  We obtain exact results on autocorrelation of the order parameter in the
nonequilibrium stationary state of a paradigmatic model of spontaneous
collective synchronization, the Kuramoto model of coupled oscillators, evolving
in presence of Gaussian, white noise. The method relies on an exact mapping of
the stationary-state dynamics of the model in the thermodynamic limit to the
noisy dynamics of a single, non-uniform oscillator, and allows to obtain
besides the Kuramoto model the autocorrelation in the equilibrium stationary
state of a related model of long-range interactions, the Brownian mean-field
model. Both the models show a phase transition between a synchronized and an
incoherent phase at a critical value of the noise strength. Our results
indicate that in the two phases as well as at the critical point, the
autocorrelation for both the model decays as an exponential with a rate that
increases continuously with the noise strength.
