Critical behavior and synchronization of discrete stochastic phase-coupled oscillators

Kevin Wood,1,2 C. Van den Broeck,3 R. Kawai,4 and Katja Lindenberg1

1Department of Chemistry and Biochemistry and Institute for Nonlinear Science, University of California at San Diego,9500 Gilman Drive, La Jolla, California 92093-0340, USA

2Department of Physics, University of California at San Diego, 9500 Gilman Drive, La Jolla, California 92093-0340, USA3Hasselt University, Diepenbeek, B-3590, Belgium

4Department of Physics, University of Alabama at Birmingham, Birmingham, Alabama 35294, USA�Received 2 June 2006; published 14 September 2006�

Synchronization of stochastic phase-coupled oscillators is known to occur but difficult to characterize be-cause sufficiently complete analytic work is not yet within our reach, and thorough numerical descriptionusually defies all resources. We present a discrete model that is sufficiently simple to be characterized inmeaningful detail. In the mean-field limit, the model exhibits a supercritical Hopf bifurcation and globaloscillatory behavior as coupling crosses a critical value. When coupling between units is strictly local, themodel undergoes a continuous phase transition that we characterize numerically using finite-size scalinganalysis. In particular, we explicitly rule out multistability and show that the onset of global synchrony ismarked by signatures of the XY universality class. Our numerical results cover dimensions d=2, 3, 4, and 5 andlead to the appropriate XY classical exponents � and �, a lower critical dimension dlc=2, and an upper criticaldimension duc=4.

DOI: 10.1103/PhysRevE.74.031113 PACS number�s�: 64.60.Ht, 05.45.Xt, 89.75.�k

I. INTRODUCTION

The role of dissipative structures and self-organization insystems far from equilibrium in the description of real andobservable physical phenomena has been undisputed sincethe experiments with the Belusov-Zhabotinsky reactions inthe early 1960s. The breaking of time translational symmetryhas since become a central and typical theme in the analysisof nonlinear nonequilibrium systems. It is somewhat surpris-ing that in the later studies of spatially distributed systems,most of the interest shifted to pattern-forming instabilities,and little attention was devoted to the phenomenon of bulkoscillation and the required spatial frequency and phase syn-chronization, especially in view of the intense interest gen-erated in the scientific and even broader community by theemergence of phase synchronization in populations of glo-bally coupled phase oscillators �1�. The synchronous firing offireflies is one of the most visible and spectacular examplesof phase synchronization. Because intrinsically oscillatingunits with slightly different eigenfrequencies underlie themacroscopic behavior of an extensive range of biological,chemical, and physical systems, a great deal of literature hasfocused on the mathematical principles governing the com-petition between individual oscillatory tendencies and syn-chronous cooperation �2–4�. While most studies have fo-cused on globally coupled units, leading to a matureunderstanding of the mean-field behavior of several models,relatively little work has examined populations of oscillatorsin the locally coupled regime �5–7�. In fact, models of lo-cally coupled oscillators typically involve a prohibitivelylarge collection of interdependent nonlinear differentialequations, thus preventing any extensive characterization ofthe phase transition to phase synchrony. Further inclusion ofstochastic fluctuations in such models typically renders themcomputationally and analytically intractable for even a mod-est number of units. As a result, the description of emergent

synchrony has largely been limited to small-scale and/or glo-bally coupled deterministic systems �8,9�, despite the factthat the dynamics of the physical systems in question likelyreflect a combination of finite-range forces and stochasticity.Two recent studies by Risler et al. �10,11� represent notableexceptions to this trend. Using an elegant renormalization-group approach, they provide analytical evidence that iden-tical locally coupled noisy oscillators belong to the XY uni-versality class, though to date there had been no empiricalverification, numerical or otherwise, of their predictions.

