Strong stochasticity threshold in nonlinear large Hamiltonian systems: Effect on mixing times

Abstract

The dynamics of high-dimensional Hamiltonian flows is extensively investigated by means of numerical simulations in the case of the Fermi-Pasta-Ulam (FPU) \ensuremath{\beta} model and classical lattice ${\mathit{cphi}}^{4}$ 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 ${\mathrm{\ensuremath{\tau}}}_{\mathit{R}}$ toward equipartition of energy is measured following the time behavior of spectral entropy. A systematic study of ${\mathrm{\ensuremath{\tau}}}_{\mathit{R}}$=${\mathrm{\ensuremath{\tau}}}_{\mathit{R}}$(\ensuremath{\varepsilon},n${\mathrm{\ifmmode \tilde{}\else \~{}\fi{}}}_{\mathrm{exc}}$) is reported, where \ensuremath{\varepsilon} is the energy per degree of freedom and n${\mathrm{\ifmmode \tilde{}\else \~{}\fi{}}}_{\mathrm{exc}}$ is the average wave number of the initially excited packet.In the FPU case, it is found that below the strong stochasticity threshold ${\mathrm{\ensuremath{\varepsilon}}}_{\mathit{c}}$, the equipartition time is an increasing function of n${\mathrm{\ifmmode \tilde{}\else \~{}\fi{}}}_{\mathrm{exc}}$, 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 ${\mathrm{\ensuremath{\varepsilon}}}_{\mathit{c}}$, 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 ${\mathit{cphi}}^{4}$ case, at \ensuremath{\varepsilon}>${\mathrm{\ensuremath{\varepsilon}}}_{\mathit{c}}$ the excitation of high-frequency modes results in exponentially increasing ${\mathrm{\ensuremath{\tau}}}_{\mathit{R}}$ as a function of n${\mathrm{\ifmmode \tilde{}\else \~{}\fi{}}}_{\mathrm{exc}}$. At \ensuremath{\varepsilon}${\mathrm{\ensuremath{\varepsilon}}}_{\mathit{c}}$, above some critical n${\mathrm{\ifmmode \tilde{}\else \~{}\fi{}}}_{\mathrm{exc}}$, ${\mathrm{\ensuremath{\tau}}}_{\mathit{R}}$ is apparently divergent. It is also shown that the crossover in the scaling behavior ${\ensuremath{\lambda}}_{1}$(\ensuremath{\varepsilon}) of the largest Lyapunov exponent occurs always at ${\mathrm{\ensuremath{\varepsilon}}}_{\mathit{c}}$ independent of the initial conditions, thus providing a good intrinsic probe of the strong stochasticity threshold.