The Fermi-Pasta-Ulam problem revisited
Abstract
The Fermi-Pasta-Ulam ``alpha'' 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 (lambda) of the FPU alpha model and that of the Toda lattice. For generic initial conditions, lambda(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 Lyapunov exponent of the FPU model appears to approach a constant, while the one of the Toda lattice appears to approach zero, consistently with its integrability. This suggests that for generic initial conditions the FPU model is chaotic and will therefore approach equilibrium and equipartition of energy. There is, however, a threshold energy density (which behaves as 1/N^2) below which trapping occurs, the dynamics appears to be non-chaotic and the approach to equilibrium - if any - takes longer than observable on any available computer. Above this threshold the system appears to behave in accordance with statistical mechanics, exhibiting an approach to equilibrium in physically reasonable times.Keywords
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