An Averaging Theorem for FPU in the Thermodynamic Limit

Consider an FPU chain composed of $Ngg 1$ particles, and endow the phase space with the Gibbs measure corresponding to a small temperature $beta ^{-1}$ . 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 $beta ^{1-a}$ , $a>0$ ) 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 $beta $ . 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 $Nrightarrow infty $ , $beta >0$ .