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 ${\mathit{E}}_{\mathit{h}}$=${\mathrm{\rho }}^2$ ${\mathit{E}}_{\mathit{h}\mathrm{-}2\gamma }$, with γ the mode in which most of the initial energy is placed and ρ=(3β${\mathit{E}}_{\gamma }$)/(4πγ). A transition occurs at R≡6β${\mathit{E}}_{\gamma }$(N+1)/${\mathrm{\pi }}^2$∼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 ${\mathrm{\Omega }}_{\mathit{b}}$∼4βγ${\mathit{E}}_{\gamma }$/${\mathit{N}}^2$. There is a second transition at a critial energy β${\mathit{E}}_{\mathit{c}}$∼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 ${\mathrm{\tau }}_{\mathit{b}}$=2π/${\mathrm{\Omega }}_{\mathit{b}}$ and a diffusive filling time.