Birth, Death and Flight: A Theory of Malthusian Flocks
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Birth, Death and Flight: A Theory of Malthusian Flocks
John Toner
Department of Physics and Institute of Theoretical Science, University of Oregon, Eugene, OR 97403 (Dated: June 11, 2018)I study “Malthusian Flocks”: moving aggregates of self-propelled entities (e.g., organisms, cy-toskeletal actin, microtubules in mitotic spindles) that reproduce and die. Long-ranged order (i.e.,the existence of a non-zero average velocity h ~v ( ~r, t ) i 6 = ~
0) is possible in these systems, even in spa-tial dimension d = 2. Their spatiotemporal scaling structure can be determined exactly in d = 2;furthermore, they lack both the longitudinal sound waves and the giant number fluctuations foundin immortal flocks. Number fluctuations are very persistent , and propagate along the direction offlock motion, but at a different speed. PACS numbers: 05.65.+b, 64.70.qj, 87.18.Gh
Flocking [1] – the coherent motion of large numbers oforganisms – spans a wide range of length scales: fromkilometers (herds of wildebeest) to microns (microorgan-isms [2, 3]; mobile macromolecules in living cells[4, 5]).It is also [6] a dynamical version of ferromagnetic or-dering. A“hydrodynamic” theory of flocking [7] showsthat, unlike equilibrium ferromagnets [8], flocks can spon-taneously break a continuous symmetry (rotation in-variance) by developing long-ranged order, (i.e., a non-zero average velocity h ~v ( ~r, t ) i 6 = ~
0) in spatial dimensions d = 2, even with only short ranged interactions.Many quantitative predictions of the hydrodynamictheory , including the stability of long-ranged order in d = 2, the existence of propagating, dispersion-less soundmodes with non-trivial direction dependence of theirspeeds, and the presence of anomalously large numberfluctuations, agree with numerical simulations[7, 9], andexperiments on self-propelled molecules[10].However, a recent re-analysis[11] of this hydrodynamictheory has cast doubt on the claim that exact scaling ex-ponents could be determined for flocks in d = 2. This isdue to the erroneous neglect in[7] of non-linearities aris-ing from the local number density dependences of variousphenomenological parameters; these non-linearities couldchange the scaling exponents from those claimed in [7].In this paper, I show that these difficulties can beavoidedin flocks without number conservation. A num-ber of real systems[12, 13] lack number conservation, in-cluding growing bacteria colonies[13], and “treadmilling”molecular motor propelled biological macromolecules ina variety of intra-cellular structures, including the cy-toskleton, and mitotic spindles[4, 5], in which moleculesare being created and destroyed as they move. Hence, hestudy of such systems is not only convenient, but experi-mentally relevant. The most obvious example of flocking- namely, actual birds - is clearly not a good exampleof a Malthusian flock. I will henceforth use the term“boid” [1] to refer to the moving, self-propelled entitiesthat make up the flock.I will treat systems with the same symmetries as wereconsidered in earlier work on immortal flocks[7]: orien-tationally ordered, translationally disordered phases (i.e.,phases with h ~v ( ~r, t ) i 6 = ~ d = 2) a fixed background medium