Near-integrable systems in high dimensions: slow equilibration and chaos

Hamiltonian integrable models are essentially the simplest volume preserving dynamical systems: their flow is quasi-periodic along invariant tori, defined by a large set of conserved functions. These systems are non-ergodic and non-chaotic. The interest in near-integrable systems--- where the set of conservation laws is slightly perturbed--- goes back to Laplace and Lagrange studies on the stability of the Solar System. While integrability breaking in low dimensions can be fully characterized by the Kolmogorov-Arnold-Moser theorem, the case of high, or infinite dimensional systems is not fully understood. I will introduce two complementary points of view to describe the dynamics of many-body, near-integrable systems: (I) by a fast ergodic dynamics tangent to the invariant tori together with a slow drift normal to it, and (II) as a weak stochastic perturbation to an integrable model. Both ideas will be illustrated via numerical simulations for the paradigmatic example of the Fermi-Pasta-Ulam-Tsingou (FPUT) chain. I will show how the former picture can be employed to devise a fast numerical integration scheme to study very long equilibration times of the system. As for point (II), I will present a simple stochastic model which explains the basic mechanism ruling the onset of chaos in large FPUT chains. Finally, I will discuss implications to other models such as the classical Solar System and quantum chains.