Noisy multistate voter model for flocking in finite dimensions
aa r X i v : . [ phy s i c s . s o c - ph ] F e b Noisy multistate voter model for flocking in finite dimensions
Ernesto S. Loscar and Gabriel Baglietto
Instituto de F´ısica de L´ıquidos y Sistemas Biol´ogicos (IFLYSIB), UNLP,CCT La Plata-CONICET, Calle 59 no. 789, B1900BTE La Plata, Argentina
Federico Vazquez ∗ Instituto de C´alculo, FCEN, Universidad de BuenosAires and CONICET, Buenos Aires, Argentina (Dated: February 5, 2021)
Abstract
We study a model for the collective behavior of self-propelled particles subject to pairwise copyinginteractions and noise. Particles move at a constant speed v on a two–dimensional space and, ina single step of the dynamics, each particle adopts the direction of motion of a randomly chosenneighboring particle, with the addition of a perturbation of amplitude η (noise). We investigate howthe global level of particles’ alignment (order) is affected by their motion and the noise amplitude η . In the static case scenario v = 0 where particles are fixed at the sites of a square lattice andinteract with their first neighbors, we find that for any noise η c > η = 0 full order is eventuallyachieved for a system with any number of particles N . Therefore, the model displays a transitionat zero noise when particles are static, and thus there are no ordered steady states for a finitenoise ( η > N as η c ∼ N − and η c ∼ ( N ln N ) − / in one and two–dimensional lattices, respectively, which is linked to knownresults on the behavior of a type of noisy voter model for catalytic reactions. When particles areallowed to move in the space at a finite speed v >
0, an ordered phase emerges, characterized bya fraction of particles moving in a similar direction. The system exhibits an order-disorder phasetransition at a noise amplitude η c > v , and that scales approximately as η c ∼ v ( − ln v ) − / for v ≪
1. These results show that the motion of particles is able to sustain astate of global order in a system with voter-like interactions. ∗ [email protected] . INTRODUCTION The study of the collective properties of systems composed by self-propelled individualshas been the focus of intense research in the last two decades [1–3]. The flocking behaviorof a large group of animals is observed in many different species such as fish, birds, bacteriaand insects, among others. From a statistical physics viewpoint, the interactions betweenparticles in a system are responsible of its collective behavior, and lead to well characterizedclasses represented by archetype models. For the case of flocking, the alignment interactionamong individuals is usually modeled as a local averaging of moving directions of nearbyindividuals, plus a noise that accounts for errors in the average process [4]. A crucial role inthe emergent behavior of the system is played by the displacement of the individuals, whichchanges dramatically its ordering properties [5].Within the context of flocking, the dynamics of collective alignment in groups of fishwas recently studied in [6]. The authors performed experiments with cichlid fish
Etroplussuratensis that swim in a circular shallow tank, in order to explore how schooling is affectedby the fish group size. The level of group alignment is quantified by a vector order param-eter M that is the average velocity of fish, also called group polarization, in such a waythat | M | ∼ | M | ∼ N = 15, 30 and 60, they found that the col-lective alignment | M | increases as N decreases. An insight into this phenomenon is givenby a phenomenological stochastic differential equation (SDE) for the time evolution of M ,where its parameters were extracted from the experimental data. It is shown that grouppolarization is the result of the interplay between the drift and the demographic (popula-tion) noise terms in the SDE, that is, the fewer the fish, the greater the demographic noiseand so the greater the alignment level. Thus, they conclude that schooling (highly polar-ized and coherent motion) is induced by the intrinsic population noise that arises from thestochasticity related to the finite number of interacting fish. They derived the SDE for M by means of a mean-field (MF) model in which particles (fish) interact by pairs and followa simple imitation dynamics: each particle either copies the direction of another randomparticle or spontaneously changes its direction, modeled as an external noise of amplitude η .They also show that other ternary or higher-order aligning interactions, including local aver-2ges like in the Vicsek-like family of models, are unnecessary to explain these experimentalresults. Therefore, they arrive to the conclusion that the minimal theoretical mechanismthat reproduces the collective alignment properties of fish observed in the experiments isthat of pairwise interactions with copying dynamics and noise. We notice that the noiselessversion of this particular alignment dynamics that induces flocking was first introduced in[7], where the authors study the collective motion of particles on a two–dimensional (2 D )space subject to voter-like interactions, that is, each particle aligns its direction of motionwith that of a random neighboring particle within an interaction radius.From the theoretical point of view, an interesting result can be inferred from the workin [6] by analyzing the SDE for the group polarization M . That is, this equation predictscomplete order ( | M | = 1) for zero noise ( η = 0) and full disorder ( | M | = 0) for any finitenoise amplitude η > N → ∞ limit. This observation is in agreement with recentanalytical results obtained in a similar model with a discrete set of S angular directions, amultistate voter model (MSVM) with external noise [8], where it is shown that the orderparameter | M | approaches 1 . − | M | ∼ η N in the η → N increases as 1 / ( η N ) for any 0 < η ≪
