期刊
PROCEEDINGS OF THE NATIONAL ACADEMY OF SCIENCES OF THE UNITED STATES OF AMERICA
卷 112, 期 14, 页码 4208-4213出版社
NATL ACAD SCIENCES
DOI: 10.1073/pnas.1404397112
关键词
alpha-Fermi-Pasta-Ulam chain; thermalization; wave-wave interactions; FPU recurrence; resonant interactions
资金
- Office of Naval Research (ONR) [214 N000141010991]
- Ministry of Education, University [PRIN 2012BFNWZ2]
- ONR [N00014-12-1-0280]
We study the original alpha-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 alpha-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/is an element of(8), where is an element of 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 (flipover) resonant interactions, typical of discrete systems. The thermodynamic limit is also briefly discussed.
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