Transitions and time scales to equipartition in oscillator chains: Low-frequency initial conditions

We study the times to equipartition $T_{\mathrm{eq}}$ in an oscillator chain, which is the discretized Klein-Gordon equation with a quartic nonlinearity $({\phi }^4$ system). The numerical results are compared to the Fermi-Pasta-Ulam (FPU) oscillator chain with quartic nonlinearity (FPU-$\beta$ system). For both chains we consider initial energies in low-frequency modes, of the linear systems. The methods previously developed to estimate the equipartition times for the FPU-$\beta$ chain are applied to the more complicated ${\phi }^4$ chain. The results indicate that the methods are still applicable, but do not give as accurate predictions of the equipartition time or the transitions between power-law and exponential behavior.