On the problem of energy equipartition for large systems of the Fermi-Pasta-Ulam type: analytical and numerical estimates

We report on some analytical and numerical results on the exchanges of energy in systems of the Fermi-Pasta-Ulam type, in the light of Nekhoroshev's theorem, with particular attention to the dependence of the estimates on the number n of degrees of freedom. For the ordinary FPU problem we look for a control of the single normal mode energies, and we find both the analytical and numerical estimates to agree in predicting that the energy exchanges of the single modes cannot be controlled in the thermodynamic limit. We consider then a modified FPU model, with alternating light and heavy particles, which appears as composed of two subsystems, of low (acoustic) frequency and of high (optical) frequency respectively. We try to control the exchange of the total energy of the high frequency modes up to times increasing exponentially with the frequency. In this case the numerical estimates are stronger than the available analytical ones, and give indications for nonequipartition with constants essentially independent of the number n of degrees of freedom.