1. Thus, the partial order obtained with voterinteractions and noise in a MF set up is only a finite size effect that eventually disappearsin the thermodynamic limit. These results suggest a peculiar order-disorder transition atzero noise, unseen in related flocking models such as the Vicsek model, where the transitionhappens at a critical noise larger than zero. However, we notice that the experimental resultsobtained in [6] correspond to fish moving on a 2 D set up (tank), while both the SDE andthe model in [8] are for a MF set up (infinite dimension), where every particle interacts withany other particle, and thus motion plays no role in the dynamics. It is natural, therefore,to wonder whether these results hold when particles move on a 2 D space. Do space andmotion affect the transition at zero noise?In this article we study a noisy multistate voter model for flocking in finite dimensions,and we investigate the order-disorder phase transition in different case scenarios. We startby analyzing the case of all-to-all interactions or MF. We then explore the static case whereeach particle occupies a site of a square lattice and interacts with its first nearest-neighbors,and we finally study the dynamic case in which particles move on a 2 D continuous space andchange their direction when they interact with other nearby particles. In the simplest casewhere particles can have only two possible angular states and interact on a MF set up, the3odel turns to be equivalent to the noisy two-state voter model (NVM) introduced in [9, 10],in which each individual of a population holds one of two states (opinions) that are updatedby either copying the state of a random neighbor or spontaneously switching state (noise).In the absence of noise, any finite population eventually reaches full order (consensus) inall dimensions, as in the original voter model [11, 12], with all individuals sharing the sameopinion. However, the addition of a weak noise leads to a bi-stable regime in which thesystem jumps between two steady states corresponding to a quasi-consensus in one or theother opinion [9, 10], while for strong noise the system remains disordered. This is in linewith the fact that adding thermal bulk noise in the voter model destroys global order inany dimension [13], even when the noise is weak. In square lattices, the NVM is equivalentto a particular limit of the catalytic reaction model with desorption originally introducedin [14] and widely studied after [15, 16], which exhibits a finite size transition induced bynoise called saturation transition [17–19]. More recently, the dynamics of the NVM hasbeen investigated in complex networks [20–22], and its version with multiple states has beenexplored in fully connected systems [8, 23]. Also, an asymmetric variant of the NVM withlong-range interactions has recently been proposed to study the competition between twospecies for territory [24].While in 2 D lattice models bulk noise inhibits the formation of long range order in thethermodynamic limit, it is known that in flocking systems the displacement of particles playsan ordering role. This ordering phenomenon is observed in the Vicsek model, thought as anon equilibrium version of the XY model with particles moving ballistically in the directionsof their spins. That is, while the Vicsek model can sustain long-range order for finite valuesof noise amplitude due to particles’ motion [5], the XY model is unable to do so [25]. Asa result, the system exhibits a thermodynamic phase transition from an ordered phase for η < η c to a disordered phase for η > η c , where η c > ρ and speed v of particles. Recently, great interest was dedicated to thismodel in the ρ → v → η → η c ∼ v σ ρ κ , with dimension dependent exponents σ and κ [26]. In summary, theextensive literature about Vicsek type models shows that the effect of the velocity is to leadto steady states associated with a new ordered phase below a transition noise η c . Basedon this paradigm we expect, in the flocking voter model (FVM) studied in this article, acompetition between the ordering mechanism generated by particles’ motion and the typical4isordering effect induced by noisy voter interactions in the thermodynamic limit. Therefore,we aim to explore whether the ordered phase observed in flocking models is still present inthe FVM, or it is rather completely suppressed by noise.The rest of the article is organized as follows. In section II we define the model. Section IIIpresents MF results, while section IV is dedicated to the static version of the model in oneand two dimensional square lattices. In section V we study the dynamic version of the modelin a continuous 2 D space. We investigate the effects of particles’ velocity in the transition,with a particular focus on the behavior at low speeds in the thermodynamic limit. Finally,in section VI we summarize and give some conclusions. II. THE MODEL
A set of N particles are allowed to move at a constant speed v on a 2 D square box of side L with periodic boundary conditions. The position and velocity of particle i ( i = 1 , , .., N ) attime t are denoted by r ti = ( x ti , y ti ) and v ti = ( v cos θ ti , v sin θ ti ), respectively, where v = | v ti | isthe particle’s speed and θ ti is its angular moving direction. The density of particles ρ = N/L is fixed at 0 . t = 1 of the dynamics,each particle i updates it position and direction according to r t +1 i = r ti + v ti ∆ t, (1a) θ t +1 i = θ tj + ξ t +1 i , (1b)where θ tj is the moving direction of a randomly chosen particle j that is inside a disk of radius R = 1 centered at r ti , and ξ ti is a random angle drawn uniformly in [ − ηπ, ηπ ) with amplitude η (0 < η < v following a given straight path andupdates its direction at integer times t = 1 , , , ... , by adopting the direction of a randomneighboring particle with an error of amplitude η . If a particle has no neighbors inside itsinteraction range R , then its direction is changed only by the noise ξ .In flocking models, noise –in its various forms– plays a fundamental role in the behaviorof the system. It is known that the amplitude of noise η induces an order-disorder phasetransition, from a phase where a large fraction of particles move in a similar direction (order)5or small η , to a phase in which particles move in