepicked direction of flock motion decay and, therefore, have oriented the simulation axes with the rapidly ͑i.e., nonhydrodynamically͒ with time. In such a axes used in our theoretical discussion; i.e., our ʈ axis equals the simulation’s periodic direction. model, it is easy to show that the nonlinearities are irrel- Alternative boundary conditions add the further complica- evant, and there are no interesting fluctuations left at long tion of having to first determine the direction of mean flock distances and times. motion before calculating correlation functions. This compli- And now a few words about parameter choices. For defi- cation is even worse for a finite flock ͑as any simulation must niteness, we will discuss in what follows the Vicsek model, treat͒, since the mean direction of motion will wander, ex- whose parameters are v0ϭS/R0 , where S is the distance the ecuting essentially a random walk that will explore the full birds travel on each time step and R0 is the radius of the circle in a time of order T ϭ2ͱN/⌬. Our results, which circle of neighbors, the mean number density 0 in units of flock d assume a constant direction of flock motion, will only apply 1/R0 where d is the dimension of the system, and the noise for time scales tӶTFlock . Even drifts of the mean flock di- strength ⌬, which is the mean squared angular error. Since rection through angles Ӷ2 can cause problems, however, the interesting nonlinear effects in our model come from 2 since most of the interesting scaling behavior is concentrated terms proportional to v0 , those effects will become impor- in a narrow window of angles qʈϳqЌӷqЌ ; i.e., near the tant at shorter length scales in a faster moving flock. That is, direction of mean flock motion. So this drift greatly compli- in, e.g., the Vicsek model, should we choose the dimension- cates the experimental analysis, and is best avoided by using less velocity as large as possible, consistent with the flock the toroidal boundary conditions just described. ordering. However, if we take v0 too big, i.e., v0ӷ1, then, Of course, it is considerably harder to produce these on each time step, each bird is likely to have a completely boundary conditions in a real experiment. Ants walking different set of neighbors. It is difficult to see how order can around a cylinder may come close, although gravity will al- develop in such a model. So, to take v0 as big as possible ways break rotation invariance on a real cylinder. Perhaps without violating v0ӷ1, we should choose v0ϳ1. The simu- the experiment could be done on the space shuttle, or with a lations of Vicsek et al. ͓1͔ took v0Ӷ1, and, hence, probably rapidly spinning cylinder producing artificial gravity that never explored ͑in their finite flocks͒ the long length scale swamps real gravity, or by using neutrally buoyant organ- regime in which our nonlinear effects become important. isms in a fluid. Alternatively, one could use a ‘‘track’’ such Now to the mean density 0 , which is, of course, just as that shown in Fig. 10, and take data only from the cross- determined by the total number of birds N and the volume V hatched region, chosen to be in the middle of the straight of the box via 0ϭN/V. We clearly want this to be large section of the track, far from the curves. enough that each bird usually finds some neighbors in its d Other, more ingenious ways to prepick the direction of neighbor sphere: This means we want 0R0уO(1). How- 4854 JOHN TONER AND YUHAI TU PRE 58 ever, if we make 0 too large, each bird has so many neigh- bors that a simulation is considerably slowed down, since the ‘‘direction picking’’ step of the Vicsek algorithm takes a time proportional to the number of neighbors ͑because we have got to average their directions͒. Thus, for simulations, one wishes to choose 0 as small as possible, consistent, again, with getting good order. Finally, we consider the noise ⌬. Here again, to see our fluctuation effects, we want ⌬ as big as possible. However, if ⌬ is too big, the flock will not order. Furthermore, even