Route to thermalization in the α-Fermi–Pasta–Ulam system

Onorato, Miguel, Vozella, Lara, Proment, Davide and Lvov, Yuri V. (2015) Route to thermalization in the α-Fermi–Pasta–Ulam system. Proceedings of the National Academy of Sciences of the United States of America (PNAS), 112 (14). 4208–4213. ISSN 1091-6490

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We study the original α-Fermi–Pasta–Ulam (FPU) system with N = 16, 32, and 64 masses connected by a nonlinear quadratic spring. Our approach is based on resonant wave–wave interaction theory; i.e., we assume that, in the weakly nonlinear regime (the one in which Fermi was originally interested), the large time dynamics is ruled by exact resonances. After a detailed analysis of the α-FPU equation of motion, we find that the first nontrivial resonances correspond to six-wave interactions. Those are precisely the interactions responsible for the thermalization of the energy in the spectrum. We predict that, for small-amplitude random waves, the timescale of such interactions is extremely large and it is of the order of 1/ϵ8, where ϵ is the small parameter in the system. The wave–wave interaction theory is not based on any threshold: Equipartition is predicted for arbitrary small nonlinearity. Our results are supported by extensive numerical simulations. A key role in our finding is played by the Umklapp (flip-over) resonant interactions, typical of discrete systems. The thermodynamic limit is also briefly discussed.

Item Type: Article
Uncontrolled Keywords: α-fermi–pasta–ulam chain,thermalization,wave–wave interactions,fpu recurrence,resonant interactions
Faculty \ School: Faculty of Science > School of Mathematics
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Depositing User: Pure Connector
Date Deposited: 11 Oct 2016 10:00
Last Modified: 03 Apr 2021 00:20
DOI: 10.1073/pnas.1404397112

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