J Stat Phys (2012) 148:1106–1134 DOI 10.1007/s10955-012-0568-9 Extensive Adiabatic Invariants for Nonlinear Chains Antonio Giorgilli · Simone Paleari · Tiziano Penati Received: 22 February 2012 / Accepted: 9 August 2012 / Published online: 28 August 2012 © Springer Science+Business Media, LLC 2012 Abstract We look for extensive adiabatic invariants in nonlinear chains in the thermody- namic limit. Considering the quadratic part of the Klein-Gordon Hamiltonian, by a linear change of variables we transform it into a sum of two parts in involution. At variance with the usual method of introducing normal modes, our constructive procedure allows us to ex- ploit the complete resonance, while keeping the extensive nature of the system. Next we construct a nonlinear approximation of an extensive adiabatic invariant for a perturbation of the discrete nonlinear Schrödinger model. The fluctuations of this quantity are controlled via Gibbs measure estimates independent of the system size, for a large set of initial data at low specific energy. Finally, by numerical calculations we show that our adiabatic invari- ant is well conserved for times much longer than predicted by our first order theory, with fluctuation much smaller than expected according to standard statistical estimates. Keywords Adiabatic invariant · Thermodynamic limit · Extensive Hamiltonian lattice · Resonant normal form · Ergodicity 1 Introduction In the celebrated report of Fermi, Pasta and Ulam (FPU) [21] the fundamental question was reopened whether a small perturbation of an integrable system could act as a trigger for a relaxation to equilibrium in the sense of Statistical Mechanics. For the model investigated in that report, namely a discretization of a non linear string, equipartition was expected among the normal modes of the chain. In contrast, the first numerical experiments showed that the energy may remain concentrated on a few modes for a long time, with no tendency to A. Giorgilli · S. Paleari ( ) · T. Penati Depart. of Mathematics “F. Enriques”, via Saldini, 50, Milan, Italy e-mail: simone.paleari@unimi.it A. Giorgilli e-mail: antonio.giorgilli@unimi.it T. Penati e-mail: tiziano.penati@unimi.it