Dynamical glass in weakly non-integrable Klein-Gordon chains

Integrable many-body systems are characterized by a complete set of preserved actions. Close to an integrable limit, a {\it nonintegrable} perturbation creates a coupling network in action space which can be short- or long-ranged. We analyze the dynamics of observables which turn into the conserved actions in the integrable limit. We compute distributions of their finite-time averages and obtain the ergodization time scale $T_E$ on which these distributions converge to $\delta$-distributions. We relate $T_E \sim (\sigma_\tau^+)^2/\mu_\tau^+$ to the statistics of fluctuation times of the observables, which acquire fat-tailed distributions with standard deviations $\sigma_\tau^+$ dominating the means $\mu_\tau^+$. The Lyapunov time $T_{\Lambda}$ (the inverse of the largest Lyapunov exponent) is then compared to the above time scales. We use a simple Klein-Gordon chain to emulate long- and short-range coupling networks by tuning its energy density. For long-range coupling networks $T_{\Lambda}\approx \sigma_\tau^+$, which indicates that the Lyapunov time sets the ergodization time, with chaos quickly diffusing through the coupling network. For short-range coupling networks we observe a {\it dynamical glass}, where $T_E$ grows dramatically by many orders of magnitude and greatly exceeds the Lyapunov time, which $T_{\Lambda} \lesssim \mu_\tau^+$. This is due to the formation of a highly fragmented inhomogeneous distributions of chaotic groups of actions, separated by growing volumes of non-chaotic regions. These structures persist up to the ergodization time $T_E$.