We theoretically investigate collective phase synchronization between
interacting groups of globally coupled noisy identical phase oscillators
exhibiting macroscopic rhythms. Using the phase reduction method, we derive
coupled collective phase equations describing the macroscopic rhythms of the
groups from microscopic Langevin phase equations of the individual oscillators
via nonlinear Fokker-Planck equations. For sinusoidal microscopic coupling, we
determine the type of the collective phase coupling function, i.e., whether the
groups exhibit in-phase or anti-phase synchronization. We show that the
macroscopic rhythms can exhibit effective anti-phase synchronization even if
the microscopic phase coupling between the groups is in-phase, and vice versa.
Moreover, near the onset of collective oscillations, we analytically obtain the
collective phase coupling function using center-manifold and phase reductions
of the nonlinear Fokker-Planck equations.Comment: 15 pages, 7 figure