Synchrony and Critical Behavior: Equilibrium Universality in Nonequilibrium Stochastic Oscillators

Transcription

1 Synhrony and Critial Behavior: Equilibrium Universality in Nonequilibrium Stohasti Osillators Kevin Wood*, C. Van den Broek^, R. Kawai** and Katja Lindenberg* * Department of Chemistry and Biohemistry, Department of Physis, and Institute for Nonlinear Siene, University of California San Diego, 9500 Oilman Drive, La Jolla, CA , USA ^Hasselt University, Diepenbeek, B-3590, Belgium "Department of Physis, University of Alabama at Birmingham, Birmingham, AL 35294, USA * Department of Chemistry and Biohemistry and Institute for Nonlinear Siene, University of California San Diego, 9500 Oilman Drive, La Jolla, CA , USA Abstrat. We review our work on a disrete model of stohasti, phase-oupled osillators that is suffiiently simple to be haraterized in omplete detail, lending insight into the universal ritial behavior of the orresponding nonequilibrium phase transition to marosopi synhrony. In the mean-field limit, the model exhibits a superritial Hopf bifuration and global osillatory behavior as oupling elipses a ritial value. The simpliity of our model allows us to perform the first detailed haraterization of stohasti phase oupled osillators in the loally oupled regime, where the model undergoes a ontinuous phase transition whih remarkably displays signatures of the XY equilibrium universality lass, verifying reent analytial preditions. Finally, we examine the effets of spatial disorder and provide analytial and numerial evidene that suh disorder does not destroy the apaity for synhronization. Keywords: stohasti phase oupled osillators, synhronization, ritial behavior, universality PACS: Ht, 05.45Xt, k Time and periodiity play a ritial role in a host of physial, biologial, and hemial systems [1, 2]. In partiular, a vast number of systems onsist of noisy individual entities haraterized by periodi (osillatory) dynamis that give rise to a potentially omplex ompetition between individual dynamis on a mirosopi sale and large sale ooperative behavior. Originating with the pioneering work of Kuramoto [3, 4], the sientifi literature is replete with studies of suh synhronization [1, 2]. The typial approah involves systems of oupled, nonlinear differential equations. As suh, work has been traditionally limited to relatively small, deterministi systems in the mean field limit [5, 6, 7, 8, 9]. While providing a mature understanding of the synhronization dynamis in oupled osillators, the numerial and analytial omplexities of these familiar models have prohibited a omplete haraterization in the large system limit. Therefore, the analogy between the ooperative behavior of these osillators with phase transitions and ritial phenomena from statistial mehanis [10, 11] has not been fully exploited. Speifially, the onset of synhronization when viewed on a long wavelength sale is haraterized by a marosopi hange in an inherently nonequilibrium symmetry: for systems below the synhronization threshold, the large-sale dynamis appears time-translationally invariant, while synhronized systems osillate olletively, breaking this time-based symmetry. With these possible analogies in mind, and with an eye towards ritial behavior, we develop a simple, disrete model of noisy phase-oupled osillators [12, 13, 14] whih displays the desired marosopi behavior while remaining suffiiently simple to be haraterized as a phase transition. Owing to the well-established notion of universality, whih renders mirosopi speifis essentially irrelevant for study of the typially long-wavelength behavior ourring near ritiality, our model of the phase transition in question must preserve the marosopi symmetry-breaking while inorporating the relevant physial details, namely, stohastiity and mirosopi periodiity. Our starting point is a three-state unit [12, 13, 14] governed by transition rates g (Fig. 1). Loosely speaking, we interpret the state designation as a generalized (disrete) phase, and the transitions between states, whih we onstrut to be unidiretional, as a phase hange and thus an osillation of sorts. The probability of going from the urrent state i to state i+ 1 in an infinitesimal time dt is gdt, with i=l,2,3 modulo 3. For an isolated unit, the transition rate is simply a onstant (g) that sets the osillator's intrinsi frequeny; for many oupled units, we will allow the transition rate to depend on the neighboring units in the spatial grid, thereby oupling neighboring phases. CP913, Nonequilibrium Statistial Mehanis and Nonlinear Physis, XV Conferene edited by O. Desalzi, O. A. Rosso, andh. A. Larrondo 2007 Amerian Institute of Physis /07/S

