Finite-Time and 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.

Figure 1

Finite-time–finite-size scaling (FTFSS): $Q(t,N,K)=f\mathbf{(}t{N}^{\overline{z}},(K-K_c)N^{1/\overline{\nu }}\mathbf{)}$ for the globally coupled Kuramoto oscillators with quenched disorder yields a smooth scaling surface with $\overline{\nu }=5/2$, $\overline{z}=2/5$, and $K_c=1.595769$. The scaling collapse is good enough to make the difference of surfaces obtained from different sizes $N=800$, 1600, 3200, and 6400 almost invisible. The thick dashed and solid lines are two cross sections of the surface at $K=K_c$ and at $t{N}^{-\overline{z}}=1.2$ displayed in Figs. 2 and 2, respectively.

Figure 2

The two-variable FTFSS form (1) in Fig. 1 at zero temperature is cross sectioned to one variable scaling form (b) $Q=f(t{N}^{-\overline{z}},0)$ at $K=K_c$ and (d) $Q=f\mathbf{(}t{N}^{-\overline{z}}=1.2,(K-K_c)N^{1/\overline{\nu }}\mathbf{)}$. The raw data [(a) and (c)] obtained for various system sizes are scaled into smooth scaling curves [(b) and (d)] with the critical exponents (b) $\overline{z}=2/5$ and (d) $\overline{\nu }=5/2$ with the critical coupling strength $K_c=1.595769$.

Figure 3

The FTFSS for the globally coupled Kuramoto oscillators with thermal disorder only: $\overline{\nu }=2$ and $\overline{z}=1/2$ are obtained at $K_c=2$ (in units of the temperature $T$). The thick dashed and solid lines are two cross sections of the surface at $K=K_c$ and at $t{N}^{-\overline{z}}=2.5$ displayed in (b) and (c), respectively.

Figure 4

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 for $\overline{z}=2/5$ and $\overline{\nu }=5/2$, confirming that the Kuramoto models in the WS network and in the globally coupled structure belong to the same universality class.

Figure 5

The FTFSS surface for the Monte Carlo dynamics of the globally coupled $q$-state clock model with (a) $q=2$, (b) $q=4$, (c) $q=6$, and (d) $q=16$. Smooth scaling surfaces are obtained for the well-known mean-field exponents ($\overline{z}=1/2$ and $\overline{\nu }=2$).