The two-stage dynamics in the Fermi-Pasta-Ulam problem: From regular to diffusive behavior

Abstract

A numerical and analytical study of the relaxation to equilibrium of both the FermiPasta-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.

Universal law of thermalization for one-dimensional perturbed Toda lattices

“…It can be seen that thermalization is faster approached in the former case as a consequence of the tiny difference of ò 4 , suggesting that the thermalization rate depends on the perturbation strength sensitively. It can also be seen that the energy of low-frequency modes remains for a long time and the energy of high-frequency modes increases very slowly, known as a signature of the metastable state [35,36,[58][59][60]. Nevertheless, the system will be thermalized eventually.…”

Stages of dynamics in the Fermi-Pasta-Ulam system as probed by the first Toda integral

“…We quote here some important numerical and theoretical main lines of approach to the FPU problem, which are related to our present study: i) Departure from the harmonic limit: the lowest order integrable approximation to the FPU is the chain of uncoupled oscillators constituting its linear normal modes. Initial conditions 'à la Fermi', i.e., low-frequency normal mode excitations, have been studied extensively [1,3,4,21,22,23,28,36,37,38]. They are well known to lead to exponentially localized energy profiles in the space of normal modes.…”

Energy Transports in Toda Lattices with Quasiperiodic On-site Potentials

“…On the other hand, the energy transport behaviors of Toda model have been found very similar to the FPU model in the short time scale [21,22,24]. But the difference is also obvious.…”

“…Since then, extensive studies on the energy transport behaviors in nonlinear lattices have been carried out . For the FPU model, there is a general agreement that two well separated time scales are present for weak nonlinearity [15,[20][21][22]24]. That is, in a relative short time scale, a kind of metastable state is appeared.…”

“…But the difference is also obvious. Energy equipartition was not achieved in Toda model, which indicates that the Toda model is integrable and nonergodic [22].…”