The difficulty with existing models of locally coupled os-cillators is that each is typically described by a nonlineardifferential equation, and the resulting systems of coupledequations are computationally extravagant, especially whenstochastic components are also included. Here we introducea far more tractable model consisting of identical and dis-crete phase-coupled oscillators whose simple structure ren-ders it amenable to extensive numerical study. The use ofsuch minimal models is common in statistical physics, wheremicroscopic details can often be disregarded in favor of phe-nomenological macroscopic variables. As Landau theory�12� reminds us, macroscopically observable changes �thosethat occur on length and time scales encompassing a mag-nificently large number of degrees of freedom� occur withoutreference to microscopic specifics. In a sense, the distin-guishing features of even highly diverse systems become ir-relevant for the description of cooperative behavior at thelevel of a phase transition; instead, the underlying statisticalsimilarities give rise to classes of universal behavior whosemembers differ greatly at the microscopic level. In the spiritof this universality, simple toy models have been devised inthe hopes of capturing the essential qualitative features ofphase transitions without concern for the microscopic struc-ture of the problem. With this in mind, we construct thesimplest model that exhibits global phase synchrony andcontains the physical ingredients listed above, namely sto-chastic variation within individual units and short-ranged in-

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teractions �13�. The simplicity of the model allows for rela-tively fast numerical simulation and thus an extensivedescription of the phase transition in question. An abbrevi-ated version of our principal results can be found in �13�.There we characterized the universality class of the transi-tion, including the critical exponents and the lower and uppercritical dimensions. Here we present considerably more de-tail as well as additional results to support our characteriza-tion.

The organization of the paper is as follows. In Sec. II, weintroduce our description of a single unit as well as the cou-pling scheme between units. Section III presents the linearstability analysis of the mean-field limit, and Sec. IV con-tains the finite-size scaling analysis that unveils the criticalbehavior of the locally coupled model. We conclude with asummary in Sec. V.

II. THREE-STATE MODEL

Our starting point is a three-state unit �14� governed bytransition rates g, as shown in Fig. 1. We interpret the statedesignation as a generalized �discrete� phase, and the transi-tions between states, which we construct to be unidirectional,as a phase change and thus an oscillation of sorts. The prob-ability of going from the current state i to state i+1 in aninfinitesimal time dt is gdt, with i=1,2 ,3 modulo 3. For asingle unit, g is simply a constant that sets the oscillator’sintrinsic frequency; for many units coupled together, we willallow g to depend on the neighboring units in the spatialgrid, thereby coupling neighboring phases. The choice ofcoupling, specified below, is not unique.

For a single unit we write the linear evolution equation

�

�tP�t� = MP�t� , �1�

where

P�t� = �P1�t�P2�t�P3�t�

� , �2�

Pi�t� is the probability of being in state i at time t, and

M = �− g 0 g

g − g 0

0 g − g� . �3�

The system clearly reaches a steady state for P1*= P2

*= P3*

=1/3. The transitions 1→2, 2→3, 3→1 occur with a roughperiodicity determined by g. The time evolution of oursimple model thus qualitatively resembles that of thediscretized phase of a generic noisy oscillator.

We are interested in the behavior that emerges when in-dividual units are coupled to one another by allowing thetransition probability of a given unit to depend on the statesof the unit’s nearest neighbors in the spatial grid. The phaseat a given site is compared with those of its neighbors, andthe phase of the given site is adjusted so as to facilitate phasecoherence. The expectation is to capture the physical natureof synchronization. It is further expected that within certainrestrictions �e.g., the coupling must surely be nonlinear�, thespecific nature of the coupling is not important �in otherwords, we expect universality� so long as we ultimatelyobserve a transition to global synchrony at some finite valueof the coupling parameter. We settle upon a particular expo-nential form below. As we shall see, linear stability analysisfor this choice confirms the existence of a Hopf bifurcationin the mean-field limit. Upon further analysis, the Hopfbifurcation is shown to be supercritical.

More concretely, we specify that each unit may transitionto the state ahead or remain in its current state depending onthe states of its nearest neighbors. For unit �, which we taketo be in state i, we choose the transition rate to state j asfollows:

gij = g exp�a�Nj − Ni�2d

�� j,i+1, �4�

where a is the coupling parameter, � is the Kronecker delta,Nk is the number of nearest neighbors in state k, and 2d is thetotal number of nearest neighbors in d dimensions. The tran-sition rate is thus determined by the number of nearest neigh-bors of unit � that are one state ahead and in the same stateas unit �. Table I shows the explicit transition rates in onedimension. While these rates are somewhat distorted by theirassumed independence of the number of nearest neighbors instate i−1 �e.g., in one dimension the transition rate from state

FIG. 1. Single three-state unit with generic transition rates g.From �13�.

TABLE I. Transition rates in one dimension.

