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Finite-Time-Finite-Size Scaling of the Kuramoto Oscillators

Phase transition in its strict sense can only be observed in an infinite system, for which equilibration takes an infinitely long time at criticality. In numerical simulations, we are often limited both by the finiteness of the system size and by the finiteness of the observation time scale. We propose that one can overcome this barrier by measuring the nonequilibrium temporal relaxation for finite systems and by applying the finite-time-finite-size scaling (FTFSS) which systematically uses two scaling variables, one temporal and the other spatial. The FTFSS method yields a smooth scaling surface, and the conventional finite-size scaling curves can be viewed as proper cross sections of the surface. The validity of our FTFSS method is tested for the synchronization transition of Kuramoto models in the globally-coupled structure and in the small-world network structure. Our FTFSS method is also applied to the Monte-Carlo dynamics of the globally-coupled q-state clock model.

Introduction -In the frame of statistical physics, it is important to find critical exponents in a system aiming at understanding of its critical behaviors. For a limited number of model systems, it might be possible to analytically obtain the exponents via the mean-field analysis, the transfer-matrix calculation, the renormalization group approach, and other analytic tools [1]. In most of realistic model systems, however, a rigorous analytic calculation of critical exponents is often a formidable task, making numerical approaches unavoidable and essential.
A phase transition manifests itself as a singularity of the free energy, which exists only in thermodynamic limit of the infinite system size. On the other hand, one can only simulate finite systems in any computational approach, which results in the so-called finite-size effects. The very finiteness of system sizes can be utilized in the form of the finite-size scaling (FSS), which has been successfully applied to further our understanding of critical behaviors [2]. In the conventional FSS, the system under study is asked to be in equilibrium, and thus the functional form of FSS lacks any time dependence. As the system size becomes larger, the time scale at criticality often diverges together with the correlation length, which gives rise to the problem of the critical slowing down. In order to circumvent this, the finite-size dynamical scaling approach [3,4] has been introduced. The basic assumption in this approach of dynamical scaling is that one can extract critical behavior by observing the temporal relaxation of the system in early times even before reaching equilibrium.
In general, one can assume that a time-varying thermodynamic quantity Q is a function of the time t, the linear size L, and the coupling strength K. As the critical point K = K c is approached, the correlation length ξ diverges following the form ξ ∼ (K − K c ) −ν and so does the relaxation time scale τ ∼ ξ z . If we further assume that Q is chosen in such a way that its anomalous dimension is null [4], the scaling form of Q is written as Q(t, L, K) = f tL −z , (K − K c )L 1/ν . In words, the first scaling variable tL −z describes the competition of the two time scales, the finite observation time t and the relaxation time τ , while the second scaling variable (K − K c )L 1/ν is for the competition of the two length scales, the finite system size L and the correlation length ξ. It is straightforward to extend the scaling form for the globally-coupled system, which reads where N is the size of the system andν andz are correlation and dynamic exponents defined for globally-coupled system. Throughout the present study, we term the scaling in Eq. (1) as finite-time-finite-size scaling (FTFSS). Precisely speaking, the phase transition is defined only in the limit of infinite time t → ∞ and infinite system size N → ∞, whereas any numerical calculation is limited by finiteness of t and N . Just like the conventional FSS aims to systematically utilizes the finite-size effect by using the scaling variable L/ξ, our FTFSS uses t/τ as well to utilize the finite-time effect introduced by the finiteness of the observation time t. Figure 1 exhibits how the FTFSS can be used to produce a smooth scaling surface for the globally-coupled Kuramoto model (see below for details).
Globally-Coupled Kuramoto Model -Synchronization phenomena are ubiquitously observed in a variety of systems such as the neuronal network and epilepsy in the brain [5], circadian rhythm [6], the collapse of the Millennium Bridge [7], power grids [8], and social behavior of humans [9]. Kuramoto model has been most popularly used to describe such synchronization phenomena, and we first study the globally-coupled N oscillators with both quenched intrinsic frequency ω i and thermal noise η i described by where K is the coupling strength and θ i is the phase of the i-th oscillator. We use the Gaussian distribution of the zero mean and the variance σ 2 for the distribution function for ω i , and the thermal noise satisfies η i = 0 and η i (t)η j (t ′ ) = 2T δ ij δ(t − t ′ ) with T being the effective temperature and · · · the ensemble average. In the zero-temperature limit of T → 0, the system corresponds to the conventional Kuramoto model for which the correlation exponentν = 5/2 [10,11] and the dynamic exponentz = 2/5 [11] have been found. In the limit of σ → 0, on the other hand, the system behaves as the globally-coupled XY model for whichν = 2 [11,12] andz = 1/2 [11] are known. The equation of motion (2) in a finite dimension can be viewed as the time-dependent Ginzberg Landau (TDGL) dynamics of the superconducting array with the dc current ω i and the thermal noise current η i [13].
