Energy transitions and time scales to equipartition in the Fermi-Pasta-Ulam oscillator chain

We study the energy transitions and time scales, in the Fermi-Pasta-Ulam oscillator chain, at which the energy E, initially in a single or small group of low-frequency modes, is distributed among modes. The energy transitions, with increasing energy, are classified. At low energy the linear parts of the energies are distributed in a geometrically decreasing series Eh=ρ2Eh-2γ, with γ the mode in which most of the initial energy is placed and ρ=(3βEγ)/(4πγ). A transition occurs at R6βEγ(N+1)/π2∼1, with N the number of oscillators and β the quartic coupling constant. Above this transition there is strong local coupling among neighboring modes with a characteristic resonant frequency Ωb∼4βγEγ/N2. There is a second transition at a critial energy βEc∼0.3, above which stochasticity among low-frequency resonances transfers energy into high-frequency resonances by the Arnold diffusion mechanism. Above this transition we numerically determine a universal scaling for the time scale to approach equipartition among the modes. The universal time scale is qualitatively explained in terms of the driving time scale τb=2π/Ωb and a diffusive filling time. © 1995 The American Physical Society.