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

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 ${\epsilon }_c\sim N^{-b}$, with $b \in [1.5, 2.5]$\mathrm{.}For\mathrm{‘}genericdat{a}^{\mathrm{\prime }}initialconditions,instead,$J(t)$ allows to quantify the continuous in time slow diffusion of the FPU trajectories in a direction transverse to the Toda tori.