Finite-size scaling in the system of coupled oscillators with heterogeneity in coupling strength

We consider a mean-field model of coupled phase oscillators with random heterogeneity in the coupling strength. The system that we investigate here is a minimal model that contains randomness in diverse values of the coupling strength, and it is found to return to the original Kuramoto model [Y. Kuramoto, Prog. Theor. Phys. Suppl. 79, 223 (1984)] when the coupling heterogeneity disappears. According to one recent paper [H. Hong, H. Chaté, L.-H. Tang, and H. Park, Phys. Rev. E 92, 022122 (2015)], when the natural frequency of the oscillator in the system is “deterministically” chosen, with no randomness in it, the system is found to exhibit the finite-size scaling exponent $\overline{\nu }=5/4$. Also, the critical exponent for the dynamic fluctuation of the order parameter is found to be given by $\gamma =1/4$, which is different from the critical exponents for the Kuramoto model with the natural frequencies randomly chosen. Originally, the unusual finite-size scaling behavior of the Kuramoto model was reported by Hong et al. [H. Hong, H. Chaté, H. Park, and L.-H. Tang, Phys. Rev. Lett. 99, 184101 (2007)], where the scaling behavior is found to be characterized by the unusual exponent $\overline{\nu }=5/2$. On the other hand, if the randomness in the natural frequency is removed, it is found that the finite-size scaling behavior is characterized by a different exponent, $\overline{\nu }=5/4$ [H. Hong, H. Chaté, L.-H. Tang, and H. Park, Phys. Rev. E 92, 022122 (2015)]. Those findings brought about our curiosity and led us to explore the effects of the randomness on the finite-size scaling behavior. In this paper, we pay particular attention to investigating the finite-size scaling and dynamic fluctuation when the randomness in the coupling strength is considered.

Figure 1

(a) Behavior of the order parameter $R$ as a function of the mean $\mu$, varying the system size $N$ from $N=100$ to $N=12800$. (b) Behavior of the order parameter $S$ plotted as a function of $\mu$, for various system size $N$. The width of the natural-frequency distribution function is set to $\mathrm{\Delta }=0.5$. The theoretical prediction about the transition point, ${\mu }_c=2\mathrm{\Delta }$, is also shown as a dotted line with the label ${\mu }_c$. The data for both $R$ and $S$ have been averaged over one hundred samples with different sets $\left\{J_j\right\}$ and different initial conditions $\left\{{\varphi }_i\left(0\right)\right\}$, where the magnitude of the error is the symbol size.

Figure 2

Binder's fourth-order cumulant for the order parameter $R$ as a function of the mean value $\mu$ for various system size $N$, for a given value of $\mathrm{\Delta }=0.5$. The theoretical prediction for the transition point, ${\mu }_c=1$, is also shown by the dotted line with the label ${\mu }_c$. The data have been averaged over one hundred samples with different sets $\left\{J_j\right\}$ and different initial conditions $\left\{{\varphi }_i\left(0\right)\right\}$, where the magnitude of the error is the symbol size.

Figure 3

Dynamic fluctuation ${\chi }_R$ as a function of the mean value $\mu$ for various system size $N$, where $\mathrm{\Delta }=0.5$. The theoretical prediction of the transition point, ${\mu }_c=1$, is also displayed as the dotted line with the label ${\mu }_c$. The data have been averaged over one hundred samples with different sets $\left\{J_j\right\}$ and different initial conditions $\left\{{\varphi }_i\left(0\right)\right\}$, where the magnitude of the error is the symbol size.

Figure 4

Order parameter $R$ versus the system size $N$ is shown in log-log scale, for three different values: $\mu =0(<\mu )$, $\mu =1(={\mu }_c)$, and $\mu =2(>{\mu }_c)$. At the transition ($\mu ={\mu }_c$), $R$ is found to behave as $R\sim N^{-0.21}$.

Figure 5

Dynamic fluctuation ${\chi }_R$ is shown as a function of the system size $N$ at the transition ($\mu ={\mu }_c$) in log-log scale. The ${\chi }_R$ is found to behave as ${\chi }_R\sim N^{0.41}$.

Figure 6

Behavior of $\sqrt{N_s}$ as a function of $\sqrt{NS}$ at the transition ($\mu ={\mu }_c$), displaying the linearly proportional relation.