Incoherent state of the Kuramoto model ($\rho (\theta \mid \omega )=1/(2\pi ),r=0$)

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.^[ref]