Equipartition threshold in nonlinear large Hamiltonian systems: The Fermi-Pasta-Ulam model

Abstract

The Fermi-Pasta-Ulam β model has been studied by integrating numerically the equations of motion for a system of N nonlinearly coupled oscillators with N ranging from 64 to 512. Multimode excitations have been considered as initial conditions; the number Δn of initially excited modes is such that the ratio Δn/N is kept constant. We can consider the system as a gas of weakly coupled phonons (normal modes), so that if we keep the ratio Δn/N constant we find an analogy with the thermodynamical limit of statistical mechanics where the ratio M/V is constant when both the volume V and the number of particles M are increased up to infinity. The relaxation towards stationary states is followed through the time evolution of a suitably defined ‘‘spectral entropy’’ which depends on the shape of the space Fourier spectrum; this spectral entropy is a good equipartition indicator: Strong evidence is reported in favor of the existence of an equipartition threshold. Its persistence at very different values of N is also clearly shown. The main result concerns the occurrence of the threshold at the same value of the energy density (i.e., of the ‘‘control parameter’’) when the number of degrees of freedom is changed. More general initial conditions are also considered and the same result is found using as a control parameter a pseudo-Reynolds-number R: The threshold occurs at the same critical value $R_c$ when N is varied. It turns out that a fully chaotic regime (equipartition) is obtained with an ‘‘average nonlinearity’’ of the system of about 3%.