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

The Fermi-Pasta-Ulam \ensuremath{\beta} 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 \ensuremath{\Delta}n of initially excited modes is such that the ratio \ensuremath{\Delta}n/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 \ensuremath{\Delta}n/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%.