  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.