random directions (disorder) for large η .To study this phenomenon in the FVM we define the order parameter (see for instance [6, 7]) ϕ ( t ) ≡ v N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N X i =1 v ti (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 1 N vuut" N X i =1 cos θ ti + " N X i =1 sin θ ti (2)that measures the level of collective alignment in the system (magnitude of the normalizedmean velocity of all particles), and the susceptibility χ ≡ N (cid:2) h ϕ i − h ϕ i (cid:3) , (3)which accounts for the amplitude of fluctuations of ϕ at the stationary state. Here h ϕ m i isthe m -th moment of ϕ , and the symbol h·i represents the average value of a given magnitudeover many realizations of the dynamics at the steady state.Our aim is to explore via computational simulations and scaling theory how space andmotion affects the phase transition in the FVM. For that, we first study the model in MF( R = L ), we then explore the static case v = 0 in lattices, and we finally investigate thedynamic case v > D . III. MEAN FIELD
In order to gain an insight into the behavior of the FVM, we start by analyzing in thissection the simplest case scenario of all-to-all interactions or MF, which corresponds to thelarge interaction range limit R → L of the model defined in section II. In this case, thedynamics of the angular directions of particles θ is independent of the positions of particles,and thus it is entirely determined by Eq. (1b). That is, each particle simply adopts thedirection of another randomly chosen particle in the system, with the addition of noise.This dynamics is equivalent to that of the multistate voter model with imperfect copyingintroduced and studied in [8], in the limit of continuum angular states. In a single timestep ∆ t = 1 /N , a particle i with state θ i is picked at random, then it copies the state θ j ofanother randomly chosen particle j , and this state is slightly perturbed: θ i ( t + ∆ t ) = θ j ( t ) + ξ i ( t + ∆ t ) . (4)We note that we are implementing here a sequential update in which only one particleupdates its state in a time step, unlike the parallel update where all N particles are updated6 -3 -2 -1 η -3 -2 -1 < ϕ > -4 -2 η N -2 -1 < ϕ > (a) N -4 -2 η N -8 -6 -4 -2 χ / N -3 -2 -1 η -8 -6 -4 -2 χ (b) FIG. 1. Results of the FVM in MF. (a) Average value of the order parameter ϕ at the stationarystate as a function of noise amplitude η for the system sizes N indicated in the legend. The insetshows the collapse of the data points when they are plotted as a function of the scaling variable x MF = η N . The dashed line has slope − /
2. (b) Susceptibility χ vs η for the same system sizesas in panel (a). Inset: collapse of the data when it is plotted vs x MF and the y –axis is rescaled by N − . Averages were done in a time window ∆ t ∼ over 10 independent realizations. at once. However, we have verified that the behavior of the macroscopic variables ϕ and χ under the parallel update is recovered by making the substitution N → N in the resultsobtained with the sequential update, as mathematically proved by Blythe and McKane in[27] for population genetic models akin to the voter model. Inversely, the transformation N → N/ t = 10 and t = 2 × , and over 10 independent realizations. In panel (a) we observe that the orderparameter ϕ continuously decreases as η increases, and that approaches the value ϕ = 1(full order) as η →
0, which corresponds to the absorbing consensus state obtained in thezero noise case η = 0, as it is known from previous works of the multistate voter model[7, 8, 28, 29]. We also see that, for a fixed value of η > ϕ vanishes as the system size N increases, suggesting that ϕ → η > N → ∞ limit. Indeed, an expression7 N -3 -2 -1 η c MF2D1D N -3 -2 -1 η c η c η c (a) ^ N χ m a x χ m a x χ m a x MF (b) FIG. 2. Results of the FVM in MF (circles) and in square lattices of dimensions d = 1 (diamonds)and d = 2 (squares). (a). Transition noise η c vs system size N . Straight lines are best power-lawfits η c = A N − α with exponents α = 0 . ± .
015 (MF), 0 . ± .
02 ( d = 1) and 0 . ± .
01 ( d = 2).Inset: η c for d = 2 (squares) and the effective noise ˆ η c = η c √− ln η c (circles). The upper solidline is the best power law fit ˆ η c ≃ B N − / , with B = 1 . ± .
04, while the bottom solid curve isthe approximation η c ≃ . N ln N ) − / from Eq. (15c). (b) Maximum value of the susceptibility χ max vs N . Best power-law fits χ max ∼ N γ (straight lines) have exponents γ = 1 . ± .
01 (MF),0 . ± .
005 ( d = 1) and 0 . ± .
01 ( d = 2). for the scaling of h ϕ i with η and N that confirms this assumption can be obtained fromanalytical results of this model recently presented in [8], for an order parameter ψ = ϕ .It was shown in [8] that h ψ i ∼ ( η N ) − for η ≪ η N &
1, and thus assuming h ϕ i ∼ h ψ i / we obtain the approximate MF behavior h ϕ i MF ∼ (cid:0) η N (cid:1) − / for η ≪ η N & . (5)In the inset of Fig. 1(a) we plot the data as a function of the scaling variable x MF ≡ η N ,where we can see that h ϕ i MF obeys the power law decay from Eq. (5) for η N & x MF , showing that the order parameter is a function of x MF , h ϕ i MF = f ( η N ), with f ( x MF ) ∼ x − / MF for x MF & η = 0 the system reaches full order8 ϕ = 1), but a tiny amount of noise η > ϕ = 0) in the thermodynamic limit, which suggests a transition at zero noise. To studythis in more detail, we show in Fig. 1(b) the behavior of the susceptibility χ with η . Weobserve that the curve for a given system size N exhibits a maximum that is an indicationof a transition that depends on N , between an ordered phase for η < η c ( N ) and a disorderedphase for η > η c ( N ), where the transition point η c ( N ) is estimated as the location of thepeak. In Fig. 2(a) we plot the transition noise η c vs N (circles), where we can see that η c vanishes as N increases following a power-law behavior N − α , with a best fitting exponent α = 0 . ± . η c ( ∞ ) = 0 in the thermodynamic limit. Inpanel (b) of Fig. 2 we see that the maximum value of the susceptibility increases with N as χ max ∼ N γ , where γ ≃ . ± .
