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We perform a stochastic model reduction of the Kuramoto-Sakaguchi model forfinitely many coupled phase oscillators with phase-frustration. Whereas in thethermodynamic limit coupled oscillators exhibit stationary states and aconstant order parameter, finite-size networks exhibit persistent temporalfluctuations of the order parameter. These fluctuations are caused by theinteraction of the synchronized oscillators with the non-entrained oscillators.We show that the collective effect of the non-entrained oscillators on thesynchronized cluster can be approximated by a Gaussian process. This allows foran effective closed evolution equations for the synchronized oscillators drivenby a two-dimensional Ornstein-Uhlenbeck process. Our reduction reproduces thestochastic fluctuations of the order parameter and leads to a simple stochasticdifferential equation for the order parameter.

A stochastic approximation for the finite-size Kuramoto-Sakaguchi model