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

Miguel Onorato, Lara Vozella, Davide Proment, Yuri V. Lvov

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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.
Original languageEnglish
Pages (from-to)4208–4213
Number of pages6
JournalProceedings of the National Academy of Sciences of the United States of America (PNAS)
Issue number14
Publication statusPublished - 7 Apr 2015


  • α-Fermi–Pasta–Ulam chain
  • thermalization
  • wave–wave interactions
  • FPU recurrence
  • resonant interactions

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