In order to carry out FTFSS, we use the initial condition θ i (t = 0) = 0 for all oscillators and measure the key quantity Q(t) defined by [4,14,15]  which satisfies Q(0) = 1 and Q(t → ∞) = 0 at any parameter values of K, σ, and T due to the rotational U (1) symmetry in equilibrium. It is to be emphasized that the value of Q(t) is not directly related with the order parameter, but it detects how much initial information at t = 0 is lost as the system approaches equilibrium which is characterized by Q = 0. The advantage of using Q lies on that it does not have the anomalous dimension [4], which allows us to use the simple FTFSS form in Eq. (1). We also emphasize that our FTFSS method uses how Q(t) evolves in time before reaching equilibrium, and thus equilibration is not a requirement to extract critical exponents [3]. However, it is to be noted that before the ensemble average · · · , each sample run returns the value either +1 or −1 at time t since only the sign is taken in Eq. (3). This means that in order to get a smooth continuous form for Q(t) as a function of time t, the ensemble average must be performed over sufficiently many samples. We use the second-order algorithm to integrate equation of motion with the discrete time step ∆t = 0.01. We first report the scaling results obtained from 100000 samples for the usual Kuramoto model with only the quenched randomness in the intrinsic frequency, corresponding to the zero-temperature limit of Eq. (2) with the thermal noise term dropped. The full FTFSS with  the two scaling variables in Eq. (1) is shown in Fig. 1 with σ = 1. All surfaces obtained for various sizes N = 800, 1600, 3200, and 6400 collapse into a single smooth surface with the critical exponentsν = 5/2,z = 2/5, and the critical coupling strength K c = 8/π = 1.595769, as expected from previous studies [11,12]. The dynamic scaling [11] is easily obtained by fixing the second scaling variable in the FTFSS (1) to null by putting K = K c as shown in Fig. 2(b). All the curves in Fig. 2(a) are shown to collapse nicely into a single smooth curve. Interesting application of the FTFSS is achieved by fixing the first scaling variable tN −z in Eq. (1), to make the FTFSS form identical to the conventional FSS form. As an example, we use tN −z = 1.2 to plot Fig. 2(d). It is clearly seen that by fixing the first scaling variable, one can successfully obtain the critical exponentν = 5/2 and K c = 1.595769. In order to get such a finite-size scaling as in Fig. 2(d), it is important to choose the observation time t systematically for the given system size N to keep the value tN −z as constant.
In other extreme case with only thermal noise, corresponding to σ = 0 in Eq. (2), the dynamics is effectively identical to the mean-field version of the TDGL equation, for which it is known thatν = νd u = 2 andz = z/d u = 1/2 beyond upper critical dimension The FTFSS of the Kuramoto oscillators in the Watts-Strogatz (WS) network at the rewiring probability p = 0.5. A good quality of scaling surface is obtained forz = 2/5 andν = 5/2, confirming that the Kuramoto models in the WS network and in the globally-coupled structure belong to the same universality class. d u = 4 [15]. In parallel to Fig. 2 where K is in units of σ, we now measure K in units of the temperature T . From the well-known result that the critical value of T /K is 1/2 for the globally-coupled XY model [16], we expect that K c = 2 in units of the temperature T . In the presence of thermal noise, integration of equation of motion takes much longer time since generation of random number is needed at each time step. We use N = 400, 800, and 1600 and the ensemble averages are performed for 20000 samples. As expected from known results of K c = 2,ν = 2, andz = 1/2, our FTFSS gives us again a good quality of scaling surface as shown in Fig. 3(a). As in Fig. 2, we also make cross sections of the scaling surface to construct scaling collapses in Fig. 3(b) and (c), in which it is shown clearly that the use ofν = 2 and z = 1/2 with K c = 2 yields scaling collapses as expected. We also apply our FTFSS to the globally-coupled Kuramoto oscillators in the presence of both quenched and thermal disorder, which again yields results compatible with existing studies [see Supplemental Material (SM)].
