Time scale to ergodicity in the Fermi–Pasta–Ulam system

We study the approach to near-equipartition in the N-dimensional Fermi-Pasta-Ulam Hamiltonian with quartic (hard spring) nonlinearity. We investigate numerically the time evolution of orbits with initial energy in some few low-frequency linear modes. Our results indicate a transition where, above a critical energy which is independent of N, one can reach equipartition if one waits for a time proportional to N(2). Below this critical energy the time to equipartition is exponentially long. We develop a theory to determine the time evolution and the excitation of the nonlinear modes based on a resonant normal form treatment of the resonances among the oscillators. Our theory predicts the critical energy for equipartition, the time scale to equipartition, and the form of the nonlinear modes below equipartition, in qualitative agreement with the numerical results.