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A numerical and analytical study of the relaxation to equilibrium of both theFermi-Pasta-Ulam (FPU) alpha-model and the integrable Toda model, when thefundamental mode is initially excited, is reported. We show that the dynamicsof both systems is almost identical on the short term, when the energies of theinitially unexcited modes grow in geometric progression with time, through asecular avalanche process. At the end of this first stage of the dynamics thetime-averaged modal energy spectrum of the Toda system stabilizes to its finalprofile, well described, at low energy, by the spectrum of a q-breather. TheToda equilibrium state is clearly shown to describe well the long-livingquasi-state of the FPU system. On the long term, the modal energy spectrum ofthe FPU system slowly detaches from the Toda one by a diffusive-like rising ofthe tail modes, and eventually reaches the equilibrium flat shape. We find asimple law describing the growth of tail modes, which enables us to estimatethe time-scale to equipartition of the FPU system, even when, at smallenergies, it becomes unobservable.

The two-stage dynamics in the Fermi-Pasta-Ulam problem: from regular to diffusive behavior