  We investigate the emergence of an astonishingly long pre-thermal plateau in
a classical phonon field, here a harmonic chain with on-site pinning.
Integrability is broken by a weak anharmonic on-site potential with strength
$\lambda$. In the small $\lambda$ limit, the approach to equilibrium of a
translation invariant initial state is described by kinetic theory. However,
when the phonon band becomes narrow, we find that the (non-conserved) number of
phonons relaxes on much longer time scales than kinetic. We establish rigorous
bounds on the relaxation time, and develop a theory that yields exact
predictions for the dissipation rate in the limit $\lambda \to 0$. We compare
the theoretical predictions with data from molecular dynamics simulations and
find good agreement. Our work shows how classical systems may exhibit phenomena
which at the first glance appear to require quantization.
