1996/09/30 by Lapo Casetti, Monica Cerruti-Sola, Marco Pettini +1 · 1 citation
Physics and Astronomy · #chao-dyn #nlin.CD
paper · pdf · doi:10.1103/physreve.55.6566
published as Phys. Rev. E 55, 6566-6574 (1997) · 24 pages, REVTeX, 8 PostScript figures. Published version
arxiv created 1998/04/21 · arxiv updated 2009/11/30
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 with decreasing energy (at fixed N). Then suddenly a bifurcation appears, which can be discussed in relation to the breakup of regular, soliton-like structures. After this bifurcation, 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 εc(N)∼ 1/N2, below which trapping occurs; here the dynamics appears to be regular, soliton-like 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. The initial conditions chosen by Fermi, Pasta and Ulam were not generic and below threshold and would have required possibly an infinite time to reach equilibrium.