if ⌬ is small enough that the flock does order, we want also to be sure that we are well below the critical value ⌬c of ⌬ at which the flock disorders. Otherwise, for distances smaller than the correlation length associated with the order- disorder transition, the scaling properties of the flock will be controlled by the fixed point that controls the order-disorder transition, not the low-temperature fixed point we have stud- FIG. 11. Illustration of the experiment to measure the mean- squared lateral wandering w2(t). One labels all of the birds some ied here. central stripe ͑of width ӶL, the channel width͒, and then measures If this transition is continuous, as it appears to be in Vic- the evolution of their mean displacements xជ (t) perpendicular to sek’s simulations ͓1͔, this correlation length diverges as ⌬ Ќ the mean direction of motion ͑which mean direction is horizontal in ⌬Ϫ . Thus, to observe scaling behavior we predict over as c this figure͒. →many decades of length scale as possible, we want to choose ⌬ substantially less than ⌬ , but as big as possible consis- c t tent with this ͑to maximize fluctuation effects͒. Choosing ⌬ Ќ Ќ Ќ xជ i ͑t͒ϭxជ i ͑0͒ϩ vជ i ͑t͒dt, ͑7.2͒ to be a little below the point at which the mean velocity ͗vˆ ͘ ͵0 starts to ‘‘saturate’’ seems like a fairly good compromise between these two competing effects. Similar considerations Ќ where vជ i (t) is the Ќ velocity of the ith bird at time t, the apply for choosing the optimal 0 , and v0 , which we want mean width is given by to be as small or big, respectively, as they can be without substantially suppressing long-ranged order. The best t t choices will probably lead to all three parameters , v , 2 ជЌ ជЌ 0 0 w ͑t͒ϭ dtЈ dtЉ͗vi ͑tЈ͒•vi ͑tЉ͒͘. ͑7.3͒ and ⌬ being, in suitably dimensionless units, O(1). ͵0 ͵0 Having chosen the appropriate parameter values and boundary conditions, what should an experimentalist or Now we need to relate the velocity of the ith bird to the simulator measure to test our theory? We have already dis- position and time-dependent velocity field vជЌ(rជ,t). This is cussed a number of such measurements in the Introduction; easily done: namely, the spatially Fourier-transformed equal-time and spatiotemporally Fourier-transformed unequal time density- vជЌ͑t͒ϭvជ ͓rជ ͑t͒,t͔ ͑7.4͒ density correlation functions C(qជ ) and C(qជ ,), respec- i Ќ i tively. Our predictions for these are given in Eqs. ͑5.38͒ and ͑5.40͒. where rជi(t) is the position of the ith bird at time t. This is One additional correlation function that can be measured given by quite easily is the mean-squared lateral displacement of a bird: ¯ ˆ ʈ ˆ Ќ rជi͑t͒ϭrជi͑0͒ϩvtxʈϩ␦xi͑t͒xʈϩ␦ជxi ͑t͒, ͑7.5͒ 2 ជЌ ជЌ 2 w ͑t͒ϵ͉͗xi ͑t͒Ϫxi ͑0͉͒ ͘ ͑7.1͒ where perpendicular to the mean direction of motion of the flock. 1 ¯vϵ vជ ͑7.6͒ This can easily be measured as a function of time in a simu- N ͚ i lation or experiment simply by labeling a set of n birds in a ͯ i ͯ ‘‘strip’’ near the center of the channel with its long axis running parallel to the mean direction of bird motion ͑see is the velocity averaged over all birds, which, as discussed Fig. 11͒ and then following their subsequent motion. It is earlier, is not to be confused with the space averaged v0 best to center the strip in the channel so as to postpone the ϵ͉͐vជ (rជ,t)ddr͉ that appears in the expression for the sound birds reaching the reflecting walls as long as possible. Once speeds cϮ(qជ ). This distinction proves to be crucial here, as 2 ʈ Ќ they do reach the walls, of course w (t ϱ) saturates at we shall see in a moment. In Eq. ͑7.5͒, ␦xi (t) and ␦xi (t) 2 → ϳLЌ , LЌ being the width of the channel. We will deal in the reflect the motion of the ith individual bird relative to the following discussion with times much smaller than that re- mean motion of the flock ͑at speed ¯v). quired for a bird at the center of the channel to wander out to Using Eq. ͑7.5͒, we see that the desired single bird auto- Ќ its edge. Since the Ќ position xជ i of each bird obeys correlation function in Eq. ͑7.3͒ is PRE 58 FLOCKS, HERDS, AND SCHOOLS: A QUANTITATIVE . . . 4855
