The analysis of is carried out in the continuum limit $N \rightarrow \infty$. Then the state of the system is described by a density function $f(\theta, \omega, t)$. Here $f$ is defined such that at time $t$, the fraction of oscillators with phases between $\theta$ and $\theta + d\theta$ and natural frequencies between $\omega$ and $\omega + d \omega$ is given by $f(\theta, \omega, t) \, d\theta \, d\omega$. Thus $$\label{normalization}
 \int_{-\infty}^\infty \int_0^{2 \pi} f(\theta, \omega,t)  \, d \theta \,d \omega  = 1$$ and $$\label{freqdist}
\int_0^{2 \pi} f(\omega,\theta,t) \, d \theta = g(\omega),$$ by definition of $g(\omega)$.

The evolution of $f$ is given by the continuity equation $$\label{continuitysimple}
\frac{\partial f}{\partial t} + \frac{\partial}{\partial \theta} (f v) = 0,$$ which expresses the conservation of oscillators of frequency $\omega$. Here $v(\theta, \omega, t)$ is the velocity field on the circle corresponding to as $N \rightarrow \infty$: $$\label{velocity}
v(\theta, \omega, t) = (\omega-\sigma) + K \int_{-\infty}^\infty  \int_0^{2 \pi} \sin(\theta'-\theta) \, f(\theta', \omega',t) \, d \theta'  \,d \omega' - F \sin \theta.$$ This expression can be written more compactly in terms of the complex order parameter $z$, which in the continuum limit becomes $$\label{z_continuum}
z(t) =  \int_{-\infty}^\infty \int_0^{2 \pi} e^{i \theta} f(\theta, \omega,t)  \,d 