Role of Toda integrals in the FPUT metastable state
In plain words
The FPUT chain is close to the Toda chain, a solvable model with as many conserved quantities as particles. The question is whether the long-lived non-thermal state of FPUT is the equilibrium of the Toda chain, and how fast its conserved quantities drift.
Precise statement
Determine whether the pre-thermal state of the $\alpha$-FPUT chain at small $\epsilon$ is described by a generalized Gibbs ensemble built from Toda integrals, and find the scaling with $\epsilon$ and $N$ of the drift time of these integrals, which sets $T_{\mathrm{eq}}$. An answer is a mechanism with a scaling law consistent with simulations.
What would settle it
Simulations tracking all Toda integrals in FPUT dynamics, with their drift rates derived from perturbation theory around Toda.
Status in the literature
Unverified note
Simulations using Toda integrals as slowly varying quantities support this picture (arXiv:2511.08149, 2025); reported scaling exponents differ between studies.