Neighbors Transition rate

i−1, i−1 g

i−1, i g exp�−a /2�i−1, i+1 g exp�a /2�i, i−1 g exp�−a /2�i, i g exp�−a�i, i+1 g

i+1, i−1 g exp�a /2�i+1, i g

i+1, i+1 g exp�a�

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i to state i+1 is the same if both nearest neighbors are instate i−1 and if one is in state i and the other in i+1�, theform �4� is simplified by this assumption and, as we shallsee, does lead to synchronization. Note also that an equallysimple model might posit a coupling that depends on Ni−1,the number of units “behind” the unit in question, rather thanNi+1, or a more complex model could be constructed thatdepends on both. We settle on our choice �4� because thephase transition we seek occurs for a smaller value of thecoupling constant a, and therefore numerical simulations canbe run with larger time steps. We have also explored cou-pling with a sigmoidal function, which shows similar quali-tative behavior �that is, a supercritical Hopf bifurcation in themean-field limit and an apparent synchronization transitionin locally coupled simulations�, but this again occurs for ahigher value of a, making simulations more cumbersome.We stress again that universality suggests that such micro-scopic details should not substantially alter the qualitativepicture of the phase transition as long as the coupling issufficiently nonlinear and favors synchronization.

III. MEAN-FIELD THEORY

To test for the emergence of global synchrony, we firstconsider a mean-field version of the model, that is, one inwhich each unit is coupled to all other units. In the large-Nlimit with all-to-all coupling, we write

gij = g exp�a�Pj − Pi��� j,i+1, �5�

where Pk is the �ensemble� probability of being in state k.Note that with all-to-all coupling, gij does not depend on thelocation of the unit within the lattice. Note also that there isan inherent assumption that we can replace Nk /N with Pk,that is, we are assuming that N, the total number of units, islarge enough that Nk /N serves as a good estimation of theensemble probability Pk. With this simplification, we arriveat an equation for the mean field P,

�

�tP�t� = M�P�t��P�t� , �6�

where

M�P�t�� = �− g12 0 g31

g12 − g23 0

0 g23 − g31� . �7�

We have explicitly noted the dependence of M on P�t� sinceeach of the matrix elements gij depends on the evolvingprobabilities. Equation �6� is thus a highly nonlinearequation.

The normalization condition P1+ P2+ P3=1 allows us toeliminate P3 and obtain a closed set of equations for P1 andP2. We can further characterize the mean-field solutions us-ing standard linear stability analysis. Specifically, we linear-ize about the fixed point �P1

* , P2*�= �1/3 ,1 /3� and obtain the

Jacobian J evaluated at �P1* , P2

*�,

J = ag − 2g − g

g ag − g . �8�

The eigenvalues of J characterize the fixed point, and theyare given by

�± =g

2�2a − 3 ± i�3� . �9�

Both cross the imaginary axis at a=1.5, indicative of a Hopfbifurcation at this value. Hence, as a increases, the meanfield undergoes a qualitative change at a=1.5 from disorder�P1= P2= P3� to global oscillations, and the desired globalsynchrony emerges.

The predictions of the linearization can be verified bynumerically solving the mean-field equations �6�. In turn,these solutions agree well with direct simulations of the mul-tiple unit model characterized by Eq. �4� if we consider all-to-all coupling rather than merely nearest-neighbor coupling�Fig. 2�. As such, the mean-field equations accurately capturethe behavior of the nearest-neighbor model in the high �spa-tial� dimensional limit.

Furthermore, the Hopf bifurcation seen in our mean-fieldmodel can be shown to be supercritical. Such an analyticalargument is formally related to the structure of the normalform for the Hopf bifurcation. Practically speaking, one mustconsider the sign of the first Lyapunov coefficient at the bi-furcation point �ac=1.5�. Following �15�, we transform ourtwo-dimensional nonlinear equation �6� to a single equationfor the complex variable z valid for small �=a−ac. The formof the equation is given by

z = ����z + f�z,z†,�� , �10�

where f�z ,z† ,�� is an O��z�2� smooth function of z, z†, and �,and ���� is an �-dependent eigenvalue of the linearizedJacobian �8� given above. We achieve such a transformationby first finding complex eigenvectors p and q given by

FIG. 2. �Color online� Simulations with 5000 globally coupledunits �bottom panel� agree well with the numerical solution of themean-field equations �top panel�. As predicted by linearization, aHopf bifurcation occurs near a=1.5.

