The mean field of weakly coupled oscillators exhibits non-smooth phase noise

The magnitude of the complex-valued mean field of N coupled oscillators is known to act as an order parameter for the synchronization transition; in this letter it is shown that the phase speed (rate of change of the phase angle) of the mean field is a non-smooth random process, as demonstrated by the existence of a $1/f$ spectrum across a range of frequencies. An exact expression for the phase speed correlation function is derived in the limit of vanishing coupling between oscillators; numerical results show the $1/f$ scaling can also persist beyond the synchronization threshold