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) {β} model and classical lattice cphi⁴ 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 τR toward equipartition of energy is measured following the time behavior of spectral entropy. A systematic study of τR=τR({ε},ñ \~exc) is reported, where {ε} is the energy per degree of freedom and ñ \~exc is the average wave number of the initially excited packet.In the FPU case, it is found that below the strong stochasticity threshold εc, the equipartition time is an increasing function of ñ \~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 Schr\"odinger equation. However, above εc, 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 cphi⁴ case, at {ε}>εc the excitation of high-frequency modes results in exponentially increasing τR as a function of ñ \~exc. At {ε}εc, above some critical ñ \~exc, τR is apparently divergent. It is also shown that the crossover in the scaling behavior λ₁({ε}) of the largest Lyapunov exponent occurs always at εc independent of the initial conditions, thus providing a good intrinsic probe of the strong stochasticity threshold.