An Averaging Theorem for FPU in the Thermodynamic Limit

Consider an FPU chain composed of N ≫ 1 particles, and endow the phase space with the Gibbs measure corresponding to a small temperature β - 1 . Given a fixed K , we construct K packets of normal modes whose energies are adiabatic invariants (i.e., are approximately constant for times of order β 1 - a , a > 0 ) for initial data in a set of large measure. Furthermore, the time autocorrelation function of the energy of each packet does not decay significantly for times of order β . The restrictions on the shape of the packets are very mild. All estimates are uniform in the number N of particles and thus hold in the thermodynamic limit N → ∞ , β > 0 .