Entropy maximisers and their stability 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 is 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. This is based on joint works with Pierre Germain (Imperial College London), Joonhyun La (KIAS) and with Miguel Escobedo (Bilbao).