Adiabatic Invariants for the FPUT and Toda Chain in the Thermodynamic Limit

Tamara Grava, Alberto Maspero*, Guido Mazzuca, Antonio Ponno

*Corresponding author for this work

Research output: Contribution to journalArticle (Academic Journal)peer-review

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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.
Original languageEnglish
Pages (from-to)811–851
Number of pages41
JournalCommunications in Mathematical Physics
Publication statusPublished - 29 Sep 2020


  • Adiabatic invariants
  • Toda lattice
  • FPUT lattice


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