that breaks Galilean invariance.Since my treatment is hydrodynamic, it only describeslarge systems at long length and time scales. However,it becomes asymptotically exact in that limit.The removal of number conservation leads to profoundchanges. The sound modes of number conserving (here-after “immortal”) flocks disappear, and are replaced bylongitudinal velocity fluctuations which drift in the di-rection of flock motion with a speed γ = v , where v isthe mean speed of the flock. While drifting, these modesalso spread diffusively along the direction of flock motion,and hyperdiffusively perpendicular to that direction.In both Malthusian and immortal flocks, anomaloushydrodynamics stabilizes long-ranged orientational order(i.e., h ~v ( ~r, t ) i 6 = ~
0) in spatial dimension d = 2. In Malthu-sian flocks, the order outlives the boids: the persistencetime diverges as the number of boids N → ∞ , while thelifetime of the boids remains finite in this limit.The scaling exponents of the hydrodynamics can be de-termined exactly in spatial dimension d = 2 for Malthu-sian flocks, while, as discussed above, those in immortalflocks cannot. These exponents are: the dynamical ex-ponent z for the scaling of timescales t ( L ⊥ ) ∝ L z ⊥ withlength scale L ⊥ perpendicular to the direction of flockmotion; an anisotropy exponent ζ for the scaling of dis-tances L k ( L ⊥ ) ∝ L ζ ⊥ parallel to the direction of motionwith L ⊥ ; and a “roughness” exponent χ relating thescale of velocity fluctuations to L ⊥ via δv ∝ L χ ⊥ . I findthat, for Malthusian flocks in spatial dimension d = 2,these exponents are: ζ = 35 , z = 65 , χ = − . (1)The velocity field can have long-ranged order;( h ~v ( ~r, t ) i 6 = ~
0) in d = 2, because the roughness exponent χ ( d = 2) < persistence : the experimentally observabledensity-density correlation function C ρ ( ~r, t ) ≡ h δρ ( ~r ′ , t ′ ) δρ ( ~r ′ + ~r, t ′ + t ) i , (2)where δρ ( ~r, t ) ≡ ρ ( ~r, t ) − ρ is the departure of the localnumber density of boids ρ ( ~r, t ) from its mean value ρ ,decays algebraically with time at a fixed point in space: C ρ ( ~r = ~ , t ) ∝ | t | − , (3)in spatial dimensions d = 2, while for a point translat-ing along the direction ˆ x k of flock motion at the “drift”speed γ the decay is even slower: in both 2 and 3 spatialdimensions, I find C ρ ( ~r = γt ˆ x k , t ) ∝ | t | − . (4)These should be contrasted with the exponential decaywith time of density fluctuations that occurs in a disor-dered Malthusian flock (i.e., one in which < ~v > = ~ C ρ and C v can be constructed from any set ofhigh-resolution images of a moving flock, as has beendone for flocks of starlings in reference[14] and for a re-lated correlation function of bacteria in[13], it should notbe difficult to test these predictions.I will now outline the derivation of these results.My starting equation of motion for the velocity is ex-actly that of an “immortal” flock[15]: ∂ t ~v + λ ( ~v · ~ ∇ ) ~v + λ ( ~ ∇ · ~v ) ~v + λ ~ ∇ ( | ~v | ) = α~v − β | ~v | ~v − ~ ∇ P − ~v (cid:16) ~v · ~ ∇ P ( ρ, | ~v | ) (cid:17) + D oB ~ ∇ ( ~ ∇ · ~v ) + D T ∇ ~v + D ( ~v · ~ ∇ ) ~v + ~f (5)where all of the parameters λ i ( i = 1 → α , β , D oB , D T, and the “pressures” P , ( ρ, | ~v | ) are, in general, functionsof the boid number density ρ and the magnitude | ~v | ofthe local