Ќ Ќ ¯ ˆ Now, we need to distinguish two cases: ͗vជ ͑tЈ͒ vជ ͑tЉ͒͘ϭ͗vជЌ͕rជi͑0͒ϩ͓vtЈϩ␦x ͑tЈ͔͒x i • i ʈ ʈ Case ͑1͒:2/ϾϪ1. In this case, which holds in dϭ2, ជ ជ ជ ¯ ˆ where ϭϪ 1 and ϭ 3 , the double integral over tЈ and tЉ is ϩ␦xЌ͑tЈ͒,tЈ͖•vЌ͓ri͑0͒ϩvtЉϩ␦xʈ͑tЉ͒xʈ 5 5 dominated, for tӷt0 , the microscopic time scale, by tЈ, tЉ, ϩជ␦xЌ͑tЉ͒,tЉ͔͘ and ͉tЈϪtЉ͉ of order tӷt0 . Hence, our calculation of Cc which used the hydrodynamic ͑i.e., long time͒ limiting forms ¯ ϭCc͓xជЌ͑tЈ͒ϪxជЌ͑tЉ͒,v͉tЈϪtЉ͉ of the correlation functions, is correct, and Eq. ͑7.13͒ holds. Changing variables to TЈϵtЈ/t and TЉϵtЉ/t, we see that ϩ␦xʈ͑tЈ͒Ϫ␦xʈ͑tЉ͒,tЈϪtЉ͔, ͑7.7͒
ជ 2 where Cc(R,t) is the real-space velocity field autocorrelation w2͑t͒ϰt2͑1ϩ/͒ϭt4/3, ϾϪ1, ͑7.14͒ function defined in the Introduction. We assume ͑and will verify a posteriori͒ that both ␦xʈ and ␦ជ xЌ are small enough compared to the average motion the last equality holding in dϭ2. Note that this behavior is ¯ ˆ ‘‘hyperdiffusive’’: the mean-squared displacement w2(t) vtxʈ that their effect on the velocity-velocity autocorrelation in Eq. ͑7.7͒ is negligible. For now neglecting them, we see grows faster than it would in a simple random walk; i.e., that we are left with the task of evaluating C (Rជ ϭ0,R faster than linearly with time t. v Ќ ʈ Case ͑2͒:2/ϽϪ1. In this case, which certainly holds ϭ¯ vt,t). for dϾ4 ͑where ϭ1Ϫd/2ϽϪ1 and ϭ1), the integral Expressing C in terms of its Fourier transform then gives c over tЉ converges as ͉tЈϪtЉ͉ ϱ. Hence, that integral is, in → fact, dominated by ͉tЈϪtЉ͉ϭO(t0), the microscopic time, ¯ ជ ϭ ϭ¯ ϭ dϪ1 i͑Ϫvqʈt͒ ជ where our hydrodynamic result Eq. ͑7.12͒ is not valid. Pre- Cc͑RЌ 0,Rʈ vt,t͒ ͵ d qЌdqʈde Cii͑q,͒. sumably, the correct ͉tЈϪtЉ͉ 0 limit of the single bird ve- ͑7.8͒ locity autocorrelation ͑7.3͒ is→finite; and, hence, so the inte- gral over tЉ in Eq. ͑7.13͒ approaches a finite limit as t ϱ. Using the fact that Cii(qជ,) is peaked at ϭcϮ(qជ )q with → z Hence, we get widths that scale like qЌ f ͓qʈl 0 /(qЌl 0) ͔, the dominating peak is at ϭϪ for v0(0)Ͼ␥(0) or at ϭϩ for v0(0) Ϫ␦ t 2 Ͻ␥(0), with heights that scale as qЌ g͓qʈl 0 /(qЌl 0) ͔ w2͑t͒ϰ dtЈϫ͑finite constant͒ϰt, ϽϪ1. with ␦ϭ2ϩzϩϩdϪ1 ͓see Eq. ͑5.56͔͒. Assuming that ͵0 v0(0)Ͼ␥(0), it is straightforward to show that, upon inte- ͑7.15͒ grating Eq. ͑7.8͒ over , we obtain We now need only verify our a posteriori assumptions that
¯ qʈl 0 zϪ␦ ␦x and ␦ជ x were negligible in the velocity-velocity autocor- C ϭ ddϪ1q dq ei[cϪ͑qជ͒qϪvqʈ]tf q . ʈ Ќ c ͵ Ќ ʈ Ϫ Ќ relation Eq. ͑7.7͒. ͩ͑qЌl ͒ ͪ 0 ͑7.9͒ First consider xជЌ ; we have just shown that the root-mean- 1ϩ/ squared ͉xជЌ(tЈ)ϪxជЌ(tЉ)͉ϰ͉tЈϪtЉ͉ . From our scaling This integral is dominated, as t ϱ,byqʈϳ(qЌl0)/l0 → expression ͑6.43͒, we see that Cc(RЌ ,Rʈ ,t)ϷCc(RЌ ӷqЌ; hence, qជ 0, and we get → ϭ0,Rʈ ,t)ifRЌӶRʈ. In Eq. ͑7.7͒, we are interested in RЌ t Ϫt 1ϩ/ and R t Ϫt ; hence, the condition R q ϰ͉ Ј Љ͉ ʈϰ͉ Ј Љ͉ Ќ dϪ1 i͑v Ϫ¯v͒q t ʈl 0 zϪ␦ C ϭ d q dq e 0 ʈ f q . ӶRʈ will be satisfied as ͉tЈϪtЉ͉ ϱ provided ϩϽ1. c ͵ Ќ ʈ Ϫ Ќ → ͩ͑qЌl ͒ ͪ 0 Since р1 and Ͻ0 for all dу2, this condition is satisfied ͑7.10͒ for all dу2. For ␦xʈ we need only show that ͉␦xʈ(tЈ) Ϫ␦x (tЉ)͉Ӷ͉tЈϪtЉ͉ as ͉tЈϪtЉ͉ ϱ. This is easily shown by We can scale the time dependence out of this integral with ʈ → ជ the change of variables using the fact, alluded to earlier, that ␦vʈ(r,t), the fluctua- tion of the velocity along the mean direction of motion, has only short-ranged temporal correlations. Using this fact, it is Qʈ Qជ Ќ q ϵ and qជ ϵ , ͑7.11͒ straightforward to show that ␦x (t) just executes a simple ʈ Ќ 1/ ʈ t t͑ ͒ random walk; that is, which give 2 ͱ͉␦xʈ͑tЈ͒Ϫ␦xʈ͑tЉ͉͒ ϰͱ͉tЈϪtЉ͉Ӷ͉tЈϪtЉ͉ ͑7.16͒ 2/ Ccϰt . ͑7.12͒ and hence these fluctuations are negligible as well. Using this in Eq. ͑7.7͒ for the single bird velocity autocorre- Unfortunately, the analogous calculation for the aniso- lation function, and then using that autocorrelation function tropic model shows that this random ‘‘transverse walk’’ is in the expression ͑7.3͒ for the mean squared random walk much less interesting: the mean-squared transverse displace- distance gives ment in the x direction ͑the direction in the ‘‘easy place’’ of the anisotropic model orthogonal to the mean direction of t t w2͑t͒ϰ dtЈ dtЉ͉tЈϪtЉ͉2/. ͑7.13͒ motion, y) is given by an expression very similar to Eq. ͵0 ͵0 ͑7.16͒ 4856 JOHN TONER AND YUHAI TU PRE 58