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J�0�q = ��0�q, J�0�Tp = ��0�†p , �11�

with J�0� the Jacobian evaluated at a=ac=1.5. One then nor-malizes p ,q�, where brackets in this context represent thestandard complex scalar product. Explicitly, we obtain

p = �1 + i�3�

�3 − i�3�,1 ,

q = 1

2�− 1 + i�3�,1 . �12�

An equation for z of the desired form �10� is formallyobtained at �=0 as

z = ��0�z + p,F�zq + z†q†,0�� , �13�

where F�x ,�� is related to our original dynamical system,i.e.,

�

�tP�t� = J���P�t� + F�P�t�,�� . �14�

From this, we may obtain the first Lyapunov coefficient L1 as

L1 =1

2�02 Re�if20f11 + �0f21� , �15�

where

�02 = �det J�a=ac

= �3 − 3ac + 3ac2� , �16�

and with f ij given by the formal Taylor expansion of f ,

f�z,z†,0� = p,F�zq + z†q†,0�� = �k+l2

1

k!l!fklz

k�z†�l.

�17�

Specifically, for �=0 we write f as an explicit function of acand the complex numbers b�3i+�3 and c� i+�3,

f�z,z†,0� =1

3b*�exp−1

2iac�b*z − �3i − 2�3�z†�

�− 2i�1 + 3z + 3z†�exp3

2i�3acz

† + �c − 6iz

− 3c*z†�exp�3

2ac�ic*z + z†�� − �6iz + c* − 3cz†�

exp�3acz + 1 +�3

2iz†��� , �18�

and upon Taylor expanding we get

L1�ac� =

Re�3ac2

b* �0�15 − 3�3 + acb*��

2�02 . �19�

As previously noted, ac=1.5, leading to

L1�1.5� = − 11.69 � 0, �20�

indicative of a supercritical Hopf bifurcation to a unique,stable limit cycle as a eclipses ac.

In what follows, we characterize the breakdown of themean-field description as spatial dimension is decreased, andcharacterize the phase transitions observed with nearest-neighbor coupling.

IV. CRITICAL BEHAVIOR OF THE LOCALLYCOUPLED MODEL

With a firm understanding of the mean-field model, wenow follow with a study of the locally coupled model. Weperform simulations in continuous time on d-dimensional cu-bic lattices of various sizes. For all simulations, we imple-ment periodic boundary conditions. Time steps dt are takento be 10 to 100 times smaller than the fastest possible localaverage transition rate, that is, dt�e−a �we set g=1 in oursimulations�. This estimate is actually quite conservative,particularly because the fastest possible transition corre-sponds to a single unit in state i with all 2d nearest neighborsin state i+1, a scenario that certainly does not dominate themacroscopic dynamics. We have ascertained that differencesbetween these simulations and others run at much smallertime steps �500–1000 times smaller than e−a� are very small.All simulations were run until an apparent steady state wasreached. Furthermore, we start all simulations from randominitial conditions, and we calculate statistics based on 100independent trials. Although the simplicity of the model al-lows for efficient numerical simulation, our results neverthe-less represent a modest computational achievement; simula-tions required approximately 5 weeks on a 28-node dualprocessor cluster.

To characterize the emergence of phase synchrony, weintroduce the order parameter �8�

r = R�, R �1

N��

j=1

N

ei j� . �21�

Here is a discrete phase, taken to be 2��k−1� /3 for statek� �1,2 ,3� at site j. The brackets represent an average overtime in the steady state and an average over all independenttrials. A nonzero value of r in the thermodynamic limit sig-nifies the presence of synchrony. We also make use of thecorresponding generalized susceptibility

� = Ld� R2� − R�2� . �22�

We begin by considering the model in two spatial dimen-sions. Here, as shown in Fig. 3, we do not see the emergenceof global oscillatory behavior. Instead, we observe intermit-tent oscillations �for very large values of a� that decreasedrastically with increasing system size. In fact, as depicted inFig. 4, r approaches zero in the thermodynamic limit, evenfor very large values of a. We conclude that the phase tran-sition to synchrony cannot occur for d=2. Interestingly,snapshots of the system reveal increased spatial clustering asa is increased as well as the presence of defect structures,perhaps indicative of Kosterlitz-Thouless-type phenomena�Fig. 5� �12�. Further studies along these lines are underway.