01 is the best fitting exponent.These scalings can be nicely verified by assuming that χ is also a function of the scalingvariable x MF = η N for ϕ in Fig. 1. Indeed, rescaling the y –axis of Fig. 1(b) by N − andplotting the resulting data vs x MF we find a good collapse of all curves for different N values(see inset), showing that the MF susceptibility behaves as χ MF = N g (cid:0) η N (cid:1) , (6)where g ( x MF ) is a smooth function of x MF . From Eq. (6) we have that at the MF transitionpoint η MF c is χ maxMF /N = g (cid:2) ( η MF c ) N (cid:3) = constant and, therefore, η MF c ∼ N − / , (7)in agreement with numerical results [Fig. 2(a)].In summary, the mean field version of the FVM exhibits an order-disorder phase transitionat zero noise η c = 0 in the thermodynamic limit, between a perfectly ordered phase where ϕ = 1 for η = 0 and a completely disordered phase where ϕ = 0 for η > IV. STATIC CASE v = 0 IN ONE AND TWO DIMENSIONS
In this section we analyze the static version of the FVM in finite dimensions. For that, weconsider that each particle occupies a site of a square lattice of length L and d dimensions( N = L d sites), and interacts with its 2 d nearest neighbors only. We have simulated thedynamics of the model under the sequential update described in section III on lattices of9 -1 η N -2 -1 < ϕ > -4 -3 -2 -1 η -2 -1 < ϕ > (a) N -4 -3 -2 -1 η -6 -5 -4 -3 -2 -1 χ -1 η N -6 -4 -2 χ / N (b) FIG. 3. Results of the static version of the FVM in one dimension. (a) Average value of ϕ atthe stationary state vs η for the system sizes N indicated in the legend. Inset: same data vs thescaling variable x = ηN showing the collapse of curves for different N values. The dashed linehas slope − /
2. (b) Susceptibility χ vs η for the same system sizes as in panel (a). Inset: x and y –axis are rescaled by N and N − , respectively, to show the collapse of the data. dimensions d = 1 and d = 2 with periodic boundary conditions. In a time step ∆ t = 1 /N ,a randomly selected particle copies the angular state of a first neighbor chosen at random,with the addition of an error of amplitude η .Figure 3 shows simulation results for the FVM in one dimension. The behavior of h ϕ i and χ are similar to those of the MF model, with a scaling variable x ≡ ηN in thisone–dimensional case. The variable x was obtained from the behavior of the transitionnoise η c with N given by the peak of χ in panel (b) of Fig. 3. We found η c ∼ N − α , with α = 0 . ± .
02 [Fig. 2(a)], while for the peak of the susceptibility we found the scaling χ max ∼ N γ , with γ = 0 . ± .
005 [Fig. 2(b)]. Therefore, assuming the scalings η c ∼ N − and (8) χ max ∼ N, (9)we arrive at the following scaling for the susceptibility: χ = N g ( ηN ) , (10)and thus the scaling variable is x = ηN , as stated above. Indeed, we can check in the10 -4 -2 η N -3 -2 -1 < ϕ > x
8 16 x
16 32 x
32 64 x x x x x -4 -2 η N -8 -6 -4 -2 χ / N -4 -2 η N -3 -2 -1 < ϕ > -4 -2 η N -8 -6 -4 -2 χ / N (a) (b)(d)(c) N ^ ^ FIG. 4. Static version of the FVM in two dimensional lattices. (a) Average order parameter h ϕ i and (b) normalized susceptibility χ/N vs the scaling variable η N . for the system sizes N indicated in the legend. (c) h ϕ i and (d) χ/N vs the scaling variable x = ˆ η N for the same systemsizes as in panels (a) and (b), with ˆ η = η √− ln η . Dashed lines in panels (a) and (c) have slopes − .
45 and − /
2, respectively. insets of Fig. 3 the collapse of the curves for different system sizes when the data is plottedvs x , and the y –axis in panel (b) is rescaled by N − . Also, in the inset of panel (a) weshow that the order parameter scales as h ϕ i ∼ x − / for x & / N and, therefore,we conclude that the static version of the FVM in one dimension exhibits an order-disordertransition at zero noise in the thermodynamic limit, as it happens in MF.We repeated the same analysis for the FVM model on two dimensional lattices. Simula-tion results are presented in Fig. 4, where the data collapse was obtained by means of twodifferent scaling variables, as we describe below. As it happens for the MF and the 1 D cases,the transition noise (given by the maximum of the susceptibility) decays as a power law withthe system size N as η c ∼ N − α [square symbols in Fig. 2(a)], with a best power-law fittingexponent α ≃ . ± .
01. Even though this exponent is different from the MF and 1 D . η c = 0 in the N → ∞ limit. The peak of the susceptibility χ max seems to increaselinearly with N as in MF and 1 D , with a best-fitting exponent γ ≃ . ± .
01 [Fig. 2(b)].Based on these results, we plot h ϕ i and χ/N as a function of η N . in Figs. 4(a) and (b),respectively, where we observe a good collapse of curves for different system sizes. For thesake of simple comparison, we have also collapsed the same data using the MF scaling vari-able η N instead, and found that the data points do not fall into a single curve but theylook rather disperse (plot not shown). Therefore, we conclude that the 2 D case appears tohave its own scaling variable, which is proportional to a non-trivial power of N .A more appealing scaling variable can be obtained from known results of the behaviorof the surface-reaction model introduced by Fichthorn, Gulari and Ziff (FGZ) in [15] andstudied later in [17–19], akin to the two-state NVM [9, 10], which be believe it belongs tothe same class of the MSVM for flocking studied here. In the FGZ model, N particles oftwo different species A and B occupy the sites of a square lattice that simulates a catalyticsubstrate. In a single step, two possible reaction events can take place. (i) With probability p d one particle is chosen at random and desorbs, and the vacant site is immediately occupiedwith a particle of species A or B with the same probability 1 /
2. This corresponds to theexternal noise of the NVM that switches the state of a particle with probability p d /
2. (ii)With the complementary probability 1 − p d a pair of neighboring sites is chosen at randomand, if it is an AB pair, both particles desorb and are replaced with an AA or a BB pair,equiprobably. This represents the copy dynamics of the NVM. The control parameter of theFGZ model is the desorption probability p d (noise amplitude). The steady state at p d = 0is a poisoned absorbing state with a coverage equals to 1 . A or B ),which is analogous to complete order for η = 0 in the FVM. For p d > .
