Validity condition of normal form transformation for the $β$-FPUT system
Boyang Wu, Miguel Onorato, Zaher Hani, Yulin Pan
TL;DR
The paper addresses the validity of a normal-form transformation that removes non-resonant cubic terms in the β-FPUT chain under random-phase spectra. It proves a rigorous probabilistic bound $β \ll 1/N^{1+ε}$, tying the importance of non-resonant interactions to the product $βN$ and supporting the result with hypercontractivity-based estimates and Wick contractions. Numerical tests with a sixth-order symplectic integrator show that the non-resonant term fraction collapses onto a universal curve when plotted against $βN$ for both thermal-equilibrium and out-of-equilibrium states, validating the theoretical condition. The findings yield a practical criterion for parameter choices and a broadly applicable methodology for normal-form analysis in Hamiltonian systems, including justified use in deriving wave-kinetic equations.
Abstract
In this work, we provide a validity condition for the normal form transformation to remove the non-resonant cubic terms in the $β$-FPUT system. We show that for a wave field with random phases, the normal form transformation is valid by dominant probability if $β\ll 1/N^{1+ε}$, with $N$ the number of masses and $ε$ an arbitrarily small constant. To obtain this condition, a bound is needed for a summation in the transformation equation, which we prove rigorously in the paper. The condition also suggests that the importance of the non-resonant terms in the evolution equation is governed by the parameter $βN$. We design numerical experiments to demonstrate that this is indeed the case for spectra at both thermal-equilibrium and out-of-equilibrium conditions. The methodology developed in this paper is applicable to other Hamiltonian systems where a normal form transformation needs to be applied.