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

Abstract

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{\ensuremath{\rho}}}^{2}$${\mathit{E}}_{\mathit{h}\mathrm{\ensuremath{-}}2\ensuremath{\gamma}}$, with \ensuremath{\gamma} the mode in which most of the initial energy is placed and \ensuremath{\rho}=(3\ensuremath{\beta}${\mathit{E}}_{\ensuremath{\gamma}}$)/(4\ensuremath{\pi}\ensuremath{\gamma}). A transition occurs at R\ensuremath{\equiv}6\ensuremath{\beta}${\mathit{E}}_{\ensuremath{\gamma}}$(N+1)/${\mathrm{\ensuremath{\pi}}}^{2}$\ensuremath{\sim}1, with N the number of oscillators and \ensuremath{\beta} the quartic coupling constant. Above this transition there is strong local coupling among neighboring modes with a characteristic resonant frequency ${\mathrm{\ensuremath{\Omega}}}_{\mathit{b}}$\ensuremath{\sim}4\ensuremath{\beta}\ensuremath{\gamma}${\mathit{E}}_{\ensuremath{\gamma}}$/${\mathit{N}}^{2}$. There is a second transition at a critial energy \ensuremath{\beta}${\mathit{E}}_{\mathit{c}}$\ensuremath{\sim}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 ${\mathrm{\ensuremath{\tau}}}_{\mathit{b}}$=2\ensuremath{\pi}/${\mathrm{\ensuremath{\Omega}}}_{\mathit{b}}$ and a diffusive filling time.