t t ational time-dependent Ginsburg Landau ͑TDGL͒ model for 2 2 ͉͗xi͑t͒Ϫxi͑0͉͒ ͘ϵw ͑t͒ϭ dtЈ dtЉ͗vix͑tЈ͒vix͑tЉ͒͘ a spin system with the number of components n of the spin ͵0 ͵0 ͑7.17͒ equal to the dimension d of the space those spins live in. We have convinced ourselves by power counting that at and a calculation so closely analogous to that just given for the transition, for dϽ4, the vertices are a relevant pertur- the isotropic model that we shall not bother to repeat it for bation to the Gaussian critical point. Whether they constitute this case shows that a relevant perturbation to the 4Ϫ⑀ TDGL fixed point, thereby changing its critical properties, can only be answered v ͑t ͒v ͑t ͒ ϰ t Ϫt 3ϪdϪ2/z. ͑7.18͒ ͗ ix Ј ix Љ ͘ ͉ Ј Љ͉ by a full-blown dynamical renormalization group analysis. As in our analysis of the isotropic case, here, too, the ques- Obviously, a similar analysis could also be done for the an- tion of whether ‘‘simple random walk’’ behavior ͓w2(t) isotropic model. ϰt͔ or ‘‘hyperdiffusive’’ behavior ͓w2(t)ϰt␥,␥Ͼ1͔ occurs ͑2͒ The shape and cohesion of an open flock, and its fluc- hinges entirely on whether the exponent in Eq. ͑7.18͒ is tuations. We have thus far focused on flocks in closed or greater or less than Ϫ1, with hyperdiffusive behavior occur- periodic boundary conditions. Real flocks are usually sur- ring in the former case ͑exponent ϾϪ1) and simple random rounded by open space. How do they stay together under walk behavior in the latter ͑exponent ϽϪ1). Using our ex- these circumstances? What shape does the flock take? How act result ͑6.43͒ for z in the anisotropic model for 2рdр4, does this shape fluctuate, and is it stable? This issue is some- we see that hyperdiffusive behavior will occur if what similar to the problems of the shapes of equilibrium and growing crystals ͑e.g., faceting, dendritic growth͒.In 2 2Ϫ2d those problems, it was important to first understand bulk pro- 3ϪdϪ ϭ ϾϪ1, ͑7.19͒ z 3 cesses ͑e.g., thermal diffusion in the case of dendritic growth͒ before one could address surface questions ͑e.g., which is satisfied only for dϽ5/2. Unfortunately, this condi- dendritic growth͒. The nontrivial aspects of the bulk pro- tion is not satisfied for either dϭ3ordϭ4. In dϭ2, the cesses in flocks ͑e.g., anomalous diffusion͒ will presumably anisotropic model is the same as the isotropic model, while radically alter the shapes and their fluctuations. for dϾ4, zϭ2 and 3ϪdϪ2/zϽϪ1. So in no case in which ͑3͒ A somewhat related question is: What happens if birds the anisotropic model is different from the isotropic one is move at different speeds? By ‘‘move at different speeds,’’ hyperdiffusive behavior observable; rather, we expect we do not mean simply that at any instant, different birds 2 w (t)ϰt for all those cases. This negative prediction could will be moving at different speeds ͓a possibility already in- be checked experimentally, although its confirmation, while cluded in our ‘‘soft spin’’ dynamical model equation ͑2.6͔͒. a nontrivial check of our theory, would clearly be less excit- Rather, we mean a model in which some birds have a differ- 2 ing than verification of our hyperdiffusive prediction w (t) ent probability distribution of speeds than others. ͑In our ϰt4/3 for the isotropic dϭ2 model. model, this distribution of the speed of any given bird is the Some of the numerical tests discussed in this section have same over a sufficiently long time, and controlled by the been carried out recently, and good agreement with our pre- values of the parameters ␣ and , with large ␣ and  lead- diction has been reached ͓12͔. ing to a distribution sharply peaked around a mean speed