  In globally coupled oscillators, it is believed that strong higher harmonics
of coupling functions are essential for multibranch entrainment (MBE), in which
there exist many stable states, whose number scales as $\sim$ $O(\exp N)$
(where N is the system size). The existence of MBE implies the non-ergodicity
of the system. Then, because this apparent breaking of ergodicity is caused by
microscopic energy barriers, this seems to be in conflict with a basic
principle of statistical physics. In this paper, using macroscopic dynamical
theories, we demonstrate that there is no such ergodicity breaking, and such a
system slowly evolves among branch states, jumping over microscopic energy
barriers due to the influence of thermal noise. This phenomenon can be regarded
as an example of slow dynamics driven by a perturbation along a neutrally
stable manifold consisting of an infinite number of branch states.
