Random Initial Data and Average Shock Time in the Fermi-Pasta-Ulam-Tsingou Chain

Abstract

We investigate the dynamics of the Fermi-Pasta-Ulam-Tsingou chain with long-wavelength random initial data. When the energy per particle is small, thermal equilibrium is not reached on a fast timescale, and the system enters prethermalization. The formation of the prethermal state is characterized by the development of a Burgers-type shock and the onset of a turbulentlike spectrum with a time dependent exponent ζ(t) in the inertial range. We perform a significant step forward by demonstrating that these features are robust under generic long-wavelength random initial conditions. By employing advanced probabilistic techniques inspired by the works of Dudley and Talagrand, we derive a sharp asymptotic expression for the average shock time in the thermodynamic limit. For large p, this time scales as (psqrt[logp])^{-1}, where p is the number of excited modes, proving that it is an intensive quantity up to a logarithmic correction in the size of the system.

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BibTeX RIS

BibTeX

@article{gallone2026random,
  title = {{Random Initial Data and Average Shock Time in the Fermi-Pasta-Ulam-Tsingou Chain}},
  author = {Matteo Gallone and Ricardo Grande and Antonio Ponno and Stefano Ruffo and Erwan Druais},
  journal = {Physical Review Letters},
  year = {2026},
  month = may,
  doi = {10.1103/qxzs-1t1l},
  eprint = {2511.07664},
  archivePrefix = {arXiv},
  url = {https://hal.science/hal-05773710v1},
  note = {CC BY},
}

RIS

TY  - JOUR
TI  - Random Initial Data and Average Shock Time in the Fermi-Pasta-Ulam-Tsingou Chain
AU  - Matteo Gallone
AU  - Ricardo Grande
AU  - Antonio Ponno
AU  - Stefano Ruffo
AU  - Erwan Druais
PY  - 2026
DA  - 2026/05/27
JO  - Physical Review Letters
DO  - 10.1103/qxzs-1t1l
UR  - https://hal.science/hal-05773710v1
AB  - We investigate the dynamics of the Fermi-Pasta-Ulam-Tsingou chain with long-wavelength random initial data. When the energy per particle is small, thermal equilibrium is not reached on a fast timescale, and the system enters prethermalization. The formation of the prethermal state is characterized by the development of a Burgers-type shock and the onset of a turbulentlike spectrum with a time dependent exponent ζ(t) in the inertial range. We perform a significant step forward by demonstrating that these features are robust under generic long-wavelength random initial conditions. By employing advanced probabilistic techniques inspired by the works of Dudley and Talagrand, we derive a sharp asymptotic expression for the average shock time in the thermodynamic limit. For large p, this time scales as (psqrt[logp])^{-1}, where p is the number of excited modes, proving that it is an intensive quantity up to a logarithmic correction in the size of the system.
SN  - 0031-9007
SN  - 1079-7114
ER  -

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