Dynamical glass in weakly nonintegrable Klein-Gordon chains

Integrable many-body systems are characterized by a complete set of preserved actions. Close to an integrable limit, a nonintegrable perturbation creates a coupling network in action space which can be short or long ranged. We analyze the dynamics of observables which become the conserved actions in the integrable limit. We compute distributions of their finite time averages and obtain the ergodization time scale 𝑇𝐸 on which these distributions converge to 𝛿 distributions. We relate 𝑇𝐸 to the statistics of fluctuation times of the observables, which acquire fat-tailed distributions with standard deviations 𝜎+ 𝜏 dominating the means 𝜇+ 𝜏 and establish that 𝑇𝐸∼(𝜎+ 𝜏)2/𝜇+ 𝜏 . The Lyapunov time 𝑇Λ (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 𝑇Λ≈𝜎+ 𝜏 , 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 dynamical glass, where 𝑇𝐸 grows dramatically by many orders of magnitude and greatly exceeds the Lyapunov time, which satisfies 𝑇Λ≲𝜇+ 𝜏 . This effect arises from the formation of highly fragmented inhomogeneous distributions of chaotic groups of actions, separated by growing volumes of nonchaotic regions. These structures persist up to the ergodization time 𝑇𝐸 .