Instability of the Rayleigh-Jeans spectrum in weak wave turbulence theory

We study the four-wave kinetic equation of weak turbulence linearized around the Rayleigh-Jeans spectrum when the collision integral is associated with short-range interactions between non-relativistic bosonic quasiparticles. The technique used for the analysis of the stability is based on the properties of the Mellin transform of the kernel in the integral equation. We find that any perturbation of the Rayleigh-Jeans distribution evolves towards low momentum scales in such a form that, when $t \to \infty$, all the particles occupy a sphere of radius arbitrary small.