An extensive adiabatic invariant for the Klein-Gordon model in the thermodynamic limit

In the quest for a mathematically rigorous foundation of Statistical Physics in general, and Statistical Mechanics in particular, despite many efforts and recent successes, a lot of work is still to be done. More specifically, if one considers an Hamiltonian system, instead of some ad hoc model, for the microscopic description of large systems, the behaviour over different long time scales is often still a challenge. One of the possible, and natural strategies, is to apply the techniques and results of Hamiltonian perturbation theory to large systems, with particular attention to the thermodynamic limit, i.e. when the number of degrees of freedom grows very large, at fixed, non vanishing, specific energy. The present report (based on paper [10]) is concerned with the existence of an adiabatic invariant for an arbitrarily large one dimensional Klein-Gordon chain, with estimates uniform in the size of the system.