2 FIGURE 1. Three-state unit with generi transition rates ; FIGURE 2. Dimensionality (d) dependent phase transition. Left olumn shows r vs L on a log-log sale for d = 2,3, 4, top to bottom. For d = 2, the urves start from a = 2,0 (top) with inrements of 0.5. For d = 3, the urves start from a = (top) with inrements of (with the exeption of a = 2.350, not shown). For the d = 4 ase, the urves are a = 1.6,1.7,1.8,2.0,2.1,2.2, top to bottom. In the limit of infinite system size, a synhronous phase emerges for d > 3 for suffiiently large a. The right olumn shows rvsa and % vs a for d = 3, L = 80 (top) and d = 4, L = 16, bottom. Flutuations peak near the ritial point, giving an estimation of a = ± and a = ± for d = 3 and d = 4, respetively. For an isolated unit we write the linear evolution equation dp{t)/dt = MP{t), where the omponents P,-(? olumn vetor P(t) = (Pi(t) Pz(t) Ps(t)) T are the probabilities of being in state i at time t, and of the M 0 (1) The system reahes a steady state for Pj* = P 2 * = P 3 * = 1/3. The transitions i > i +1 (with j +1 = 1 when i = 3) our with a rough periodiity determined by g; that is, the time evolution of our simple model qualitatively resembles that of the disretized phase of a generi noisy osillator. We implement mirosopi oupling by allowing the transition probability of a given unit to depend on the states of the unit's nearest neighbors in a spatial grid. Speifially, we hoose a funtion whih ompares the phase at a given site with its neighbors, and adjusts the phase at the given site so as to failitate phase oherene. We settle on the following form of the transition rate from state i to state j: gij = gexp a{nj-ni) 2d 8 JM (2) 50

3 Here the onstant a is the oupling parameter, g is a parameter related to the intrinsi frequeny of eah osillator, and 8 is the Kroneker delta. Nk is the number of nearest neighbors in state k, and 2d is the total number of nearest neighbors in d dimensional ubi latties. While this hoie is by no means unique and these rates are somewhat distorted by their independene of the number of nearest neighbors in state i - 1, the form (2) does lead to synhronization and allows for fast numerial simulation of large latties near the ritial regime [12, 13]. To verify the emergene of global synhrony, we first onsider a mean field version of the model. In the large N limit with all-to-all oupling we write gij = gexp [a(pj - Pi)} 8jj+i. (3) Note that in the mean field limit gij does not depend on the loation of the unit within the lattie. Also, there is an inherent assumption that we an replae Nt/N with P k. With this simplifiation we arrive at a nonlinear equation for the mean field probability, dp(t)/dt = M[P(t)]P(t), with f-gn 0 g3i \ M[P{t)} = g n (4) \ 0 g 2 3 -gilj Normalization allows us to eliminate P3 and obtain a losed set of equations for Pi and Pi. We an further haraterize the mean field solutions by linearizing about the fixed point (Pf,/^*) = (1/3,1/3). The omplex onjugate eigenvalues of the Jaobian evaluated at the fixed point, A± = g{2a 3 ± iv3)/2, ross the imaginary axis at a = 1.5, indiative of a Hopf bifuration at this value, whih following a more detailed analysis [12, 13] an be shown to be superritial. Hene, as a inreases, the mean field undergoes a qualitative hange from disorder to global osillations, and the desired breaking of time translational symmetry emerges. Numerial solutions onfirm this behavior, yielding results that agree with simulations of an all-to-all oupling array [12, 13]. We now proeed with a detailed numerial haraterization of the loally oupled ase. We performed simulations of the loally oupled model in ontinuous time on rf-dimensional ubi latties with periodi boundary onditions. Time steps were 10 to 100 times smaller than the fastest loal average transition rate, i.e., dt <C e~ a (we set g=l). We find that muh smaller time steps lead to essentially the same results. Starting from random initial onditions, all simulations were run until an apparent steady state was reahed, and statistis are based on 100 independent trials. Following other works on phase synhronization, we introdue the order parameter [3,4] r=(r), R=- N x h Here 0/ is the disrete phase 2?r(f-l)/3 for state k e {1,2,3} at site j. The brakets represent an average over time in the steady state and over all independent trials. Nonzero r in the thermodynami limit indiates synhrony. We also alulate the generalized suseptibility % = L d [(R 2 ) (R) 2 }. As shown in Fig. 2, the model undergoes a dimension-dependent transition marked by harateristis of a typial phase transition, inluding a marosopi hange in r in the infinite system limit, a peak in flutuations at the ritial point, and a diverging spatial orrelation length [12, 13] (not shown). Speifially, in d = 2 we do not see the emergene of a synhronous phase. Instead, we observe intermittent osillations (for very large values of a) that derease drastially with inreasing system size. In fat, r > 0 in the thermodynami limit, even for very large values of a. We onlude that the phase transition to synhrony annot our for d = 2. In ontrast to the d=2 ase, whih serves as the lower ritial dimension, a lear thermodynami-like phase transition ours in three dimensions. Figure 2 shows expliitly that for a < a when d = 3, r > 0 as system size is inreased, and a disordered phase persists in the thermodynami limit. For a > a, the order parameter approahes a finite value as the system size inreases. In addition, Fig. 2 shows the behavior of the order parameter as a is inreased for the largest system studied (L=80); the upper left inset shows the peak in % at a = ± 0.005, thus providing an estimate of the ritial point a. While the auray of our urrent estimation of the ritial point is modest, it nonetheless suffies to determine the universality lass of the transition. Similarly, for d = 4 we estimate the transition oupling to be a = 1.900±0.025 (Fig. 2), and for d = 5 we see a transition to synhrony at a = 1.750±0.015 [12,13] (not shown). To further haraterize this transition, we use finite size saling analysis by assuming the standard saling N i = l r = L-vF[(a-a )Lv]. (6) Here F{x) is a saling funtion that approahes a onstant as x > 0. To test our numerial data against different universality lasses we hoose the appropriate ritial exponents for eah, reognizing that there are variations in the (5) 51

