Freezing of the optical-branch energy in a diatomic Fermi Pasta Ulam chain

We aim at showing that the dynamics of nonlinear dynamical systems with many degrees of freedom (possibly infinitely many) can be similar to that of ordered systems in a surprising fashion. To this aim, in the literature techniques from perturbation theory are typically used, such as KAM theorem or Nekhoroshev theorem. Unfortunately they are known to be ill-suited for obtaining results in the case of many degrees of freedom. We present here a probabilistic approach, in which we focus on some observables of physical interest (obtained by averaging on the probability distribution on initial data) and for several models we get results of stability on long times similar to Nekhoroshev estimates. We present the example of a nonlinear chain of particles with alternating masses, a hyper-simplified model of diatomic solid. In this case, which is similar to the celebrated Fermi-Pasta-Ulam model and is widely studied in the literature, we show the progress with respect to previous results, and in particular how the present approach permits to obtain theorems valid in the thermodynamic limit.