Kuramoto Model in the Small-World Network -We next investigate the Kuramoto oscillators in the Watts-Strogatz (WS) small-world network structure [17,18], for which equations of motion read with the element of the adjacency matrix A ij = 1(0) if i and j are (dis)connected. Here, the thermal noise term  [10,11]. We show in SM thatν = 5/2 gives better scaling thanν ≈ 2.0 suggested in [18]. Globally-Coupled q-State Clock Model -We finally apply our FTFSS method to the Monte-Carlo (MC) dynamics of the globally-coupled q-state clock model [19] with the Hamiltonian where S i = (cos θ i , sin θ i ) with θ i = 2πn i /q (n i = 0, ..., q − 1), and J is the coupling strength. In the limit of q → ∞, the q-state clock model becomes identical to the XY model, while the clock model at q = 2 is exactly the same as the Ising model. It is well-known that except for q = 3 the system shows the continuous phase transition of the mean-field nature [19]. In the MC simulations we use the heat-bath update algorithm [20], and the time t is measured in units of one MC step for which the number of tries of spin flipping is N . Our key quantity Q(t) in Eq. (3) is measured near the critical point for different system sizes N = 100, 200, 400, 800, and 1600 to construct the scaling plots in Fig. 5 for q = 2, 4, 6, and 16, confirming the well-known mean-field exponentsz = 1/2 andν = 2 (see SM for more details). Conclusion and Summary -We have suggested the finite-time-finite-size scaling (FTFSS) method which takes into account the finiteness of the observation time scale as well as the finiteness of the system size. We have applied the FTFSS to construct the scaling surface defined on the two-dimensional parameter space. Proper cross sections of the FTFSS surface give us conventional finite-size scaling curves. We have applied our FTFSS to the globally-coupled Kuramoto model with quenched and thermal randomness, to the Kuramoto oscillators in the Watts-Strogatz small-world network, and to the Monte-Carlo dynamics of the globally-coupled qstate clock models. We have found that our FTFSS successfully and unanimously produces scaling surfaces with previously known correlation exponentν and the dynamic critical exponentz. Although we have focused only on globally-coupled systems in the present study, our FTFSS can directly be applied for systems in a finite dimension.  [1]). It has been suggested that the upper critical dimension d u = 5 for the Kuramoto model with quenched randomness [2], and d u = 4 has been agreed for the globally-coupled XY model [3], which explains the values obtained in the main paper (ν = 5/2,z = 2/5 for quenched, andν = 2,z = 1/2 for thermal disorder, respectively) on equal footing viā ν = νd u andz = z/d u with ν = 1/2 and z = 2. We apply the FTFSS to our key quantity Q in the same way as in the main paper to find the critical exponents along the phase boundary, and compare with the results obtained from the standard finite-size scaling (FSS) of the Kuramoto order parameter r defined by where · · · is for both the sample average and the temporal average in the steady state.
The standard FSS form r = N −β/ν g (T − T c )N 1/ν is then used with the critical exponent β = 1/2 [1] for the order parameter. In Fig. 2, as an example, we display the standard FSS of r measured in equilibrium at zero temperature, corresponding to Figs. 1 and 2 in the main paper. A good quality of the scaling collapse is achieved atν ≈ 2.3 at β = 1/2.
In Table I we compareν obtained from the FTFSS of Q and from the FSS of r, along the phase boundary presented in Fig. 1 of [1]. When the randomness in the system is purely thermal (σ = 0) for which the phase boundary crosses T /K axis at 1/2, the valueν ≈ 2 is observed both from Q and r. In the zero-temperature limit (T /K = 0), on the other hand, the correct valueν = 5/2 is well obtained from Q, but the value is rather inaccurate if r is used instead. Another advantage of using the FTFSS of Q is that it allows us to obtain other exponentz, which is also listed in Table I. It is noteworthy that along the full phase boundary,z ≈ 1/ν is found regardless of the value of T /K, which appears to suggest that zν = (z/d u )(νd u ) = zν = 2 · (1/2) = 1 remains constant along the phase boundary. In spite of the advantage of using the FTFSS of Q that only initial stage of nonequilibrium shorttime relaxation is enough to detect universality class, we point out that the calculation of Q requires average over a larger number of samples. In contrast, the calculation of r, for large  systems in particular, does not require extensive sample average, thanks to the self-averaging property.
Standard universality class argument suggests that as soon as thermal disorder is added to the Kuramoto system, the correlation exponent is expected to change fromν = 5/2 tō ν = 2, and the dynamic exponentz = 2/5 toz = 1/2. Although our results in Table I do not show such abrupt changes ofν andz, we believe that this can be a finite-size artifact which might be remedied if much bigger system sizes are used. In order to check this, we fix T /K = 0.2 and compute Q(t) at the critical point K c for larger sizes N = 8000, 10000, 20000, and 40000 to compute the dynamic critical exponentz. As displayed in Fig. 3, the bigger system sizes yield the larger value ofz, which appears to suggest that the expected valuez = 1/2 can be obtained if much bigger system sizes are used for the FTFSS analysis.  Fig. 4(a)].

CLOCK MODEL
To confirm the validity of FTFSS, we carry out similar procedure for the globally-coupled q-state clock model, in which each planar spin S i = (cos θ i , sin θ i ) has a discrete direction θ i = 2πn i /q with n i = 0, ..., q − 1 and the Hamiltonian is written as with the coupling strength J. In the limit of q → ∞, the model corresponds to the XY model, and when q = 2 it is exactly identical to the Ising model. In the q-state clock model, the competition between thermal fluctuation and ferromagnetic interaction produces the phase transition from disordered to ordered phase [6,7]. It is well known that for q = 3 the system exhibits the continuous phase transition of the mean-field nature. When q = 3, on the other hand, there is a discontinuous phase transition [8].
In contrast to the Kuramoto model, dynamics of the q-state clock model is not solely specified by the Hamiltonian and the Monte-Carlo (MC) dynamics depends on the choice of the spin update rule. We use the the heat-bath algorithm and measure the time t in units of one MC step, which corresponds to N MC tries of the spin update. We present our FTFSS for the globally-coupled q-state clock model with q=2, 4, 6, and 16 in Fig. 5 of the main paper. In Fig. 5(b) in the present SM, we display the scaling for q = 6 along the cross section denoted in Fig. 5(a). The choice of the well-known mean-field exponentsz = 1/2 andν = 2 gives us again a good quality of the scaling collapse, validating our FTFSS method also for MC dynamics.