4 A L=4 L=8 L=12 L= d=4 / -10 Ax d = 5 A A x A C A»* A Ax >AD» «^ , 10 (a/a -1)L 1/V» A L=4 L=6 L=8 L= :0 FIGURE 3. Finite size saling analysis for d = 3, d = 4, d = 5: Data ollapse using ansatz (6) with meanfieldexponents. reported values of these exponents [15]. For the XY universality lass we use the exponents /3=0.34 and v=0.66 [17]. For the Ising exponents we use /3=0.31 and v=0.64 [11]. In Fig. 3, we see quite onviningly a ollapse when exponents from the XY lass are used, verifying reent analyti preditions [18, 19]. For omparison, we also show the data ollapse with d = 3 Ising exponents (note the sale differenes). Beause we expet d = 4 to be the upper ritial dimension in aordane with XY/Ising behavior, we antiipate a slight breakdown of the saling relation (6). Nevertheless, as shown in Fig. 4, the data ollapse is very good with the mean field exponents. As suh, our simulations suggest that d = 4 serves as the upper ritial dimension. The ase d = 5, where the data ollapse with the mean field exponents is exellent, is also shown in Fig. 3, and further supports the laim that d u =4. Having observed onviningly equilibrium-like ritial behavior in systems of inherently nonequilibrium osillators, we now turn to the question of spatial disorder. That is, when units are no longer idential (g not equal for all units), is the apaity for synhronization destroyed? To address this question, we onsider first a dihotomously disordered population onsisting of two subpopulations of osillators, eah haraterized by a different frequeny parameter g = Yi and g = ji. We assume a modified form of the inter-unit oupling given by gij=ge*p a{nj-ni i-d Oj,i+h (7) where n is the number of osillators to whih unit v is oupled, and Nk is the number of units among the n that are in state k. This form maintains a loser marosopi analogy with oupled osillators far above the frequeny threshold [16], and while more time onsuming for in depth studies of the ritial regime, it proves more suitable for the urrent disorder studies. In the mean field limit, our dihotomously disordered array an be haraterized by a six dimensional equation for Pi t7l and Pi t72, i 1,2,3, where, for example, P\ i7l represents the probability for a unit with transition rate parameter 52

5 3 timp timp (a) (b) FIGURE 4. (a.) Upper panel: The ritial surfae a is shown for a dihotomously disordered system. Speifially, the boundary given by the ontour ReA + = 0 is plotted in (71,yz,a) spae. This ontour indiates the ritial point, where the Hopf bifuration ours and the disordered solution beomes unstable. The region above the ontour represents the synhronized phase. Lower panel: Stability boundary in terms of relative width parameter, (b.) Time evolution snapshots of a dihotomously disordered system with 71 = 0.5 and 72 = 1.5 are shown for a = 3.5 < a (left) and a = 4.1 > a (right). Eah olor represents the state of the osillator. 71 to be in state 1. Following normalization, the set redues to four equations whih an be linearized about the nonsynhronous fixed point (1/3,1/3,1/3,1/3). The eigenvalues of the relevant Jaobian are given by: where ReA ± _ 1 n + 72 ~ 8 ImA± 1 7i + 72 ~ 8 a 6±Z?(a,/l)os(C(a,/l))], V3(a + 2)±B(a,ii)sm(C(a,ii)) B(a, li) = y/l [a 4-6a 2 \i 2 + 3ji 4 (a 2,1/4 + 3) C(a,iL)\ tan -V3(a 2 -(a + 3)fl 2 ) S a 2 + 3(a-l)il 2 (8) (9) In fat, one pair of eigenvalues rosses the imaginary axis (Hopf bifuration) at a ritial value a = a, but the other pair shows no qualitative hange as a is varied. Aside from an overall fator ( ), Eqs. (8) depend only on the relative width variable M - ^ - ^ (10) (71+72) ( 2 < /I < 2), and in fat the Hopf bifuration ours at a single value of a (\l). As shown in Fig. 4, the surfae a separates regions of synhronous and asynhronous phases. A small-/i expansion leads to an estimate of a to 0(/i 2 ), I fl2 + 3M 2 + \/3A/( 12 + M 2 )(4 + 3M 2 ) (11) a result that exhibits these trends expliitly. In partiular, we note the inrease in a with inreasing /I. Figure 4 also shows snapshots from simulations of a globally oupled, dihotomously disordered population (N = 5000 units, or 2500 with 71 = 0.5 and 2500 with 72 = 1.5), onfirming the emergene of marosopi synhrony for suffiiently strong oupling. 53

