Energy equipartition starting from high-frequency modes in the Fermi-Pasta-Ulam β oscillator chain

Abstract

We study the approach to equipartition in the Fermi-Pasta-Ulam oscillator chain with quartic nonlinearity Fermi-Pasta-Ulam--(\ensuremath{\beta} system) starting from generic high-frequency-mode initial conditions. Typically 90% of the energy is placed in one high-frequency mode, with 10% in adjacent modes. The mode energy is found to distribute itself into first a number of localized structures which coalesce over time into a single localized structure, a chaotic breather (CB). Over longer times the CB is found to break up, with energy transferred to lower frequency modes which do not have the breather symmetry. A transition with decreasing initial mode frequency is found such that the CB does not form, as expected from the loss of breather symmetry. The scaling of CB formation time with energy density, $E/N,$ is found to be ${T}_{b}\ensuremath{\propto}{(E/N)}^{\ensuremath{-}1},$ and the scaling of equipartition time found to be ${T}_{\mathrm{eq}}\ensuremath{\propto}{(E/N)}^{\ensuremath{-}2}.$ The scaling of ${T}_{\mathrm{eq}}$ can be predicted from an argument which postulates stochastic diffusion from high-frequency-mode chaotic beat oscillations to the low-frequency modes. The theory also predicts that a miminum value of $E/N$ exists below which ${T}_{\mathrm{eq}}$ should increase more rapidly with $E/N$ than in the power law range, and this transition has been found numerically.