v0ϭͱ␣/, while small ␣ and  lead to a broader distribu- VIII. FUTURE DIRECTIONS tion͒. More generally, one could imagine two ͑or many͒ dif- ferent species of birds, ͑labeled by k) all flying together, k In this paper, we have only scratched the surface of a very each with different mean speeds v0 . What would the bulk deep and rich new subject. We have deliberately focused on dynamics of such a flock be? Would there be large scale the most limited possible question: what are the properties of spatial segregation, with fast birds moving to the front of the a flock far from its boundaries, and deep within its ordered flock, and slow birds moving to the back? If so, how would state? Every move away from these restricting simplifica- such segregation affect the shape of the flock? Would it elon- tions opens up new questions. To name a few that we hope to gate along the mean direction of motion? Would this elon- address in the coming millennium: gation eventually split the flock into fast and slow moving ͑1͒ The transition from the ordered ͑moving͒ to disor- flocks? dered ͑stationary, on average͒ phase of the flock. This can be ͑4͒ At the other extreme, one could consider flocks in studied by analyzing the ͑unstable͒ fixed point at which the confined geometries; e.g., inside a circular reflecting wall in renormalized ␣ of our original model ͑2.6͒ is zero. The dy- dϭ2. In such a case, the time averaged velocity of the flock namical RG analysis of this point would be technically simi- ͗vˆ (rជ,t)͘t could not be spatially homogeneous but would lar to the one we have presented here for the low-temperature have to circulate around the center of the circle; i.e., phase, with a few crucial differences: ͑a͒ All components of ˆ ͗vˆ (rជ,t)͘tϭ f (r). The spatially inhomogeneous pattern of vជ , not just the Ќ components, become massless at the tran- velocity and density that resulted could be predicted by our sition. ͑b͒ The fixed point will be isotropic, since no special continuum equations. This problem is potentially related to directions are picked out by ͗vជ ͘, since ͗vជ ͘ still ϭ0 at the the previous one, since one way a flock containing, say, transition. ͑c͒ The ͉vជ͉2vជ term becomes another relevant some very fast birds and other very slow birds, could stay vertex. We know, by power counting, that at the transition, together would be for the fast birds to fly in circles inside the this vertex becomes relevant in dϭ4. Indeed, if we ignore essentially stationary volume of space filled by the slow the vertices, our model simply reduces to a purely relax- birds. It would be very interesting to make the connection PRE 58 FLOCKS, HERDS, AND SCHOOLS: A QUANTITATIVE . . . 4857 between our continuum theory and the recently observed cir- direct transitions between them and the stationary flock cular motion of Dictyostelium cells in a confined geometry phase, and between each other, would also be of great inter- ͓13͔. est. ͑5͒ One could relax the constraint on conservation of bird We should point out here that these models differ consid- number, by allowing birds to be born, and die, ‘‘on the erably from recently considered models of moving flux lat- wing.’’ Numerical studies of such models, which may be tices ͓15͔ and transversely driven charge density waves appropriate to bacteria colonies, where reproduction and ͓15,16͔ in that here the direction of motion of the lattice is death are rapid, as well as the migration of, e.g., huge herds not picked out by an external driving force, but, rather, rep- of caribou over thousands of miles and many months, have resents a spontaneously broken continuous symmetry. already been undertaken ͓14͔; it should be straightforward to ͑7͒ Finally, we would like to study the problem of the modify our equations by adding a source term to the bird growth of order in flocks. This is a phenomenon we have all number conservation equation. seen every time we walk onto a field full of geese: eventu- ͑6͒ It is possible that phase transitions other than that from ally, our approach