In this talk we consider the four-waves spatially homogeneous kinetic equation arising in weak wave turbulence theory from one-dimensional microscopic oscillator chains. This equation is sometimes referred to as the Phonon Boltzmann Equation. I will discuss the entropy maximisation problem, the collisional invariants, and properties of solutions of the kinetic equation near the Rayleigh-Jeans (RJ) thermodynamic equilibria, in the case where the microscopic model is the Fermi-Pasta-Ulam-Tsingou (FPUT) chain. This is based on joint works with Miguel Escobedo (Bilbao), Pierre Germain (Imperial College London) and Joonhyun La (KIAS).