STRONG STOCHASTICITY THRESHOLD IN NONLINEAR LARGE HAMILTONIAN-SYSTEMS - EFFECT ON MIXING TIMES

The dynamics of high-dimensional Hamiltonian flows is extensively investigated by means of numerical simulations in the case of the Fermi-Pasta-Ulam (FPU) beta-model and classical lattice phi-4 model; both are considered at N = 512 degrees of freedom. This work aims at investigating the major consequences on the dynamical phenomenology of the existence of a strong stochasticity threshold. This threshold corresponds to a transition from two different diffusion regimes in phase space: slow diffusion (along resonances) at low-energy density and fast diffusion (across resonances) at high-energy density. Wave packets are initially excited. The relaxation time tau-R toward equipartition of energy is measured following the time behavior of spectral entropy. A systematic study of tau-R = tau-R (epsilon, n-approximately(exc)) is reported, where epsilon is the energy per degree of freedom and n-approximately(exc) is the average wave number of the initially excited packet. In the FPU case, it is found that below the strong stochasticity threshold epsilon-c, the equipartition time is an increasing function of n-approximately(exc), i.e., high-frequency modes tend to freeze compared to low-frequency modes. This is in qualitative agreement with the predictions of a so-called narrow-packet approximation in which the FPU model is approximated by a nonlinear Schrodinger equation. However, above epsilon-c, the situation is reversed, and initial excitation of high-frequency modes yields quicker mixing. Also, this is in qualitative agreement with some analytical predictions. In the phi-4 case, at epsilon > epsilon-c the excitation of high-frequency modes results in exponentially increasing tau-R as a function of n-approximately(exc). At epsilon < epsilon-c, above some critical n-approximate(exc), tau-R is apparently divergent. It is also shown that the crossover in the scaling behavior lambda-1(epsilon) of the largest Lyapunov exponent occurs always at epsilon-c independent of the initial conditions, thus providing a good intrinsic probe of the strong stochasticity threshold.