In contrast to the d=2 case, which serves as the lowercritical dimension, a clear thermodynamic-like phase transi-tion occurs in three spatial dimensions. We see the emer-

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gence of global oscillatory behavior, which persists in thelimit of large system size, as a increases past a critical valueac �Figs. 6 and 7�. This is consistent with the predictions ofthe mean-field theory. For a�ac, r approaches zero as sys-tem size is increased, and a disordered phase persists in thethermodynamic limit. As expected, for a�ac the steady-statedynamics of P1 and P2 exhibit smooth temporal oscillations�see the lower insets in Fig. 7� similar to the mean-field casebeyond the Hopf bifurcation point. In addition, Fig. 7 showsthe behavior of the order parameter as a is increased for thelargest system studied �L=80�; the upper left inset shows the

peak in � at a=2.345±0.005, thus providing an estimate ofthe critical point ac where fluctuations are largest. Strictlyspeaking, we must extrapolate this peak to obtain a result inthe limit of infinite system size, but we see no change in thevalue ac at which the peak occurs as system size is increasedbeyond L=40, indicating that finite-size effects are small inthe determination of ac in systems beyond this size. At anyrate, such finite-size effects are within the range of our esti-mation. We tried to apply the Binder cumulant crossingmethod �16� for determining ac more precisely, but residualfinite-size effects and statistical uncertainties in the data pre-vent us from determining the crossing point with more pre-cision than that stated above. In any case, we are only inter-ested in determining the critical point with sufficientaccuracy to determine the universality class of the transition.For this, as we show below, our current estimation suffices in

FIG. 3. �Color online� Absence of synchronization in 2D. Topleft: a=1.5. Top right: a=2.5. Bottom left: a=3.5. Bottom right:a=4.5. L=100 for all plots. Even for very large values of the cou-pling, highly synchronous oscillatory behavior is not present. Asdiscussed in the text and shown in the next figure, the intermittentoscillations apparent for high values of a result from finite-sizeeffects.

FIG. 4. Log-log plot of r vs L−1 for d=2. The order parameter rtends to 0 as system size increases, verifying the absence of a tran-sition in two dimensions. Even for large values of the coupling,synchronous oscillations die away in the limit of infinite systemsize.

FIG. 5. Snapshots of the system in d=2 are shown for a=1, 2.5,4.5, and 6.5. Upon close inspection, one can discern vortexlikestructures, particularly for the higher values of a. The three shadesof gray represent units in the three possible states.

FIG. 6. Log-log plots of r vs L−1 for d=3. For a�ac, the orderparameter r approaches a finite value, even as the system size in-creases indefinitely. For a�ac, r approaches zero in the thermody-namic limit.

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three dimensions as well as in higher dimensions. In addi-tion, we note that the transition to synchrony appears to be asmooth, second-order phase transition. To rule out potentialmultistability �and thus a discontinuous first-order transi-tion�, we show histograms of r for d=4 given over all inde-pendent trials in Fig. 8. The histograms show no evidencewhatsoever of multiple peaks beyond the statistical fluctua-tions expected for the relatively small sample size, and thuswe can safely rule out a discontinuous transition, in agree-ment with the findings of the mean-field analysis. Similarlypeaked histograms are found in d=3 �less sharply peaked butdistinctly unimodal� and d=5 �more sharply peaked�.

To further characterize this transition, we use a systematicfinite-size scaling analysis. We start by assuming thestandard form for r in a finite system,

r = L−�/�F„�a − ac�L1/�… , �23�

where F�x� is a scaling function that approaches a constantas x→0. This ansatz suggests that near the critical point weshould plot rL�/� versus ��a /ac�−1�L1/�, and data from dif-ferent system sizes should collapse onto a single curve. Totest our numerical data against different universality classes,we choose the appropriate critical exponents for each, recog-nizing that there are variations in the reported values of theseexponents. For the XY universality class we use the expo-nents reported in �17� ��=0.34 and �=0.66�. For the Isingexponents we use those given in �18� ��=0.31 and �=0.64�.In Fig. 9, we see quite convincingly a collapse when expo-nents from the XY class are used. For comparison, we alsoshow the data collapse when 3D Ising exponents are used�Fig. 10�. Our data suggest that the model falls within the XYuniversality class, though the very small differences betweenXY and Ising exponents make it impossible to entirely ruleout Ising-like behavior. We should point out that while somereported values of the Ising critical exponents differ from theXY values by more than those used above, others differ byless �see �19� for an exhaustive collection of estimates�. Notethat this scaling procedure was attempted for many values acwithin the stated range of accuracy. In all cases in which adistinction could be made, the XY exponents provided abetter collapse than the corresponding Ising exponents.