velocity. I will expand P , ( ρ, | ~v | ) about ρ : P i ( ρ ) = P i + P ∞ n =1 σ i,n ( | ~v | ) δρ n , where i = 1 , β , D oB , D and D T are all positive, while α < α > α and β terms give ~v a nonzero magnitude v = q αβ in the ordered phase. The diffusion constants D B,T, reflect the tendency of “boids” to follow their neighbors.The ~f term is a random Gaussian white noise, mimickingerrors made by the boids, with correlations: < f i ( ~r, t ) f j ( ~r ′ , t ′ ) > = ∆ δ ij δ d ( ~r − ~r ′ ) δ ( t − t ′ ) (6)where ∆ = constant, and i, j label vector components.The “anisotropic pressure” P ( ρ, | ~v | ) in (5) is only al-lowed due to the non-equilibrium nature of the flock; in an equilibrium fluid such a term is forbidden by Pascal’sLaw. In earlier work [7] this term was ignored.Note that (5) is not Galilean invariant; it holds only inthe frame of the fixed medium through or on which thecreatures move.I now need an equation of motion for ρ . In immor-tal flocks, this is just the usual continuity equation ofcompressible fluid dynamics. For Malthusian flocks, itmust also include the effects of birth and death. I willassume that the death rate goes up as the density goesup (the “Malthusian” assumption[16]), so that the dif-ference between the two - that is, the net, local growthrate of number density in the absence of motion, whichI’ll call g ( ρ ) - vanishes at some fixed point density ρ ,with larger densities decreasing (i.e., g ( ρ > ρ ) < g ( ρ < ρ ) > ∂ t ρ + ∇ · ( ~vρ ) = g ( ρ ) . (7)Note that in the absence of birth and death, g ( ρ ) = 0, andequation (7) reduces to the usual continuity equation, asit should, since “boid number” is then conserved.Since birth and death quickly restore the fixed pointdensity ρ , I will write ρ ( ~r, t ) = ρ + δρ ( ~r, t ) and expandboth sides of equation (7) to leading order in δρ . Thisgives ρ ~ ▽· ~v ∼ = g ′ ( ρ ) δρ, where I’ve dropped the ∂ t ρ termrelative to the g ′ ( ρ ) δρ term since I’m interested in thehydrodynamic limit, in which the fields evolve extremelyslowly. This equation can be readily solved to give δρ ∼ = ρ ~ ∇ · ~vg ′ ( ρ ) ≡ − ∆ D (1) B σ , ( ~ ∇ · ~v ) (8)where ∆ D (1) B is a positive constant, and σ , is the firstexpansion coefficient for P . I can now insert this solution(8) for δρ in terms of ~v into the isotropic pressure P ; theresulting equation of motion for ~v is: ∂ t ~v + λ ( ~v · ~ ∇ ) ~v + λ ( ~ ∇ · ~v ) ~v + λ ~ ∇ ( | ~v | ) = α~v − β | ~v | ~v − ~v (cid:16) ~v · ~ ∇ P ( ρ, | ~v | ) (cid:17) + D (1) B ~ ∇ ( ~ ∇ · ~v ) + D T ∇ ~v + D ( ~v · ~ ∇ ) ~v + ~f , (9)where I’ve defined D (1) B ≡ D oB + ∆ D (1) B . Taking the dotproduct of both sides of (9) with ~v itself, and defining U ( | ~v | ) ≡ α ( | ~v | , ρ ) − β ( | ~v | , ρ ) | ~v | , I obtain:12 (cid:16) ∂ t | ~v | + ( λ + 2 λ )( ~v · ~ ∇ ) | ~v | (cid:17) + λ ( ~ ∇ · ~v ) | ~v | = U ( | ~v | ) | ~v | − | ~v | ~v · ~ ∇ P + D (1) B ~v · ~ ∇ ( ~ ∇ · ~v )+ D T ~v · ∇ ~v + D ~v · (cid:16) ( ~v · ~ ∇ ) ~v (cid:17) + ~v · ~f , (10)In the ordered state (i.e., in which h ~v ( ~r, t ) i = v ˆ x k ), and I can expand the ~v equation of motion for smalldepartures δ~v ( ~r, t ) of ~v ( ~r, t ) from uniform motion withspeed v : ~v ( ~r, t ) = ( v + δv k )ˆ x k + ~v ⊥ ( ~r, t ) , (11)where, henceforth k and ⊥ denote components along andperpendicular to the mean