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

Matteo Gallone1,*, Ricardo Grande1,†, Antonio Ponno2,‡, Stefano Ruffo3,1,4,§, and Erwan Druais5,∥

  • *Contact author: mgallone@sissa.it
  • †Contact author: rgrandei@sissa.it
  • ‡Contact author: ponno@math.unipd.it
  • §Contact author: ruffo@sissa.it
  • ∥Contact author: erwan.druais@ens-lyon.fr
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Phys. Rev. Lett. 136, 217201 – Published 27 May, 2026

DOI: https://doi.org/10.1103/qxzs-1t1l

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 (plogp)−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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Phys. Rev. Lett. 136, 217201– Published 27 May, 2026
  • Received 11 September 2025
  • Accepted 1 May 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

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