Measuring FPUT thermalization with Toda integrals

We assess the ergodic properties of the Fermi-Pasta-Ulam-Tsingou-$\alpha$ model for generic initial conditions using a Toda integral. It serves as an adiabatic invariant for the system and a suitable observable to measure its equilibrium time. Over this timescale, the onset of action diffusion results in ergodic temporal fluctuations. We compare this timescale with the inverse of the maximum Lyapunov exponent $\lambda$ and its saturation time, which are systematically shorter. The Toda integral ergodization/equilibrium time is system size independent for long chains, but show dramatic growth when the system size is smaller than a critical one, whose value depends on the energy density. We measure the dependence of energy density on the critical system size and relate this observation to the possible emergence of a Kolmogorov-Arnold-Moser regime. We numerically determine the critical energy density of this regime, finding that it approximately decays as $1/N^2$ with the number of particles N.