  Super-critical Kuramoto oscillators with distributed frequencies separate
into two disjoint groups: an ordered one locked to the mean field, and a
disordered one consisting of effectively decoupled oscillators -- at least so
in the thermodynamic limit. In finite ensembles, in contrast, such clear
separation fails: The mean field fluctuates due to finite-size effects and
thereby induces order in the disordered group. To our best knowledge, this
publication is the first to reveal such an effect, similar to noise-induced
synchronization, in a purely deterministic system. We start by modeling the
situation as a stationary mean field with additional white noise acting on a
pair of unlocked Kuramoto oscillators. An analytical expression shows that the
cross-correlation between the two increases with decreasing ratio of natural
frequency difference and noise intensity. In a deterministic finite Kuramoto
model, the strength of the mean field fluctuations is inextricably linked to
the typical natural frequency difference. Therefore, we let a fluctuating mean
field, generated by a finite ensemble of active oscillators, act on pairs of
passive oscillators with a microscopic natural frequency difference between
which we then measure the cross-correlation, at both super- and sub-critical
coupling.