0, depending on the values of p d and N , similarly to the partial order in theFVM.It turns out that the scaling variables that we obtained for the FVM in MF and 1 D are the same as those of the FGZ model, by making a suitable change of variables. In theFGZ model they obtained analytically the scaling variables X MF = p d N in MF ( d = 3) and X = p / d N in 1 D [17, 18], while in the FVM are x MF = η N in MF and x = ηN in1 D . Thus, the scaling variables of both models match if we make the substitution p d → η .Finally, 2 D is a marginal dimension in the FGZ model, with a scaling variable similar to12hat of MF with a logarithmic correction in p d , that is, X = p d ln(1 /p d ) N . Therefore, forthe FVM in 2 D we expect a scaling variable x = ˆ η N , where we have defined an effectivenoise amplitude ˆ η ≡ η √− ln η .Panels (c) and (d) of Fig.4 show h ϕ i and χ/N plotted as a function of the scaling variable x , where we see a good data collapse. Even though this collapse with x seems as good asthat with η N . [panels (a) and (b)], the advantage of using x = ˆ η N is two fold: we arenot fitting any parameter and we recover the linear dependence on N found in MF and 1 D scaling variables x MF and x . Additionally, Fig. 4(c) shows that the order parameter scalesas h ϕ i ∼ x − / for x & D . In comparison, h ϕ i decays as a power law of η N . with a non-trivial exponent − .
45 [dashed line in Fig. 4(a)]. Finally, from the scaling relation for the susceptibility χ = N g (ˆ η N ) , (11)where g is a smooth function of x [see Fig. 4(d)], we obtain the effective transition noiseˆ η c ≃ B N − / (12)in 2 D , where B is a proportionality constant. Interestingly, the exponent ˆ α ≡ / D case agrees with that of the MF case [Eq. (7)]. In the inset of Fig. 2(a) we compare theeffective transition noise ˆ η c = η c p − ln η c (13)from simulations (circles) with the approximate scaling given by Eq. (12) (upper solid line),with a best fitting constant B = 1 . ± .
04. The good agreement between simulations andEq. (12) shows that the transformation of the original noise η c into the effective noise ˆ η c leads to power-law decay in N with a MF exponent ˆ α = 1 / η c ≃ A N − α as found numerically [squares in Fig. 2(a)], wherethe exponent α depends on N and A ≃ .
96 is a constant obtained from the fitting of thedata. Starting from the relation Eq. (13) between the effective and original noise, we applythe logarithm at both sides and replace ln ˆ η c by ln B − ln N from Eq. (12) and ln η c byln A − α ln N , which leads to(2 α −
1) ln N − A/B ) − ln (ln N ) − ln α = 0 , (14)13fter doing some algebra and rearranging terms. We have also considered the expansionln ( − ln A + α ln N ) = ln α +ln(ln N )+ O [(ln A ) / ( α ln N )] to zero-th order in (ln A ) / ( α ln N ) ≪
1, as we can check for N & , A ≃ .
96, and α & /
2. Then, as we expect α to be similarto 1 / α ≃ .
56 from the fitting of the 2 D data in Fig. 2), we replace ln α in Eq. (14)by the Taylor expansion ln α ≃ ln(1 /
2) + 2 α −
1, and solve for α . We finally arrive at thefollowing approximate scaling for the transition noise with N : η c ≃ A N − α , with (15a) α ( N ) ≃
12 + ln h AB (cid:0) ln N (cid:1) / i ln N − η c ≃ . N ln N ) − / for N ≫ , (15c)using B ≃ .
3. In the inset of Fig. 2 we can see that the approximation from Eq. (15c) (bot-tom solid curve) reproduces very well the behavior of η c vs N from simulations (squares).The second term in the exponent α ( N ) [Eq. (15b)] leads to a very slow curvature in log-logscale with an effective exponent α ≃ .
56 in the shown range of N , which approaches veryslowly to the value 1 / N increases. Finally, from Eq. (15c) we can see that the transitionpoint η c vanishes in the N → ∞ limit.Summarizing the results of this section, the static version of the FVM in one and two–dimensional lattices exhibits an order-disorder transition at zero noise in the thermodynamiclimit. V. DYNAMIC CASE v > IN TWO DIMENSIONS
When particles are allowed to move over the space, their speed v becomes a relevantparameter that drastically changes the behavior of the system respect to the static caseanalyzed in section IV, as we shall see below. Simulations were done on a two–dimensionalcontinuous space (square box) using the parallel dynamics defined in section II.In Fig. 5 we plot the susceptibility χ and the order parameter h ϕ i (inset) vs noise am-plitude η for speeds v = 0 . v = 1 . D and 2 D static cases studied previously where h ϕ i decays monotonically with η , and χ exhibits a maximum at a value η c that decreases with N , as we can clearly see for v = 0 .
1. However, an inspection of the v = 1 . -3 -2 -1 η -1 < ϕ > -3 -3 -2 -2 -1 -1 η -3 -2 -1 χ
250 5001000200040008000 N (a) -2 -1 η -2 < ϕ > x -2 x -2 x -2 x -2 x -1 x -1 η -1 χ (b) FIG. 5. Results of the dynamic version of the FVM in a two–dimensional continuous space withparticles’ speed v = 0 . v = 1 . χ vs noiseamplitude η for the system sizes indicated in the legend. The insets show the average of the orderparameter h ϕ i vs η . Vertical dashed lines indicate the estimated location of the transition noise η c (maximum of χ ). η c appears to decrease and saturate at a minimum value η c ≃ .