6 ^s^* * % * V *9 W T T T V «Kg)> = 1-5 <0(g)> = 2.5 T <0(g)> = 3.5 * <0(g)> = 4.5 I -j ** T * I i^m^ T 1^ ^ % W If H --- «Kg)> = 1-5 <0(g)> = 2.5 <0(g)> = 3.5 '"*'" < 0(g) > = 4.5 " " Width 0(g) time (a) (b) FIGURE 5. (a.) As the width of the 0(g) distribution inreases, a ritial width is reahed beyond whih synhronization is destroyed. The oupling is hosen to be a = 3.2, and the four urves represent the steady state, time-averaged order parameter for distributions with different means. As the mean of the 0 (g) distribution inreases, the transition to disorder ours at a greater width. The insets at the right show the long-time behavior of an entire population of mean transition rate parameter 3.5 (orresponding to the triangle order parameter data) and widths of 0.6, 4.0, and 6.2. (b.) The same data is plotted against /x, the relative width parameter. Finally, we show that these trends arry over to fully disordered systems, where Jt is drawn from a uniform distribution haraterized by a relative width variable /I. Expliitly, /i is the ratio of the distribution mean to the distribution width. As shown in Fig. 5, synhronization still ours in these systems and furthermore, the ritial oupling a is essentially determined by the distribution parameter /I. We find that, analogous to the dihotomously disordered system, synhronization ours at a single value of a whih depends ruially on /I. In onlusion, we have demonstrated the remarkable result that a fundamentally nonequilibrium transition, namely, a phase transition that breaks the symmetry of translation in time, is desribed by an equilibrium universality lass. By utilizing a simple disrete model for ative noisy osillators, we have shown ompelling numerial and analytial evidene that the emergene of synhronous osillations in these systems ontains signatures of an equilibrium phase transition, inluding diverging flutuations at ritiality, a marosopi hange in the order parameter, and lassi exponents belonging to the XY universality lass. In addition, we show that synhronization an and does our in both dihotomously and fully disordered populations, leading to large-sale ooperativity in spite of the non-idential nature of the onstituents. ACKNOWLEDGMENTS This work was partially supported by the National Siene Foundation under Grant No. PHY REFERENCES 1. S. H. Strogatz, Nonlinear Dynamis and Chaos, Westview Press, A. Pikovsky, M. Rosenblum, and J. Kurths, Synhronization, Cambridge Univ. Press, Cambridge, S. H. Strogatz, Physia D 143, 1-20 (2000). 4. J. A. Aebon et al,rev. Mod. Phys. 77, (2005). 5. H. Sakaguhi, S. Shinomoto, and Y. Kuramoto, Prog. Theor. Phys. 77, (1987). 6. H. Daido, Phys. Rev. Lett. 61, (1988). 7. S. H. Strogatz and R. E. Mirollo, /. Phys. A 21, L699-L706 (1988). 8. S. H. Strogatz and R. E. Mirollo, Physia D 31, (1988). 54

7 9. H. Hong, H. Park, and M. Choi, Phys. Rev. E 71, (2004). 10. N. Goldenfeld, Letures on Phase Transitions and the Renormalization Group, Westview Press, K. Huang, Statistial Mehanis Seond Edition, Wiley, New York, K. Wood, C. Van den Broek, R. Kawai, and K. Lindenberg, Phys. Rev. Lett. 96, (2006). 13. K. Wood, C. Van den Broek, R. Kawai, and K. Lindenberg, Phys. Rev. E 74, (2006). 14. T. Prager, B. Naundorf, and L. Shimansky-Geier, PhysiaA325, (2003). 15. A. Pelissetto and E. Viari, Phys. Rep. 368, (2002). 16. K. Wood, C. Van den Broek, R. Kawai, and K. Lindenberg, to appear in Phys. Rev. E (2007). 17. A. P. Gottlob and M. Hasenbush, Nul. Phys. B Suppl. 30, (1993). 18. T. Risler, J. Prost, E Juliher, Phys. Rev. Lett. 93, (2004). 19. T. Risler, J. Prost, F. Juliher, Phys. Rev. E 72, (2005). 55