Validity condition of normal form transformation for the ${\hat{I}}^2$-FPUT system

Abstract

In this work, we provide a validity condition for the normal form transformation to remove the non-resonant cubic terms in the ${\hat{I}}^2$-FPUT system. We show that for a wave field with random phases, the normal form transformation is valid by dominant probability if ${\hat{I}}^2\ll 1/N^{1+\hat{I}\mu }$, with $N$ the number of masses and $\hat{I}\mu$ 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 ${\hat{I}}^{2}N$. 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.

12 pages, 1 figure