05 as N increases, unlike inMF and the static cases where η c vanishes with N . Also, if we compare the level of order h ϕ i and its fluctuations χ for the two speeds, we can see a larger order with smaller fluctuationsfor the largest speed v = 1 .
0, suggesting that the speed has an ordering effect.To look at this in more detail, we plot in Fig. 6 the transition noise η c ( v, N ) vs the systemsize N for different speeds. Indeed, for a given speed v & .
2, we can see that η c exhibitsa decay similar to a power law for small values of N , and saturates at a minimum value η c ( v, ∞ ) > N , which decreases as v decreases. We also plot for comparison thetransition noise η c ( N ) for the static case v = 0 in two–dimensional lattices (empty circles).For the sake of clarity, the dashed line has been shifted in the y -axis to match the estimatedasymptotic behavior of η c ( v, N ) in the zero speed limit v →
0, as we do not expect η c ( N )and η c (0 , N ) to be exactly the same. This is because some macroscopic magnitudes of thedynamic model ( h ϕ i , χ and η c ) depend on other variables besides v and N , such as thedensity of particles ρ .The numerical results described above show that, in the thermodynamic limit, there is15 N -3 -2 -1 η c v = 1.00v = 0.75v = 0.50v = 0.35v = 0.20v = 0.10 v -2 -1 η c (v, ∞ ) η c (v, ∞ ) ^ (a) -1 v N -1 η c v - N -2 -1 η c (b) ^ FIG. 6. (a) Transition noise η c vs system size N for the speeds v indicated in the legend. Theempty circles correspond to the two–dimensional static case ( v = 0) on square lattices. Thehorizontal dashed lines indicate the asymptotic values η c ( v, ∞ ) for large N . Inset: transition noise η c ( v, ∞ ) vs v (circles) obtained from the main panel, and effective transition noise ˆ η c ( v, ∞ ) vs v (squares). The straight line is best power-law fit C v β of ˆ η c ( v, ∞ ) for v ≤ .
75, with resultingconstant C = 0 . ± .
01 and exponent 1 . ± .
02. (b) Collapse of the curves for the differentspeeds of panel (a) by means of ˆ η c ( v, N ). The exponents z = 2 and β = 1 in the x and y –axis,respectively, correspond to the scaling Eq. (22). The dashed line with slope − / v N .
2. The inset shows η c vs N for v = 0 .
1. The dashed line has slope − / an order-disorder transition at a finite noise amplitude η c > v . To study this transition in more detail, we investigate below the scaling behavior of η c with the speed and the system size.Since we have learned in section IV that working with an effective noise ˆ η in 2 D latticesleads to scalings with simple MF exponents, it seems reasonable to explore the data of Fig. 6for an effective transition noiseˆ η c ( v, N ) ≡ η c ( v, N ) p − ln η c ( v, N ) , (16)which incorporates a correction factor √− ln η c to the original noise η c . The approximatepower-law decay of η c for small N and its saturation for large N [Fig. 6(a)] suggests thatthe scaling behavior of ˆ η c ( v, N ) could be described by the following standard Family-Vicsek16unction with two independent exponents β and z [30]:ˆ η c ( v, N ) ∼ v β f ( v z N ) , (17)where f is a scaling function with the asymptotic properties f ( x ) ∼ x − α for x ≪ , constant for x ≫ . (18)We can check that Eq. (17) exhibits the two limiting behaviorsˆ η c ( v, N → ∞ ) ∼ v β (19)in the thermodynamic limit, and ˆ η c ( v → , N ) ∼ N − α (20)in the zero speed limit, where the exponent α satisfies the relation β = z α. (21)By means of the scaling relation Eq. (17) we can collapse the data points of Fig. 6 into asingle curve. For that, we first estimate the exponents β , α and z . From the plot ˆ η c ( v, ∞ )vs v in the inset of Fig. 6 (squares) we find the best power-law fitting C v β (straight line),where C = 0 . ± .
01 and β = 1 . ± .
02. Then, in the zero speed limit we assume that α takes the value α = α = 1 / D static case, and thus we obtain z = 2 . ± . η c ( v, N ) ∼ v f (cid:0) v N (cid:1) , (22)with f ( x ) ∼ x − / for x ≪ f ( x ) ∼ const for x ≫
1. Figure 6(b) shows a gooddata collapse obtained with the scaling Eq. (22). Remarkably, this result only required theestimation of the best fitting exponent β of the ˆ η c ( v, ∞ ) vs v data, and assuming that thescaling of the transition noise with N in the zero speed limit is the same as that of the 2 D static case.The effective transition noise given by Eq. (22) scales linearly with the speed in thethermodynamic limit, ˆ η c ( v, ∞ ) ≃ C v, (23)17here C = 0 .
095 is the best fitting constant for low speeds v . .
