= Incoherent state of the Kuramoto model
{title2=$\rho(\theta\mid\omega)=1/(2\pi),\quad r=0$}
The <incoherent state of the Kuramoto model> has a uniform conditional phase density, $\rho(\theta\mid\omega)=1/(2\pi)$, in the continuum population description. Its <Kuramoto order parameter> vanishes because $\int_0^{2\pi}e^{i\theta}\,d\theta=0$: individual phases cancel even though the oscillators keep moving.
The mean-field coupling then vanishes, so a uniform density remains uniform while each frequency class rotates. This state must be retained separately when deriving a self-consistency equation by dividing by a nonzero order parameter. A finite random sample generally has a small nonzero empirical order parameter; this fluctuation is not a contradiction of the continuum definition.<From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators by Steven H. Strogatz>[Sections 3–5, continuum density and incoherence]{ref}
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