velocity, respectively.In this hydrodynamic approach, I’m interested only influctuations ~δv ( ~r, t ) ≡ δv k ˆ x k + ~v ⊥ ( ~r, t ) and δρ ( ~r, t ) thatvary slowly in space and time. Hence, terms involvingspatiotemporal derivatives of ~δv ( ~r, t ) and δρ ( ~r, t ) are al-ways negligible, in the hydrodynamic limit, compared toterms involving the same number of powers of fields withfewer spatiotemporal derivatives. Furthermore, the fluc-tuations ~δv ( ~r, t ) and δρ ( ~r, t ) can themselves be shown tobe small in the long-wavelength limit. Hence, we needonly keep terms in [10] up to linear order in ~δv ( ~r, t ) and δρ ( ~r, t ). The ~v · ~f term can likewise be dropped.These observations can be used to eliminate manyterms in equation [10], and solve for the quantity U ≡ ( α ( ρ, | ~v | ) − β ( ρ, | ~v | ) | ~v | ); I obtain: U = λ ~ ∇ · ~v + ~v · ~ ∇ P .Inserting this expression for U back into equation [9], Ifind that P and λ cancel out of the ~v equation of mo-tion, leaving, ignoring irrelevant terms: ∂ t ~v + λ ( ~v · ~ ∇ ) ~v + λ ~ ∇ ( | ~v | ) = D T ∇ ~v + D (1) B ~ ∇ ( ~ ∇ · ~v ) + D ( ~v · ~ ∇ ) ~v + ~f , (12)This can be made into an equation of motion for ~v ⊥ in-volving only ~v ⊥ ( ~r, t ) itself by projecting perpendicularto the direction of mean flock motion ˆ x k , and eliminat-ing δv k using U = λ ~ ∇ · ~v + ~v · ~ ∇ P and the expansion U ≈ − Γ δv k − Γ δρ , where I’ve defined Γ ≡ − (cid:16) ∂U∂ | ~v | (cid:17) ρ and Γ ≡ − (cid:16) ∂U∂ρ (cid:17) | ~v | , with super- or sub-scripts 0 de-noting functions of ρ and | ~v | evaluated at ρ = ρ and | ~v | = v . Doing this, and using (8) for ρ , I obtain: ∂ t ~v ⊥ + γ∂ k ~v ⊥ + λ ( ~v ⊥ · ~ ∇ ⊥ ) ~v ⊥ + λ ~ ∇ ⊥ (cid:0) | ~v ⊥ | (cid:1) = D T ∇ ⊥ ~v ⊥ + D B ~ ∇ ⊥ ( ~ ∇ ⊥ · ~v ⊥ ) + D k ∂ k ~v ⊥ + ~f ⊥ , (13)where I’ve defined γ ≡ λ v , D B ≡ D B + 2 v λ ( λ − Γ ∆ D (1) B /σ ) / Γ and D k ≡ D T + D v .Changing co-ordinates to a new Galilean frame ~r ′ mov-ing with respect to our original frame in the direction ofmean flock motion at speed γ - i.e., ~r ′ ≡ ~r − γt ˆ x k - gives ∂ t ~v ⊥ + λ ( ~v ⊥ · ~ ∇ ⊥ ) ~v ⊥ = D T ∇ ⊥ ~v ⊥ + D B ~ ∇ ⊥ ( ~ ∇ ⊥ · ~v ⊥ )+ D k ∂ ′ k ~v ⊥ + ~f ⊥ . (14)Ignoring the non-linear term λ in this equation of mo-tion gives a noisy, anisotropic, vectorial diffusion equa-tion. This can be readily solved for the mode structureand fluctuations by spatiotemporal Fourier transforma-tion, and has d − d . These separate into d − ~v ⊥ perpendicular to ~q ⊥ , all with thesame imaginary eigenfrequency: − iω T = D T | ~q ⊥ | + D k q k .The remaining diffusive mode (the only mode in d = 2)is “longitudinal” (i.e., has ~v ⊥ along ~q ⊥ ), with frequency − iω L = D ⊥ | ~q ⊥ | + D k q k , where D ⊥ ≡ D B + D T .Because the dynamics described above is in theGalileanly boosted frame, the dynamics in the originalreference frame ~r will have a steady drift at velocity γ superposed on the diffusive motion described above; thatis, both eigenfrequencies get γq k added to them.I can also calculate the real-space velocity fluctuations (cid:10) | ~v ⊥ ( ~r, t ) | (cid:11) in this linearized approximation; I find that, in this approximation , this diverges in all d ≤
