ABSTRACT:

The FPU atomic chain is a basic Hamiltonian many-particle system which has a huge physics literature (starting with the work of F, P and U in 1947 to scrutinize some predictions of statistical mechanics) and many interesting scaling regimes connecting it to both hyperbolic conservation laws and dispersive dynamics depending on length and time scale. I will discuss rigorous results on both ``coherent'' and ``irreversible'' phenomena, including the radiative decay of coherent initial data under linear dynamics and the existence of open sets of initial data which stay coherent and exhibit recurrent Fourier spectrum for all times under nonlinear dynamics.