Normal modes on average for purely stochastic systems

We study a class of non-integrable systems, linear chains with homogeneous attractive potentials and periodic boundary conditions, which are not perturbations of the harmonic chain. In particular, we deal with the system H-4 with a purely quartic potential, which may be shown to be stochastic without any transition. For this model we prove the following pseudo-harmonic properties: (1) the existence of a spectrum of frequencies which are proportional to the harmonic ones, according to a well defined law; (2) the separability on average of the Hamiltonian function among normal modes with these frequencies. Moreover, as far as stochasticity and pseudo-harmonicity are concerned, H-4 is the limit of the Fermi-Past-Ulam (FPU) chain when the energy density tends to infinity. In this frame, the same results as previously obtained for the FPU chain at high energy density are proven to be independent of the presence of the harmonic potential, and to hold at arbitrarily high energies. As a byproduct, we have a stochasticity indicator based on correlations which proves to be very efficient and reliable.