2. Thisis analogous to the Mermin-Wagner theorem[8] in equi-librium magnets. However, as in immortal flocks[7], this“Mermin-Wagner” result, and all the linearized scalinglaws, are invalidated for d ≤ λ term in (14).To show this here, I’ll analyze equation (14) using thedynamical Renormalization Group(RG)[17].The dynamical RG starts by averaging the equationsof motion over the short-wavelength fluctuations: i.e.,those with support in the “shell” of Fourier space b − Λ ≤| ~q | ≤ Λ, where Λ is an “ultra-violet cutoff”, and b is anarbitrary rescaling factor. Then, one rescales lengths,time, and ~v ⊥ in equation (14) according to ~v ⊥ = b χ ~v ′ ⊥ , ~r ⊥ = b~r ′⊥ , r ′k = b ζ ( r ′k ) ′ , and t = b z t ′ to restore the ultra-violet cutoff to Λ. This leads to a new equation of motionof the same form as (14), but with “renormalized” values(denoted by primes below) of the parameters given by: D ′ B,T = b z − ( D B,T + graphs) , (15) D ′k = b z − ζ ( D k + graphs) , (16)∆ ′ = b z − ζ − χ +1 − d (∆ + graphs) , (17) λ ′ , = b z + χ − ( λ , + graphs) , (18)where “graphs” denotes contributions from integratingout the short wavelength degrees of freedom. If we ig-nore these graphical corrections (valid for λ , small), andchoose z , ζ , and χ to keep the linear parameters D B,T, k and ∆ fixed, equation (18) implies that an initially small λ will grow for all d ≤
4, meaning the linearized theoryIt is possible to get exact exponents in d = 2. Thisis because the nonlinearities - the λ and λ terms - in(14) add up to a total derivative in d = 2 (specifically, (cid:0) λ + λ (cid:1) ∂ ⊥ v ⊥ ), since the ⊥ subspace is one dimensionalin d = 2. In contrast, in immortal flocks, v - ρ non-linearities arising from the ρ -dependence of λ cannot bewritten as total derivatives, making it impossible to ob-tain exact exponents, a fact missed by [7]. Here, becausethe λ term is a total ⊥ -derivative, it can only graph-ically renormalize terms involving ⊥ -derivatives them-selves. Hence, the graphical corrections to D k and ∆ inequations (16) and (17) vanish. Hence, at a fixed point,in d = 2, z − ζ = 0 , z − ζ − χ + 1 − d = z − ζ − χ − . (19)There are no graphical corrections λ either, becausethe equation of motion (14) remains unchanged by thetransformation: ~r ⊥ → ~r ⊥ − λ ~v t , ~v ⊥ → ~v ⊥ + ~v for arbitrary constant vector ~v ⊥ ˆ x k . This exact sym-metry must continue to hold upon renormalization, withthe same value of λ . Hence, λ cannot be graphicallyrenormalized. Requiring that λ ′ = λ in (18), and set-ting graphs = 0, implies χ = 1 − z in all d ≤
4. Thisand (19) forms three independent equations for the threeunknowns χ , z , and ζ , whose solution in d = 2 is (1).The scaling exponents z , ζ , and χ determine the scalingform of the velocity-velocity autocorrelation function inarbitrary dimension d through the scaling relation[7]: C v ( ~r, t ) ≡ h ~v ⊥ ( ~ , · ~v ⊥ ( ~r, t ) i = | ~r ⊥ | χ G r ′k | ~r ⊥ | ζ , t | ~r ⊥ | z ! = | ~r ⊥ | χ G (cid:18) r k − γt | ~r ⊥ | ζ , t | ~r ⊥ | z (cid:19) , (20)where the second equality follows from scaling argumentsapplied to the boosted equation of motion (14), and thethird arises from undoing the boost. Here G ( u, w ) is ascaling function, with scaling arguments u ≡ r k − γt | ~r ⊥ | ζ and w ≡ t | ~r ⊥ | z . The asymptotic limits of G ( u, w ) and C v ( ~r, t )can be obtained by the following arguments.When r k − γt → t → C v ( ~r, t ) must clearlydepend only on r ⊥ , and should not vanish. Hence G ( u ≪ , w ≪ → constant = 0. This in turn implies that C v ( ~r, t ) ∝ | ~r ⊥ | χ for | ~r ⊥ | ζ ≫ | r k − γt | , t ζz . Similarly,if ~r → t →
