Analysis of the kinetic equation arising from the one-dimensional FPUT chain

We briefly review the main results from Germain–La–Menegaki (2024) and Escobedo–Germain–La–Menegaki (2025) which (i) proves nonlinear stability of the non-singular Rayleigh–Jeans equilibria for the kinetic wave equation associated with the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>β</mml:mi> </mml:math> -FPUT chain and (ii) classifies the entropy maximizers for a more general class of kinetic equations arising from such discrete systems. We briefly explain the main ideas and strategies for the proofs.