Mathematics in Engineering, 2019, 1(2): 359-377. doi: 10.3934/mine.2019.2.359

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Stages of dynamics in the Fermi-Pasta-Ulam system as probed by the first Toda integral

Helen Christodoulidi, Christos Efthymiopoulos

Research Center for Astronomy and Applied Mathematics, Academy of Athens, Soranou Efesiou 4Athens, GR-11527, Greece
$^\dagger$This contribution is part of the Special Issue: Hamiltonian Lattice Dynamics
  Guest Editors: Simone Paleari; Tiziano Penati
  Link: http://www.aimspress.com/newsinfo/1165.html

Received: , Accepted: , Published:

Special Issues: Hamiltonian Lattice Dynamics

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We investigate the long term evolution of trajectories in the Fermi-Pasta-Ulam (FPU) system, using as a probe the first non-trivial integral $J$ in the hierarchy of integrals of the corresponding Toda lattice model. To this end we perform simulations of FPU-trajectories for various classes of initial conditions produced by the excitation of isolated modes, packets, as well as `generic' (random) initial data. For initial conditions corresponding to localized energy excitations, $J$ exhibits variations yielding `sigmoid' curves similar to observables used in literature, e.g., the `spectral entropy' or various types of `correlation functions'. However, $J(t)$ is free of fluctuations inherent in such observables, hence it constitutes an ideal observable for probing the timescales involved in the stages of FPU dynamics. We observe two fundamental timescales: i) the `time of stability' (in which, roughly, FPU trajectories behave like Toda), and ii) the `time to equilibrium' (beyond which energy equipartition is reached). Below a specific energy crossover, both times are found to scale exponentially as an inverse power of the specific energy. However, this crossover goes to zero with increasing the degrees of freedom $N$ as $\varepsilon _c \sim N^{-b}$, with $b \in [1.5, 2.5]$. For `generic data' initial conditions, instead, $J(t)$ allows to quantify the continuous in time slow diffusion of the FPU trajectories in a direction transverse to the Toda tori.
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