0, then C v ( ~r, t ) should depend onlyon | r k − γt | . This implies G ( u, w ) ∝ u χζ for u ≫ w,
1, in order to cancel off the | ~r ⊥ | χ prefactor in Equation(20). This in turn implies that C v ( ~r, t ) ∝ | r k − γt | χζ for | r k − γt | ≫ | ~r ⊥ | ζ , t ζz . Similar reasoning implies that C v ( ~r, t ) ∝ | t | χz for t ≫ | ~r ⊥ | z , | r k − γt | zζ . Hence, usingthe exact exponents (1) in d = 2, C v ( ~r, t ) ∝ r − ⊥ , | r ⊥ | ≫ | r k − γt | , t ( r k − γt ) − , | r k − γt | ≫ | x | , t t − , | t | ≫ | r ⊥ | , | r k − γt | . (21)This correlation function can be measured directly inboth simulations[7, 9], and experiments [14].The relation (8) between density and velocity impliesthat density correlations should obey the same sort ofscaling law, but with an additional power of | r ⊥ | − forevery power of δρ ; hence, in d = 2: C ρ ( ~r, t ) ≡ | r ⊥ | − G ρ (cid:18) r k − γt | r ⊥ | , t | r ⊥ | (cid:19) ∝ | r ⊥ | − , | r ⊥ | ≫ | r k − γt | , t ( r k − γt ) − , | r k − γt | ≫ | r ⊥ | , t t − , | t | ≫ | r ⊥ | , | r k − γt | . (22)The last line holds in d = 3 as well, because χ = 1 − z ,does. The last two lines of (22) directly imply equations(3) and (4). It can also be shown [11] that at equal times C ρ ( ~r, t = 0)decays sufficiently rapidly that there are nogiant number fluctuations in Malthusian flocks.I thank the Institut Poincare, the ESPCI, the Univer-site Pierre et Marie Curie, the KITP (UCSB), CUNY,the Lorentz Center, University of Leiden, and the MPI-PKS, Dresden, for their hospitality; S. Ramaswamy, H.Chate, A. Cavagna, I. Giardina, M. Rao, Y. Tu, and F.Ginelli for valuable discussions; and K. Toner for a care-ful reading of the manuscript. [1] See, e.g., C. Reynolds, Computer Graphics , 25 (1987).[2] See, e.g., W. Loomis, The Development of Dictyosteliumdiscoideum (Academic, New York, 1982).[3] W.J. Rappel et. al., Phys. Rev. Lett., (6), 1247 (1999).[4] R. Voituriez, J. F. Joanny, and J. Prost, EurophysicsLetters, :404, (2005).[5] K. Kruse, J. F. Joanny, F. Julicher, J. Prost, and K.Sekimoto, European Physical Journal E, , 1226 (1995).[7] J. Toner and Y.-h. Tu, Phys. Rev. Lett. , 4326 (1995);Y.-h. Tu, M. Ulm and J. Toner, Phys. Rev. Lett. ,4819 (1998); J. Toner and Y.-h. Tu, Phys. Rev. E ,4828 (1998); J. Toner, Y.-h. Tu, and S. Ramaswamy,Ann. Phys. , 170 (2005).[8] N. D. Mermin and H. Wagner, Phys. Rev. Lett. , 1133(1966); P. C. Hohenberg, Phys. Rev. , 383 (1967).[9] See, e.g., G. Gregoire, H. Chate, and Y.-h. Tu, Phys. Rev. Lett., , 556 (2001).[10] K. Gowrishankar, S. Ghosh, S. Saha, S. Mayor and M.Rao, (unpublished).[11] J. Toner, unpublished.[12] S. Mishra, R. A. Simha, and S. Ramaswamy, J. Stat.Mech. P02003 (2010)[13] W. Mather et.al., Phys. Rev. Lett., 104, 208101 (2010).[14] M. Ballerini et. al., Animal Behaviour , 201 (2008).[15] The P term was left out by all of reference [7], as wasthe density dependence of λ . The latter oversight lead tothe erroneous conclusion that one could determine exactexponents for immortal flocks in d = 2.[16] T. R. Malthus, An essay on the principle of population ,(J. Johnson, St. Paul’s Churchyard, London, 1798).[17] See, e.g., D. Forster, D. R. Nelson, and M. J. Stephen,Phys. Rev. A16