Correlations, fluctuations, and stability of a finite-size network of coupled oscillators

The incoherent state of the Kuramoto model of coupled oscillators exhibits marginal modes in mean field theory. We demonstrate that corrections due to finite size effects render these modes stable in the subcritical case, i.e., when the population is not synchronous. This demonstration is facilitated by the construction of a nonequilibrium statistical field theoretic formulation of a generic model of coupled oscillators. This theory is consistent with previous results. In the all-to-all case, the fluctuations in this theory are due completely to finite size corrections, which can be calculated in an expansion in $1∕N$, where $N$ is the number of oscillators. The $N\to \infty$ limit of this theory is what is traditionally called mean field theory for the Kuramoto model.

Figure 1

Diagrammatic (Feynman) rules for the fluctuations about the mean. Time moves from right to left, as indicated by the arrow. The bare propagator $P_0(x,t\mid x^{\prime },t^{\prime })$ [see Eq. (31)] connects points at $x^{\prime }$ to $x$, where $x\equiv \{\theta ,\omega \}$. Each branch of a vertex is labeled by $x$ and $x^{\prime }$ and is connected to a factor of the propagator at $x$ or $x^{\prime }$. Each vertex represents an operator given to the right of that vertex. The “⋯” on which the derivatives act only include the incoming propagators, but not the outgoing ones. There are integrations over $\theta$, ${\theta }^{\prime }$, $\omega$, ${\omega }^{\prime }$, and $t$ at each vertex.

Figure 2

Diagrams for the connected two-point function at tree level (a) and to one loop (b).

Figure 3

The diagrams contributing to the propagator at one loop order, organized by topology. We consider (d) to be of different topology than (c) because it is equivalent to a tree level diagram with an additional factor of $1∕N$, due to the initial state vertex.

Figure 4

Diagrammatic equation for the propagator. The double lines represent the summation of the entire series in $1∕N$ for the propagator.

Figure 5

(Color online) The large time $(>200\mathrm{s})$ decay constants with zero driving frequency for the perturbed oscillator. Lines are the predictions given by Eq. (86). Solid line and circles represent $K=0.03$. Dashed line and boxes represent $K=0.05$.

Figure 6

(Color online) $Z_1\left(t\right)$ vs $t$ for various values of $N$ and $K=0.3K_c$. Each graph shows $N=\{10,50,100,500,1000\}$. Note that as $N\to \infty$ the curve approaches the tree level value. From top to bottom: Black line represents tree level. X’s and violet line represent $N=1000$. Triangles and blue line represent $N=500$. Diamonds and green line represent $N=100$. Boxes and red line represent $N=50$. Circles and purple line represent $N=10$.

Figure 7

(Color online) $Z_1\left(t\right)$ vs $t$ for various values of $N$ and $K=0.5K_c$. Each graph shows $N=\{10,50,100,500,1000\}$. Note that as $N\to \infty$ the curve approaches the tree level value. Symbols as in Fig. 6.

Figure 8

(Color online) $Z_1\left(t\right)$ vs $t$ for various values of $N$ and $K=0.7K_c$. Each graph shows $N=\{10,50,100,500,1000\}$. Note that as $N\to \infty$ the curve approaches the tree level value. Symbols as in Fig. 6.

Figure 9

(Color online) $Z_1\left(t\right)$ vs $t$ for various values of $N$ and $K=0.9K_c$. Each graph shows $N=\{10,50,100,500,1000\}$. Note that as $N\to \infty$ the curve approaches the tree level value. Symbols as in Fig. 6.

Figure 10

(Color online) As the previous figures, but with $K=0.3K_c$ and ${\omega }_0=0.05$. Solid line represents tree level. Dashed blue line represents the one loop calculation. Circles represent the simulation data.

Figure 11

(Color online) As the previous figures, but with $K=0.5K_c$ and ${\omega }_0=0.05$. Symbols as in Fig. 10.

Figure 12

(Color online) $\delta Z_3\left(t\right)$ vs $t$ for $K=0.3K_c$ (top) and $K=0.5K_c$ (bottom). Symbols as in Fig. 6.