Time-Scales to Equipartition in the Fermi-Pasta-Ulam Problem: Finite-Size Effects and Thermodynamic Limit

We investigate numerically the common alpha+beta and the pure beta FPU models, as well as some higher order generalizations. We consider initial conditions in which only low-frequency normal modes are excited, and perform a very accurate systematic study of the equilibrium time as a function of the number N of particles, the specific energy epsilon, and the parameters alpha and beta. While at any fixed N the equilibrium time is found to be a stretched exponential in 1/epsilon, in the thermodynamic limit, i.e. for N -> a at fixed epsilon, we observe a crossover to a power law. Concerning the (usually disregarded) dependence of T (eq) on alpha and beta, we find it is nontrivial, and propose and test a general law. A central role is played by the comparison of the FPU models with the Toda model.