The Fermi-Pasta-Ulam problem revisited: stochasticity thresholds in nonlinear Hamiltonian systems

Details The Fermi-Pasta-Ulam problem revisited: stochasticity thresholds in nonlinear Hamiltonian systems The Fermi-Pasta-Ulam problem revisited: stochasticity thresholds in nonlinear Hamiltonian systems

1996-09-25

arxiv.org/abs/chao-dyn/9609017v3

Phys. Rev. E 55, 6566-6574 (1997)

Nonlinear Sciences

Chaotic Dynamics

24 pages, REVTeX, 8 PostScript figures. Published version

Scientific paper

10.1103/PhysRevE.55.6566

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 $\lambda_{max}$ of the FPU $\alpha$-model and that of the Toda lattice. For generic initial conditions, $\lambda_{max}(t)$ 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 $\lambda_{max}$ of the FPU model appears to approach a constant, while the $\lambda_{max}$ of the Toda lattice appears to approach zero, consistent with its integrability. This suggests that for generic initial conditions the FPU $\alpha$-model is chaotic and will therefore approach equilibrium and equipartition of energy. There is, however, a threshold energy density $\epsilon_c(N)\sim 1/N^2$, 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.

Affiliated with

Also associated with

No associations

Rating

The Fermi-Pasta-Ulam problem revisited: stochasticity thresholds in nonlinear Hamiltonian systems does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with The Fermi-Pasta-Ulam problem revisited: stochasticity thresholds in nonlinear Hamiltonian systems, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and The Fermi-Pasta-Ulam problem revisited: stochasticity thresholds in nonlinear Hamiltonian systems will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-461023