  We consider the original $\beta$-Fermi-Pasta-Ulam-Tsingou ($\beta$-FPUT)
system; numerical simulations and theoretical arguments suggest that, for a
finite number of masses, a statistical equilibrium state is reached
independently of the initial energy of the system. Using ensemble averages over
initial conditions characterized by different Fourier random phases, we
numerically estimate the time scale of equipartition and we find that for very
small nonlinearity it matches the prediction based on exact wave-wave resonant
interactions theory. We derive a simple formula for the nonlinear frequency
broadening and show that when the phenomenon of overlap of frequencies takes
place, a different scaling for the thermalization time scale is observed. Our
result supports the idea that Chirikov overlap criterium { identifies} a
transition region between two different relaxation time scaling.
