Relaxation to different stationary states in the Fermi-Pasta-Ulam model

The Fermi-Pasta-Ulam model has been studied following the time evolution of the space Fourier spectrum through the numerical integration of the equations of motion for a system of 128 non-linearly coupled oscillators. One-mode and multimode excitations have been considered as initial conditions; in the former case, an approximate analytic technique has been applied to describe the "short-time" behavior of the system, which fits well the experiment. The main result in both cases is the presence of different stationary states towards which the system is evolving: a $\frac{1}{k^2}$ spectrum (corresponding to the equipartition of energy) or an exponential spectrum can be reached, depending on the value of some parameter, which takes into account the relative weight of the nonlinear to the linear term of the equations of motion.