  We study the slow relaxation of isolated quasi-integrable systems, focusing
on the classical problem of Fermi-Pasta-Ulam-Tsingou (FPU) chain. It is
well-known that the initial energy sharing between different linear-modes can
be inferred by the integrable Toda chain. Using numerical simulations, we show
explicitly how the relaxation of the FPU chain toward equilibration is
determined by a slow drift within the space of Toda's integrals of motion. We
analyze the whole spectrum of Toda-modes and show how they dictate, via a
Generalized Gibbs Ensemble (GGE), the quasi-static states along the FPU
evolution. This picture is employed to devise a fast numerical integration,
which can be generalized to other quasi-integrable models. In addition, the GGE
description leads to a fluctuation theorem, describing the large deviations as
the system flows in the entropy landscape.
