According to a recently given ergodic condition for Hermitian many-body models the thermodynamic limit and irreversibility are necessary but by themselves not sufficient. The sufficient condition turns out to be the existence of a zero-frequency mode. It is measured by an infinite product of the recurrants from the recurrence relations method, which solves the Heisenberg equation of motion in Hermitian models. This condition has been tested with a variety of assemblies of nearest-neighbor coupled harmonic oscillators. The results provide a physical insight into why the ergodic hypothesis is valid and when it fails.