The Fermi-Pasta-Ulam problem revisited

The Fermi-Pasta-Ulam α-model of harmonic oscillators with cubic anharmonic interactions is studied from a statistical mechanical point of view. Systems of N = 32 to 128 oscillators appear to be large enough to suggest statistical mechanical behavior. A key element has been a comparison of the maximum Lyapounov coefficient λmax of the FPU α-model and that of the Toda lattice. For generic initial conditions, λmax(t) is indistinguishable for the two models up to times that increase by decreasing energy (at fixed N). Then suddenly a bifurcation occurs, after which the λmax of the FPU model appears to approach a constant, while the λmax of the Toda lattice appears to approach zero, consistent with its integrability. This suggests that for generic initial conditions the FPU α-model is chaotic and will therefore approach equilibrium and equipartition of energy. There is, however, a threshold energy density ǫc(N) ∼ 1/N2, below which trapping occurs, the dynamics appears to be non-chaotic and the approach to equilibrium if any takes