ABSTRACT
A numerical and analytical study of the relaxation to equilibrium of both the Fermi-Pasta-Ulam (FPU) α-model and the integrable Toda model, when the fundamental mode is initially excited, is reported. We show that the dynamics of both systems is almost identical on the short term, when the energies of the initially unexcited modes grow in geometric progression with time, through a secular avalanche process. At the end of this first stage of the dynamics, the time-averaged modal energy spectrum of the Toda system stabilizes to its final profile, well described, at low energy, by the spectrum of a q-breather. The Toda equilibrium state is clearly shown to describe well the long-living quasi-state of the FPU system. On the long term, the modal energy spectrum of the FPU system slowly detaches from the Toda one by a diffusive-like rising of the tail modes, and eventually reaches the equilibrium flat shape. We find a simple law describing the growth of tail modes, which enables us to estimate the time-scale to equipartition of the FPU system, even when, at small energies, it becomes unobservable.
Subject(s)
Algorithms , Diffusion , Models, Chemical , Nonlinear Dynamics , Computer SimulationABSTRACT
We study energy localization in a finite one-dimensional phi(4) oscillator chain with initial energy in a single oscillator of the chain. We numerically calculate the effective number of degrees of freedom sharing the energy on the lattice as a function of time. We find that for energies smaller than a critical value, energy equipartition among the oscillators is reached in a relatively short time. On the other hand, above the critical energy, a decreasing number of particles sharing the energy is observed. We give an estimate of the effective number of degrees of freedom as a function of the energy. Our results suggest that localization is due to the appearance, above threshold, of a breather-like structure. Analytic arguments are given, based on the averaging theory and the analysis of a discrete nonlinear Schrödinger equation approximating the dynamics, to support and explain the numerical results.
ABSTRACT
We address the problem of equipartition in a long Fermi-Pasta-Ulam (FPU) chain. After giving a precise relation between FPU and Korteweg-de Vries we use the latter equation to show that, corresponding to initial data a la Fermi, the time average of the energy on the kth mode decreases exponentially with kN. The result persists in the thermodynamic limit.
ABSTRACT
We consider an infinitely extended Fermi-Pasta-Ulam model. We show that the slowly modulating amplitude of a narrow wave packet asymptotically satisfies the nonlinear Schrödinger equation (NLS) on the real axis. Using well known results from inverse scattering theory, we then show that there exists a threshold of the energy of the central normal mode of the packet, with the following properties. Below threshold the NLS equation presents a quasilinear regime with no solitons in the solution of the equation, and the wave packet width remains narrow. Above threshold generation of solitons is possible instead and the packet of normal modes can spread out. Analogous results are obtained for the straight phi(4) model. We also give an analytical estimate for such thresholds. Finally, we make a comparison with the numerical results known to us and show that, they are in remarkable agreement with our estimates.
