\theta \,d \omega.$$ Using the identity mentioned in the Introduction, we note that the double integral in simplifies to $(K z e^{-i \theta})$. Hence the continuity equation becomes $$\label{Cont}
\frac{\partial f}{\partial t} + \frac{\partial}{\partial \theta}\left( f \left[ (\omega-\sigma) + \frac{1}{2 i} \left\{ (K z + F) e^{-i \theta}-(K z + F)^* e^{i \theta} \right\} \right] \right) = 0,$$ where the asterisk denotes complex conjugation.

Normally one would try to solve by expanding $f$ as a Fourier series in $\theta$: $$\label{fourier}
f(\theta, \omega, t) = \frac{g(\omega)}{ 2 \pi} \left[ 1 + \sum_{n=1}^\infty f_n(\omega, t) e^{i n \theta} + \rm{c.c.} \right],$$ where c.c. denotes complex conjugate. Substitution of into and would generate an infinite set of coupled nonlinear ordinary differential equations for the amplitudes $f_n(\omega, t)$. Unfortunately the dynamics of this infinite-dimensional system would likely be difficult to analyze further.

It was at this point that Ott and Antonsen \[2008\] noticed something wonderful. They restricted attention to the special family of densities $f$ for which $$\label{trick}
f_n(\omega, t) = \left[\alpha(\omega, t) \right]^n,$$ for all $n \ge 1$. In other words, they assumed that all the amplitudes $f_n$ are $n^{\rm th}$ powers of the *same* function $\alpha(\omega, t)$. Amazingly, this ansatz satisfies the 