To complete the analogy with the equilibrium phase tran-sition, we explore spatial correlations in d=3. Specifically,we calculate C�l�, the spatial correlation function, given by

C�l� =��j=1

N

�n=1

3

exp�i j�exp�− i j+ln�� − r2. �24�

Here j again indicates the discrete phase of the oscillator atsite j, and ln denotes the Cartesian components in the x, y,

FIG. 7. �Color online� Onset of synchronization in d=3. Globaloscillatory behavior emerges as a is increased beyond ac, as indi-cated by the increasing value of r. The system size is L=80. Upperleft inset: Fluctuations peak near the critical point, giving an esti-mation of ac=2.345±0.005. Right insets: P1 and P2 undergosmooth temporal oscillations for large a �upper right�, while a lowervalue of a decreases temporal coherence �lower left�.

FIG. 8. �Color online� Lack of multistability in d=4: Histo-grams over all independent trials show only single peaks of varyingwidths, consistent with the expectations for a second-order phasetransition.

FIG. 9. �Color online� Exponents in d=3: Data collapse of rL�/�

vs �a /ac−1�L1/�. With ac=2.345, we show the data collapse usingthe theoretical XY exponents in 3D. The collapse is excellent, sug-gesting that the model is in the XY universality class. The insetsshow a closer view near the critical point.

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and z directions at distance l from site j. The correlationfunction depends only on this distance. As seen in Fig. 11,correlations develop for values of a near the critical point,while this correlation is absent away from ac. The functionalform of C�l� as a approaches ac is similar to that seen inequilibrium transitions. Indeed, the lower inset is at the criti-cal point �a=2.345� and explicitly shows a power-law decayof the correlation function. The upper inset is far from thecritical point �a=1.8� and shows exponential decay.

In four spatial dimensions, we also see a transition tosynchrony characterized by large fluctuations at the criticalpoint. Here we estimate the transition coupling to beac=1.900±0.025 by again considering the peak in � �see

Figs. 12 and 13�. Because we expect d=4 to be the uppercritical dimension in accordance with XY/Ising behavior, weanticipate a slight breakdown of the scaling relation �23�. Analternate scaling ansatz valid at duc is given by Eq. �23� withthe transformation L→ ln�L�L1/4 �20�. A priori it is not clearhow strongly Eq. �23� will be violated in d=4, nor is it clearthat the modified ansatz will better serve our purposes; there-fore, we will use both forms of scaling in testing for themean-field exponents �=1/2 and �=1/2.

As shown in Fig. 14, the data collapse is very good withthe mean-field exponents regardless of which scaling ansatzis used. As such, our simulations suggest that d=4 serves asthe upper critical dimension; additionally, it appears that cor-rections to finite-size scaling at d=4 are not substantial,though a much more precise study would be needed toinvestigate such corrections in greater detail.

To further support the claim that duc=4, we consider thecase d=5. We see a transition to synchrony that occurs at

FIG. 10. �Color online� Exponents in d=3. Data collapse ofrL�/� vs �a /ac−1�L1/�. With ac=2.345, we show the data collapseusing theoretical Ising 3D exponents. The collapse is reasonablygood, but still poor compared with that seen with exponents fromthe XY class. Insets show a closer view near the critical point.

FIG. 11. �Color online� Spatial correlations in d=3. As a ap-proaches the critical value ac, evidence of long-range correlationsdevelops, indicative of a diverging correlation length at the criticalpoint. The lower inset shows the power-law decay of the correlationfunction at the critical point, while the upper inset shows that thecorrelation function decays exponentially far from the critical point.

FIG. 12. �Color online� Transition in d=4: The behavior of theorder parameter near the transition point is shown for various sys-tem sizes. The inset shows the generalized susceptibility, �, whichpeaks at a=1.900±0.025, giving an estimation of ac.

FIG. 13. Log-log plots of r vs L−1 for d=4. The order parameterr clearly approaches a finite, nonzero value for a�ac and ap-proaches 0 for a�ac.

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ac=1.750±0.015 �see Figs. 15 and 16�. As expected, thisvalue for ac is considerably closer than the critical couplingin four dimensions to the value ac=1.5 calculated by linearstability analysis in mean-field theory.