75 [straight line in theinset of Fig. 6(a)]. An approximate power-law scaling η c ( v, ∞ ) ≃ D v β for the originalnoise can be obtained by following the same approach described in section IV to obtainthe scaling of η c with N [Eq. (15)]. For that, we start from the relation between ˆ η c and η c in logarithmic scale ln ˆ η c = ln η c + (1 /
2) ln( − ln η c ) and replace ln ˆ η c by ln C + ln v [Eq. (23)] and ln η c by ln D + β ln v . After rearranging terms and making the approximationln ( − ln D − β ln v ) ≃ ln β + ln( − ln v ) to zero-th order in (ln D ) / ( β ln v ) < β −
1) ln v − C/D ) + ln( − ln v ) + ln β = 0 . (24)As we expect β to be similar to 1 . β ≃ β − β . We finally obtain the followingapproximate expressions for the transition noise: η c ( v, ∞ ) ≃ D v β , with (25a) β ( v ) ≃ h CD ( − ln v ) − / i ln v + 1 / η c ( v, ∞ ) ≃ C v ( − ln v ) − / for v ≪ . (25c)The second term in Eq. (25b) gives an effective exponent β ( v ) & v decreases. Equations (25) are only valid for lowspeeds due to the fact that the approximate expansion of the logarithm that we used inEq. (24) assumes that (ln D ) / ( β ln v ) <
1, which happens for v . .
08. Unfortunately, thecomparison of Eq. (25) with simulation results is not possible because to obtain the numericalvalue η c ( v, ∞ ) for speeds v < . η c ( v, N ) ∼ N − / for v N ≪ , (26)which is confirmed in Fig. 6(b), where the collapsed data exhibits an approximate powerlaw decay with exponent − / v N .
2, denoted by the dashed line. Finally, in theinset of Fig. 6(b) we compare the curve η c vs N for the lowest speed v = 0 . N − / scaling (dashed line). A good agreement is observed only at intermediate values of N , whilefor small or large sizes a deviation from the slope − / N is due to the absence of the logarithmic correction √− ln η c η c decreases, while for large N we expect that η c reachesa saturation at a minimum value η c (0 . , ∞ ) >
0. This asymptotic value of η c (0 . , N ) isreached for system sizes outside the shown range and, in general, the approximate systemsize from where we start to see a plateau in η c seems to diverge as v approaches zero [seeFig. 6(a)]. An insight into this can be given in terms of the crossover size N cross that separatesthe two limiting behaviors of η c ( v, N ) for small and large N . For N ≪ N cross the effectivetransition noise decays with N as ˆ η c ∼ N − / , while for N ≫ N cross is ˆ η c ∼ v . At thecrossover size, these two limiting scalings should match, leading to N cross ∼ v − . This simplerelation shows that, as v approaches zero, the crossover size diverges very fast, and so weneed to run simulations in very large systems to observe the asymptotic value of η c ( v, N ).In summary, we showed in this section that the FVM in a 2 D continuous space exhibitsand order-disorder phase transition at a finite noise amplitude η c > v of particles. For low speeds, η c is linear in v with a logarithmic correction thatleads to an effective power law with a v –dependent exponent slightly larger than 1. Thus,the transition at a finite noise η c > VI. SUMMARY AND CONCLUSIONS
We studied a model for the flocking dynamics of self-propelled particles with pairwisecopying interactions and noise. This model can be considered as a version of the noisy votermodel with infinite number of angular states, which also incorporates the motion of particlesover the space. We focused on the ordering properties of the system by exploring the orderparameter ϕ that measures the global level of alignment of particles. We found that thesystem undergoes a transition as the noise amplitude η overcomes a threshold η c , from anordered phase for η < η c where a fraction of particles are aligned and thus ϕ >
0, to adisordered phase for η > η c characterized by each particle moving in a random direction,leading to ϕ = 0. We performed a numerical analysis to investigate how the speed ofparticles, the space and its dimension affect the order-disorder phase transition. We startedby the simplest case of all-to-all interactions or infinite dimension or MF, followed by thestatic case of fixed particles on one and two–dimensional square lattices, and ending withthe dynamic case of particles moving on a bounded continuous two–dimensional space. The19ransition point η c was determined by the location of the peak of the susceptibility, whichdepends on the system size N . By doing suitable finite size scaling analysis we were able toinfer the scaling behavior of the relevant magnitudes in the thermodynamic limit, includingthe transition noise.In the MF case we showed that the transition noise vanishes with N as η MF c ∼ N − / ,which is related to known analytical MF results of the MSVM. In the static case ( v = 0)we found the scalings η c ∼ N − in 1 D and ˆ η c ∼ N − / in 2 D , where ˆ η c = η c p − ln η c is an effective noise amplitude. This effective noise with a logarithmic correction in η c wasfound by drawing an analogy between our FVM and the FGZ model for catalytic reactionswith desorption probability p d , and making the transformation p d → η . Our scaling resultson MF and lattices are compatible with those predicted theoretically for the FGZ model,which is a version of the noisy two-state voter model.We therefore conclude that, in MF and 1 D and 2 D static cases, the FVM displays anorder-disorder transition at zero noise in the thermodynamic limit. This result means thatany finite noise suppresses completely any level of order in the thermodynamic limit. Thatis, even a tiny amount of noise is enough to bring the system to complete disorder.The behavior of the model in the dynamic case, where particles move at a finite speed v > D box, is very different to that of the MF and static cases. We observed that,for a fixed density of particles ρ = 0 . η >
