Shock waves and time scales to reach equipartition in the Fermi-Pasta-Ulam model

In a specific continuum limit at intermediate energy, the Fermi-Pasta-Ulam (FPU)–β chain can be described by a nonlinear partial differential equation whose solutions are shock waves. Proper long-wavelength initial conditions of the discrete model show a time evolution in numerical simulations that agrees with the solution of the continuum model where it is single valued. The breakdown times for the occurrence of the shock, when starting from a smooth initial condition, are shown to be relevant time scales for the transition to equipartition of energy, by an analysis of the time evolution of the spectral entropy. A simple time scale ${\mathit{t}}_{\mathit{B}}$∼${\mathit{N}}^2$/(βkE) is derived in the continuum limit for mode k initial excitations with energy E and N particles. This time scale is tested numerically in the FPU chain.