Finally, it is interesting to test the suggestion of Jones andYoung �20� that above the critical dimension, dduc, it isappropriate to modify the finite-size scaling ansatz �23� bythe transformation L→Ld/4. We test this suggestion ford=5. As indicated in Fig. 17, the data collapse is excellentfor both the original scaling and the modified form of theansatz. The collapse of the data with mean-field exponentsseems slightly better using the modified ansatz, though amuch more precise study would be required to accuratelycapture the form of the modified scaling in d�duc. In anycase, our data suggest that the model exhibits mean-fieldbehavior in d=5, verifying that d=4 serves as the uppercritical dimension.

V. SUMMARY

We have introduced a simple discrete model for studyingphase coherence in spatially distributed populations of noisycoupled oscillators. This model lends itself to numericalstudy even in the case of nearest-neighbor coupling becauseeach oscillator is a simple three-state system rather than oneof the usual continuum choices. The coupled system is there-fore much simpler than the usual set of coupled nonlineardifferential equations.

A mean-field treatment combined with linear stabilityanalysis shows that the globally coupled model undergoes aHopf bifurcation to macroscopic synchrony as the couplingparameter a is increased. We are able to determine the mean-field critical coupling constant analytically. For locally

FIG. 14. �Color online� Exponents in d=4: Data collapseof original ansatz �23� with mean-field exponents. Inset: Datacollapse with modified scaling ansatz r�ln�L�L1/4��/� vs��a /ac�−1��ln�L�L1/4�1/� with mean-field exponents.

FIG. 15. �Color online� Transition in d=5: The behavior of theorder parameter near the transition point is shown for various sys-tem sizes. The inset shows the generalized susceptibility, �, whichpeaks at a=1.750±0.015, giving an estimation of ac.

FIG. 16. Log-log plots of r vs L−1 in d=5. The order parameterr clearly approaches a finite, nonzero value for a�ac and ap-proaches 0 for a�ac. The value of ac appears to fall betweena=1.8 and 1.7.

FIG. 17. �Color online� Exponents in d=5: Data collapse oforiginal ansatz �23� with mean-field exponents. Inset: Data collapseof rLd�/4� vs �a /ac−1�Ld/4� with mean-field exponents. The col-lapse of the data is quite convincing when the exact mean-fieldexponents are used.

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coupled units, numerical solution of the system shows theemergence of a thermodynamic synchronous phase ford�2, indicating that the lower critical dimension is dlc=2.As d is increased, the numerically established critical valueac approaches that predicted by the mean-field treatment ofthe model. For d=3, we give strong numerical evidence thatthe model falls into the 3D XY universality class, while ford=4 the critical exponents are those predicted by mean-fieldtheory. The exponents in d=5 also take on the mean-fieldvalues, thus verifying that d=4 corresponds to the uppercritical dimension duc.

In conclusion, while nonequilibrium phase transitionshave a much wider diversity in universality classes thanequilibrium ones �21�, it is remarkable that the prototype of anonequilibrium transition, namely a phase transition thatbreaks the symmetry of translation in time, is described, atleast for the critical exponents investigated in this paper, byan equilibrium universality class. In particular, the Mermin-Wagner theorem, stating that continuous symmetries cannotbe broken in dimension 2 or lower, appears to apply. Further-more, the XY model is known to display a Kosterlitz-Thouless transition, in which, beyond a critical temperature,

vortex pairs can unbind into individual units creating long-range correlations. Preliminary results indicate that a similartransition occurs in our model. Finally, a note of cautionconcerning the discreteness of the phase is in order. We firstnote that microscopic models often feature discrete degreesof freedom. For example, our model is reminiscent of thetriangular reaction model introduced by Onsager �22�, on thebasis of which he illustrated the concept of detailed balanceas a characterization of equilibrium. Continuous phase mod-els appear in a suitable thermodynamic limit. We stress thatthe breaking of time translational symmetry can occur inde-pendently of whether the phase is a discrete or continuousvariable. It is, however, not evident whether continuous anddiscrete phase models belong to the same universality class.The results found here seem to support the latter thesis, but arenormalization calculation confirming this hypothesis wouldbe welcome.

ACKNOWLEDGMENTS

This work was partially supported by the NationalScience Foundation under Grant No. PHY-0354937.

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