0, increasing the speed leads toa larger value of ϕ with smaller fluctuations (smaller susceptibility χ ), eventually inducing astationary state of collective order for high enough speeds. We understand that this orderingeffect produced by particles’ motion is analogous to that found in Vicsek type models and, asa consequence, the system exhibits an ordered phase below a finite transition noise amplitude η c ( v ) > η c = η c √− ln η c with v and N is well described by a scaling function with two simpleexponents. On the one hand, this leads to the scaling behavior ˆ η c ∼ N − / for v N ≪ D static case, and also with the theory developed for thesaturation transition in the FGZ model [17, 18]. On the other hand, the effective noisereaches an asymptotic value as N increases, which behaves as ˆ η c ∼ v in the N → ∞ limit.This results in a transition noise with a superlinear dependence on the speed of the form η c ∼ v ( − ln v ) − / for v ≪
1, in the thermodynamic limit. For the sake of comparison, itwas recently found that in the Vicsek model the transition noise scales as η c ∼ v . in the20ow density and low speed regime [26]. We also note that the transition noise for a givenspeed and density ρ = 0 . D spacewith noisy voter interactions exhibits an order-disorder transition at a finite noise amplitude η c proportional to the speed of particles. This is a surprising result within the literature ofthe voter model, as it is known that adding an external noise to the copying dynamics of themodel wipes up collective order in the thermodynamic limit, and in this article we showedthat order can indeed be sustained by particles’ motion.It seems that the effect of motion is to correlate distant particles generating a state ofglobal order, as it happens in the Vicsek model. Thus, it might be interesting to study thecorrelations between particles’ velocities and positions in order to understand the mecha-nisms that lead to flocking in the model. We also note that the MF approximation, whichpredicts a transition at zero noise, fails for the full version of the FVM with particles movingat a finite speed, showing the importance of taking into account the space and motion ofparticles in real life situations, as it happens for instance in the recent experiments with fish[6] described in section I. It would be worthwhile to develop a mathematical description ofthe FVM that goes beyond MF and accounts for correlations between particles, which couldcorrectly capture the ordering effect of motion. Finally, within the context of the experi-ments in [6], the results we obtained in the present article suggests that a group of fish couldeventually reach an asymptotic polarized state when the group size increases, depending onthe relation between the amplitude of the spontaneous directional change (noise) of fish andtheir speed. ACKNOWLEDGMENTS
We acknowledge financial support from CONICET (PIP 11220150100039CO) and (PIP0443/2014). We also acknowledge support from Agencia Nacional de Promoci´on Cient´ıficay Tecnol´ogica (PICT-2015-3628) and (PICT 2016 Nro 201-0215). [1] T. Vicsek, and A. Zafiris,
Phys. Rep. , 71 (2012).
2] M. C. Marchetti, J. F. Joanny, S. Ramaswamy, T. B. Liverpool, J. Prost, Madan Rao, and R.Aditi Simha,
Rev. Mod. Phys , , 1143, (2013).[3] A. M. Menzel, Phys. Rep. , 1 (2015).[4] T. Vicsek, A. Czir´ok, E. Ben-Jacob, I. Cohen, and O. Shochet,
Phys. Rev. Lett. , 1226(1995).[5] J. Toner, and Y. Tu, Phys. Rev. Lett. 75, 4326 (1995).[6] Jitesh Jhawar, Richard G. Morris, U. R. Amith-kumar, M. Danny Raj, Tim Rogers, Harikr-ishnan Rajendran, Vishwesha Guttal, Nat. Phys. , 488–493 (2020).[7] Gabriel Baglietto and Federico Vazquez, J. Stat. Mech. (2018) 033403.[8] Federico Vazquez, Ernesto S. Loscar, and Gabriel Baglietto. Phys. Rev. E , 042301 (2019).[9] A. Kirman, Quarterly J. Econ. 108, 137 (1993).[10] B. L. Granovsky and N. Madras, Stochastic Process. Applicat. 55, 23 (1995).[11] P. Clifford, and A. Sudbury, Biometrika , 581 (1973).[12] R. Holley and T. M. Liggett, Ann. Probab. , 195 (1975).[13] M. Henkel, H. Hinrichsen, and S. L¨ubeck, Non-Equilibrium Phase Transitions, Volume I:Absorbing Phase Transitions , Springer (2008).[14] K. Fichthorn, E. Gulari and R. Ziff, Chemical Engineering Science , 1411 (1989).[15] K. Fichthorn, E. Gulari and R. Ziff, Phys. Rev. Lett. , 1527 (1989).[16] D. Considine, S. Redner, and H. Takayasu, Phys. Rev. Lett. 63, 2857 (1989).[17] E. Clement, P. Leroux-Hugon, and L.M. Sander, Phys.Rev.Lett. , 1661 (1991).[18] E. Clement, P. Leroux-Hugon, and L.M. Sander, Journal o f Statistical Physics, , 925(1991).[19] C. Flament, E Clment, P. Leroux Hugon, and L M Sander Phys. A: Math. Gen. , L1311-Ll322 (1992).[20] A. Carro, R. Toral, and M. San Miguel, Sci. Rep. , 24775 (2016).[21] A. F. Peralta, A. Carro, M. S. Miguel, and R. Toral, New J. Phys. , 103045 (2018).[22] A. F. Peralta, A. Carro, M. San Miguel, and R. Toral, Chaos , 075516 (2018).[23] Francisco Herrer´ıas-Azcu´e and Tobias Galla, Phys. Rev. E , 022304 (2019).[24] Ricardo Martinez-Garcia, Crist´obal L´opez, Federico Vazquez, arXiv:2011.07982 (2020).[25] N. D. Mermin and H. Wagner, Phys. Rev. Lett., 17, 1133 (1966).
26] M. Leticia Rubio Puzzo, Andr´es De Virgiliis, and Tom´as S. Grigera, Phys. Rev. E , 052602(2019).[27] R. A. Blythe and A. J. McKane, J. Stat. Mech. p. P07018 (2007).[28] M. Starnini, A. Baronchelli, and R. Pastor-Satorras, J. Stat. Mech. , 10027 (2012).[29] W. Pickering and C. Lim, Phys. Rev. E , 032318 (2016).[30] Fereydoon Family and Tam´as Vicsek, J. Phys. A: Math. Gen. , L75-L81 (1985)., L75-L81 (1985).