Finite times to equipartition in the thermodynamic limit

We study the time scale T to equipartition in a $1D$ lattice of N masses coupled by quartic nonlinear (hard) springs (the Fermi-Pasta-Ulam β model). We take the initial energy to be either in a single mode γ or in a package of low-frequency modes centered at γ and of width δγ, with both γ and δγ proportional to N. These initial conditions both give, for finite energy densities $E/N,$ a scaling in the thermodynamic limit (large N), of a finite time to equipartition which is inversely proportional to the central mode frequency times a power of the energy density $\mathrm{}(E/N).$ A theory of the scaling with $\mathrm{}(E/N)\mathrm{}$ is presented and compared to the numerical results in the range $0.03<~E/N<~\mathrm{0.8.}$