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We investigate the emergence of an astonishingly long pre-thermal plateau ina 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 atranslation invariant initial state is described by kinetic theory. However,when the phonon band becomes narrow, we find that the (non-conserved) number ofphonons relaxes on much longer time scales than kinetic. We establish rigorousbounds on the relaxation time, and develop a theory that yields exactpredictions for the dissipation rate in the limit $\lambda \to 0$. We comparethe theoretical predictions with data from molecular dynamics simulations andfind good agreement. Our work shows how classical systems may exhibit phenomenawhich at the first glance appear to require quantization.

Pre-thermalization in a classical phonon field: slow relaxation of the number of phonons