Localization of energy in FPU chains

We revisit the celebrated model of Fermi, Pasta and Ulam with the aim of investigating, by numerical computations, the trend towards equipartition in the thermodynamic limit. We concentrate our attention on a particular class of initial conditions, namely, with all the energy on the first mode or the first few modes. We observe that the approach to equipartition occurs on two different time scales: in a short time the energy spreads up by forming a packet involving all low–frequency modes up to a cutoff frequency ωc, while a much longer time is required in order to reach equipartition, if any. In this sense one has an energy localization with respect to frequency. The crucial point is that our numerical computations suggest that this phenomenon of a fast formation of a natural packet survives in the thermodynamic limit. More precisely we conjecture that the cutoff frequency ωc is a function of the specific energy ε = E/N, where E and N are the total energy and the number of particles, respectively. Equivalently, there should exist a function εc(ω), representing the minimal specific energy at which the natural packet extends up to frequency ω. The time required for the fast formation of the natural packet is also investigated.