startles the geese, and they take off en the moving to the nonmoving state occur in flocks. For ex- masse. Initially, they fly in random directions, but quickly ample, in some preliminary simulations of microscopic mod- the flock orders, and flies away coherently. The dynamics of els in which birds try to avoid getting too close to their this process is clearly in many ways similar to, e.g., the neighbors, rather than merely following them, we have ob- growth of ferromagnetic order after a rapid quench from an served ͑literally by eye͒ what appears to be a ‘‘flying crys- initial high temperature TiϾTc , the Curie temperature, to a tal’’ phase of flocks: the birds appear to lock themselves onto final temperature T f ϽTc , a problem that has long been stud- the sites of a crystalline lattice, which then appears to move ied ͓17͔ and proven to be very rich and intriguing. In flocks, coherently. It would be very interesting to test numerically where, as we have seen, even the dynamics of the completely whether this optical appearance reflects true long-range ordered state is very nontrivial, the growth of order seems translational order, by looking for a nonzero expectation likely to be even richer. value of the translational order parameters. Even this list of potential future problems, representing, as it does, probably another ten years of research for several ជ ជ iG•ri͑t͒ Gជ ͑t͒ϭ͗⌺ie /N͘, ͑8.1͒ groups, clearly represents only a narrow selection of the pos- sible directions in which this embryonic field can go. We which will become nonzero in the thermodynamic (N ϱ) → have not even mentioned, for example, the intriguing prob- limit at a set of reciprocal lattice vectors Gជ if such long- lem of one-dimensional flocking, with its applications to traf- ranged order actually develops. It will also be extremely in- fic flow ͑and traffic jams͒, a topic clearly of interest. This teresting to include the possibility of such long-ranged order problem has recently been studied ͓18͔ and found to also in our analytic model, and study the interplay between this show a nontrivial phase transition between moving and non- translational order and the anomalous hydrodynamics that moving states. we have found here for ‘‘fluid’’ flocks. Will anomalous hy- We expect flocking to be a fascinating and fruitful topic drodynamics suppress the ‘‘Mermin-Wagner’’ fluctuations of research for biologists, computer scientists, and both ex- of translational order, just as it does those of orientational perimental and theoretical physicists ͑at least these two͒ for order, and lead to true long-ranged translational order, even many years to come. in dϭ2? Will the crystallization suppress orientational fluc- tuations, and thereby slow down the anomalous diffusion ACKNOWLEDGMENTS that we found in the fluid case? And in any case, what are the We profusely thank T. Vicsek for introducing us to this temporal fluctuations of Gជ (t)? It should be noted that this problem potentially has all the problem. We are also grateful to A. Csirok, E. V. Albano, richness of liquid crystal physics: in addition to ‘‘crystal- and A. L. Barabasi for communicating their work to us prior to publication, to M. Ulm and S. Palmer for performing some ជ line’’ phases, in which the set ͕G͖ of reciprocal lattice vec- inspirational simulations, and for equally inspirational dis- tors in Eq. ͑8.1͒ spans all d dimensions of space, one could cussions, to J. Sethna and K. Dahmen for pointing out the imagine ‘‘smectic’’ phases in which all the Gជ ’s lay in the existence of the 2 and 3 terms, and to P. McEuen for same direction, and ‘‘discotic’’ phases in dϭ3, in which the suggesting the go-carts. J.T. thanks the Aspen Center for Gជ ’s only spanned a two-dimensional subspace of this three- Physics for their hospitality at several stages of this work. dimensional space. The melting transitions between these J.T.’s work was supported in part by the National Science phases and the ‘‘fluid,’’ moving flock, as well as possible Foundation under Grant No. DMR-9634596.
