The two-stage dynamics in the Fermi-Pasta-Ulam problem: From regular to diffusive behavior

Ponno, A., Christodoulidi, Helen, Skokos, Ch. and Flach, S. (2011) The two-stage dynamics in the Fermi-Pasta-Ulam problem: From regular to diffusive behavior. Chaos: An Interdisciplinary Journal of Nonlinear Science, 21 (4). 043127. ISSN 1054-1500

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A numerical and analytical study of the relaxation to equilibrium of both the Fermi-Pasta-Ulam (FPU) α-model and the integrable Toda model, when the fundamental mode is initially excited, is reported. We show that the dynamics of both systems is almost identical on the short term, when the energies of the initially unexcited modes grow in geometric progression with time, through a secular avalanche process. At the end of this first stage of the dynamics, the time-averaged modal energy spectrum of the Toda system stabilizes to its final profile, well described, at low energy, by the spectrum of a q-breather. The Toda equilibrium state is clearly shown to describe well the long-living quasi-state of the FPU system. On the long term, the modal energy spectrum of the FPU system slowly detaches from the Toda one by a diffusive-like rising of the tail modes, and eventually reaches the equilibrium flat shape. We find a simple law describing the growth of tail modes, which enables us to estimate the time-scale to equipartition of the FPU system, even when, at small energies, it becomes unobservable.

Keywords:Fermi-Pasta-Ulam, q-breathers, Energy localization
Subjects:G Mathematical and Computer Sciences > G121 Mechanics (Mathematical)
F Physical Sciences > F340 Mathematical & Theoretical Physics
G Mathematical and Computer Sciences > G120 Applied Mathematics
Divisions:College of Science > School of Mathematics and Physics
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ID Code:37110
Deposited On:16 Sep 2019 10:14

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