### Abstract

We consider the Fermi–Pasta–Ulam–Tsingou (FPUT) chain composed by

N ≫ 1 particles and periodic boundary conditions, and endow the phase space with the Gibbs measure at small temperature β−1. Given a fixed 1 ≤ m ≪ N, we prove that the first m integrals of motion of the periodic Toda chain are adiabatic invariants of FPUT (namely they are approximately constant along the Hamiltonian flow of the FPUT) for times of order β, for initial data in a set of large measure. We also prove that special linear combinations of the harmonic energies are adiabatic invariants of the FPUT on the same time scale, whereas they become adiabatic invariants for all times for the Toda dynamics.

N ≫ 1 particles and periodic boundary conditions, and endow the phase space with the Gibbs measure at small temperature β−1. Given a fixed 1 ≤ m ≪ N, we prove that the first m integrals of motion of the periodic Toda chain are adiabatic invariants of FPUT (namely they are approximately constant along the Hamiltonian flow of the FPUT) for times of order β, for initial data in a set of large measure. We also prove that special linear combinations of the harmonic energies are adiabatic invariants of the FPUT on the same time scale, whereas they become adiabatic invariants for all times for the Toda dynamics.

Original language | English |
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Number of pages | 40 |

Journal | Communications in Mathematical Physics |

DOIs | |

Publication status | Accepted/In press - 23 Jun 2020 |

### Keywords

- Adiabatic invariants
- Toda lattice
- FPUT lattice

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## Profiles

## Professor Tamara Grava

Person: Academic , Member

## Cite this

Grava, T., Maspero, A., Mazzuca, G., & Ponno, A. (Accepted/In press). Adiabatic Invariants for the FPUT and Toda Chain in the Thermodynamic Limit.

*Communications in Mathematical Physics*. https://doi.org/10.1007/s00220-020-03866-2 C