͓1͔ T. Vicsek, Phys. Rev. Lett. 75, 1226 ͑1995͒; A. Czirok, H. E. and E. Hoffmann ͑Springer-Verlag, Berlin, 1990͒, pp. 577– Stanley, and T. Vicsek, J. Phys. A 30, 1375 ͑1997͒. 590. We thank D. Rokhsar for calling these references to our ͓2͔ B. L. Partridge, Sci. Am. 246͑6͒, 114 ͑1982͒. attention. ͓3͔ C. Reynolds, Comput. Graph. 21,25͑1987͒; J. L. Deneubourg ͓4͔ J. Toner and Y. Tu, Phys. Rev. Lett. 75, 4326 ͑1995͒. and S. Goss, Ethology, Ecology, Evolution 1, 295 ͑1989͒;A. ͓5͔ N. D. Mermin and H. Wagner, Phys. Rev. Lett. 17, 1133 Huth and C. Wissel, in Biological Motion, edited by W. Alt ͑1966͒. 4858 JOHN TONER AND YUHAI TU PRE 58
͓6͔ D. Forster, D. R. Nelson, and M. J. Stephen, Phys. Rev. A 16, ͓9͔ See, e.g., M. Kardar, G. Parisi, and Y.-C. Zhang, Phys. Rev. 732 ͑1977͒. Lett. 56, 889 ͑1986͒. ͓7͔ Smectic-A liquid crystals show even stronger damping at long ͓10͔ We thank Stephanie Palmer for suggesting this alternative ap- wavelengths than that found here for flocks. For a theoretical proach to us. treatment, see G. F. Mazenko, S. Ramaswamy, and J. Toner, ͓11͔ We thank Paul McEuen for suggesting this realization to us. Phys. Rev. Lett. 49,51 1982 ; Phys. Rev. A 28, 1618 1983 . ͑ ͒ ͑ ͒ ͓12͔ Y. Tu, J. Toner, and M. Ulm, Phys. Rev. Lett. 80, 4819 ͑1998͒. Experimental confirmation of this theory is given by S. Bhat- ͓13͔ H. Levine ͑private communication͒. tacharya and J. B. Ketterson, Phys. Rev. Lett. 49, 997 ͑1982͒. ͓14͔ E. V. Albano, Phys. Rev. Lett. 77, 2129 ͑1996͒. See also the discussion in P. G. deGennes and J. Prost, The ͓15͔ L. Balents, M. C. Marchetti, and L. Radzihovsky, Phys. Rev. Physics of Liquid Crystals, 2nd ed. ͑Clarendon Press, Oxford, Lett. 78, 751 ͑1997͒. 1993͒, pp. 457–465. ͓16͔ L. Radzihovsky and J. Toner ͑unpublished͒. ͓8͔ We thank Karen Dahmen and Jim Sethna for pointing out the existence of this term to us ͑although, given all the difficulty ͓17͔ See, e.g., M. Mondello and N. Goldenfeld, Phys. Rev. E 47, this term has caused us, it is unclear whether thanks are really 2384 ͑1993͒. the appropriate response͒. ͓18͔ A. Czirok, A. L. Barabasi, and T. Vicsek ͑unpublished͒.