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 beta model). We take the initial energy to be either in a single mode gamma or in a package of low-frequency modes centered at gamma and of width deltagamma, with both gamma and deltagamma 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 (E/N). A theory of the scaling with (E/N) is presented and compared to the numerical results in the range 0.03<or=E/N<or=0.8.