Quasi-harmonicity and power spectra in the FPU model

We evaluate the power spectra of the time series for the following simple observables in the Fermi-Pasta-Ulam model: harmonic energy kinetic energy, microcanonical density, Frenet-Serret curvature and the Lyapunov variable. For some of these observables, also in the stochastic regime, the spectra show a well defined quasi-harmonic structure, with harmonic frequencies shifted with a single rescaling factor, as calculated in a previous paper. Even higher frequencies are excited: as replicas of the harmonic window at low energy, to end up with a smooth distribution at high energy, showing a power law behaviour (flicker noise). In the intermediate region the shape depends on the observables, but in all cases the crossover is the maximum of the shifted harmonic spectrum. This establishes an intrinsic short-time scale depending only on the energy density, as does the frequency rescaling factor For the curvature, we also evaluate the standard deviation: above threshold, at increasing energy, it decreases exactly as the inverse of the rescaling factor. This can be interpreted as a focalization around 'effective tori' of a harmonic-like regime which apparently coexist with the chaotic motion.