Kinetic Theory of Coupled Oscillators

We present an approach for the description of fluctuations that are due to finite system size induced correlations in the Kuramoto model of coupled oscillators. We construct a hierarchy for the moments of the density of oscillators that is analogous to the Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy in the kinetic theory of plasmas and gases. To calculate the lowest order system size effect, we truncate this hierarchy at second order and solve the resulting closed equations for the two-oscillator correlation function around the incoherent state. We use this correlation function to compute the fluctuations of the order parameter, including the effect of transients, and compare this computation with numerical simulations.

Figure 1

(a) Simulated and predicted $N⟨r^2⟩$ vs $K/K_c$ evaluated at $\tau =6/(K_c-K)$ for various values of $N$ and averaged over 1000 different initial conditions and frequencies. The frequency distribution is Lorentz with $\gamma =0.05$. The simulation was performed with a time step of 0.05. The initial distribution of angles was uniform. (b) Time evolution of $⟨r^2(\tau )⟩$ vs $\tau$ for various values of $K$ and for $N=100$. At each time point the values are averaged over 10 000 different initial conditions and frequencies. All other parameters as above.

Figure 2

$C(\omega ,{\omega }^{\prime },\theta -{\theta }^{\prime })$ integrated over $\omega$ and ${\omega }^{\prime }$ versus $\theta -{\theta }^{\prime }$ for $N=100$ and various values of $K$. Frequency distribution is Lorentz with $\gamma =0.05$ and the initial angle distribution is uniform. Results are averaged over 100 000 samples in a 2D histogram with $100\times 100$ bins. The data are then averaged over angle differences and then put into a one-dimensional histogram with bins of width 10. The time step for the evolution is 0.05.