  We demonstrate the application of the circular cumulant approach for
thermodynamically large populations of phase elements, where the Ott-Antonsen
properties are violated by a multiplicative intrinsic noise. The infinite
cumulant equation chain is derived for the case of a sinusoidal sensitivity of
the phase to noise. For inhomogeneous populations, a Lorentzian distribution of
natural frequencies is adopted. Two-cumulant model reductions, which serve as a
generalization of the Ott-Antonsen ansatz, are reported. The accuracy of these
model reductions and the macroscopic collective dynamics of the system are
explored for the case of a Kuramototype global coupling. The Ott-Antonsen
ansatz and the Gaussian approximation are found to be not uniformly accurate
for non-high frequencies.
