Universal route to thermalization in weakly-nonlinear one-dimensional chains

Lorenzo Pistone; Sergio Chibbaro; Miguel D. Bustamante; Yuri V. Lvov; Miguel Onorato; Lorenzo Pistone; Sergio Chibbaro; Miguel D. Bustamante; Yuri V. Lvov; Miguel Onorato

doi:10.3934/mine.2019.4.672

Mathematics in Engineering

2019, Volume 1, Issue 4: 672-698. doi: 10.3934/mine.2019.4.672

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Universal route to thermalization in weakly-nonlinear one-dimensional chains

1.
Dipartimento di Fisica, Università di Torino, via Pietro Giuria 1, 10125, Torino, Italy
2.
Sorbonne Université, CNRS, Institut Jean Le Rond D’Alembert, F-75005 Paris, France
3.
School of Mathematics and Statistics, University College Dublin, Belfield, Dublin 4, Ireland
4.
Department of Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, NY 12180
5.
INFN, Sezione di Torino, Via P. Giuria, 1-Torino, 10125, Italy

Received: 12 February 2019 Accepted: 07 July 2019 Published: 08 August 2019

Assuming that resonances play a major role in the transfer of energy among the Fourier modes, we apply the Wave Turbulence theory to describe the dynamics on nonlinear one-dimensional chains. We consider α and β Fermi-Pasta-Ulam-Tsingou (FPUT) systems, and the discrete nonlinear Klein-Gordon chain. We consider both the thermodynamic limit and the discrete regime and we conjecture that all the systems thermalize for large times, and that the equipartition time scales as a power-law of the strength of the nonlinearity, at least for a range of values of the nonlinear parameter. We perform state of the art numerical simulations and show that the results are mostly consistent with theoretical predictions. Some observed discrepancies are discussed. We suggest that the route to thermalization, based on the presence of exact resonance, has universal features. Moreover, a by-product of our analysis is the asymptotic integrability, up to four wave interactions, of the discrete nonlinear Klein-Gordon chain.
- FPUT,
- Klein-Gordon,
- nonlinear interactions,
- Wave Turbulence,
- thermalization
Citation: Lorenzo Pistone, Sergio Chibbaro, Miguel D. Bustamante, Yuri V. Lvov, Miguel Onorato. Universal route to thermalization in weakly-nonlinear one-dimensional chains[J]. Mathematics in Engineering, 2019, 1(4): 672-698. doi: 10.3934/mine.2019.4.672

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Abstract

Assuming that resonances play a major role in the transfer of energy among the Fourier modes, we apply the Wave Turbulence theory to describe the dynamics on nonlinear one-dimensional chains. We consider α and β Fermi-Pasta-Ulam-Tsingou (FPUT) systems, and the discrete nonlinear Klein-Gordon chain. We consider both the thermodynamic limit and the discrete regime and we conjecture that all the systems thermalize for large times, and that the equipartition time scales as a power-law of the strength of the nonlinearity, at least for a range of values of the nonlinear parameter. We perform state of the art numerical simulations and show that the results are mostly consistent with theoretical predictions. Some observed discrepancies are discussed. We suggest that the route to thermalization, based on the presence of exact resonance, has universal features. Moreover, a by-product of our analysis is the asymptotic integrability, up to four wave interactions, of the discrete nonlinear Klein-Gordon chain.

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