Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2510.04831 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866914077633150976 |
|---|---|
| author | Wu, Boyang Onorato, Miguel Hani, Zaher Pan, Yulin |
| author_facet | Wu, Boyang Onorato, Miguel Hani, Zaher Pan, Yulin |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_04831 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Validity condition of normal form transformation for the $β$-FPUT system Wu, Boyang Onorato, Miguel Hani, Zaher Pan, Yulin Mathematical Physics Numerical Analysis 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. |
| title | Validity condition of normal form transformation for the $β$-FPUT system |
| topic | Mathematical Physics Numerical Analysis |
| url | https://arxiv.org/abs/2510.04831 |