Stochasticity thresholds in a lattice field theory

Multistage slow relaxation in a Hamiltonian system: The Fermi-Pasta-Ulam model

The relaxation process toward equipartition of energy among normal modes in a Hamiltonian system with many degrees of freedom, the Fermi-Pasta-Ulam (FPU) model is investigated numerically. We introduce a general indicator of relaxation σ which denotes the distance from equipartition state. In the time evolution of σ, some long-time interferences with relaxation, named "plateaus," are observed. In order to examine the details of the plateaus, relaxation time of σ and excitation time for each normal mode are measured as a function of the energy density ε0=E0/N. As a result, multistage relaxation is detected in the finite-size system. Moreover, by an analysis of the Lyapunov spectrum, the spectrum of mode energy occupancy, and the power spectrum of mode energy, we characterize the multistage slow relaxation, and some dynamical phases are extracted: quasiperiodic motion, stagnant motion (escaping from quasiperiodic motion), local chaos, and stronger chaos with nonthermal noise. We emphasize that the plateaus are robust against the arranging microscopic state. In other words, we can often observe plateaus and multistage slow relaxation in the FPU phase space. Slow relaxation is expected to remain or vanish in the thermodynamic limit depending on indicators.

Analytical estimate of stochasticity thresholds in Fermi-Pasta-Ulam and straight phi(4) models

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.