Thermodynamic limit of the first-order phase transition in the Kuramoto model

In the Kuramoto model, a uniform distribution of the natural frequencies leads to a first-order (i.e., discontinuous) phase transition from incoherence to synchronization, at the critical coupling parameter $K_c$. We obtain the asymptotic dependence of the order parameter above criticality: $r-r_c\propto {(K-K_c)}^{2∕3}$. For a finite population, we demonstrate that the population size $N$ may be included into a self-consistency equation relating $r$ and $K$ in the synchronized state. We analyze the convergence to the thermodynamic limit of two alternative schemes to set the natural frequencies. Other frequency distributions different from the uniform one are also considered.

Figure 1

Average frequencies $\left({\overline{\omega }}_j\right)$ as a function of the coupling for different population sizes. The natural frequencies are taken according to Eq. (4) with $\gamma =1∕2$. By $K_s$ we denote the value of $K$ for the first splitting bifurcation. The critical point $K_c$ in the thermodynamic limit $(N\to \infty )$ is located at the dotted line.

Figure 2

Log-log dependence of $r$ on $K$ in a neighborhood of criticality for $\gamma =1∕2$. The circles correspond to numerical solutions of the self-consistency condition (8), the solid line depicts Eq. (17), and the dotted straight line arises taking the leading $\delta K^{2∕3}$ term only.

Figure 3

Dependence of $r$ on $K$ for finite populations: $N=$ (a) 10 and (b) 100; $\gamma =1∕2$ in both cases. Values obtained from a direct computation of the Kuramoto model are marked by $\times$. The first frequency splitting is observed at $K_s$ (for $K<K_s,r$ is no longer constant in time). The solid line is obtained solving Eq. (22) numerically. It matches the computed values, but fails when approaching the line $r=\gamma ∕K$. As a reference, the solution for an infinite population is shown with circles [after numerically solving Eq. (8)]; the critical point $(K_c,r_c)$ is marked with a ● symbol.

Figure 4

Log-log dependence of the distance from $K_s$ to $K_c$ on the population size $(\gamma =1∕2)$. Squares and diamonds correspond to different arrangements of the natural frequencies, Eqs. (4, 24), respectively. In the first case, the last two decades were fitted with a straight line, yielding a slope $-\mu =-1.502$. The dashed straight line ${\log }_{10}(4\gamma ∕\pi )-x$ arises from Eq. (25).