The microcanonical Gross-Pitaevskii (also known as the semiclassical Bose-Hubbard) lattice model dynamics is characterized by a pair of energy and norm densities. The grand canonical Gibbs distribution fails to describe a part of the density space, due to the boundedness of its kinetic energy spectrum. We define Poincaré equilibrium manifolds and compute the statistics of microcanonical excursion times off them. The tails of the distribution functions quantify the proximity of the many-body dynamics to a weakly nonergodic phase, which occurs when the average excursion time is infinite. We find that a crossover to weakly nonergodic dynamics takes place inside the non-Gibbs phase, being unnoticed by the largest Lyapunov exponent. In the ergodic part of the non-Gibbs phase, the Gibbs distribution should be replaced by an unknown modified one. We relate our findings to the corresponding integrable limit, close to which the actions are interacting through a short range coupling network.

Models of classical Josephson junction chains turn integrable in the limit of large energy densities or small Josephson energies. Close to these limits the Josephson coupling between the superconducting grains induces a short range nonintegrable network. We compute distributions of finite time averages of grain charges and extract the ergodization time TE which controls their convergence to ergodic δ-distributions. We relate TE to the statistics of fluctuation times of the charges, which are dominated by fat tails. TE is growing anomalously fast upon approaching the integrable limit, as compared to the Lyapunov time TΛ -the inverse of the largest Lyapunov exponent -reaching astonishing ratios TE/TΛ ≥ 10 8 . The microscopic reason for the observed dynamical glass is routed in a growing number of grains evolving over long times in a regular almost integrable fashion due to the low probability of resonant interactions with the nearest neighbors. We conjecture that the observed dynamical glass is a generic property of Josephson junction networks irrespective of their space dimensionality.Ergodicity is a core concept of statistical physics of many body systems. It demands infinite time averages of observables during a microcanonical evolution to match with their proper phase space averages [1]. Any laboratory or computational experiment is however constrained to finite averaging times. Are these sufficient or not? How much time is needed for a trajectory to visit the majority of the available microcanonical states, and for the finite-time average of an observable to be reasonably close to its statistical average? Can we define an ergodization time scale T E on which these properties manifest? What is that ergodization time depending on? Doubts on the applicability of the ergodic hypothesis itself were discussed for such simple cases as a mole of Ne at room temperature (see [2] and references therein). Glassy dynamics have been reported in a large variety of Hamiltonian systems [3][4][5][6]. Further, spin-glasses [7] and stochastic Levy processes [8][9][10][11][12][13] reveal that the ergodization time (and even ergodicity itself) may be affected by heavytailed distributions of lifetimes of typical excitations. The aim of this work is to address the above issues using a simple and paradigmatic dynamical many-body system testbed.Josephson junction networks are devices that are known for their wide applicability over various fields such as superconductivity, cold atoms, optics and metamaterials, among others [14-16] (for a recent survey on experimental results, see [17]): synchronization has been studied in Ref. [18,19], discrete breathers were observed and studied in Ref. [20][21][22][23], qubit dynamics was analyzed in Ref. [24,25] and the thermal conductivity was computed in Ref. [26,27]. In particular, a recent study conducted by Pino et.al.[28] showed the existence of a non-ergodic/bad metal region in the high-temperature regime of a quantum chain of Josephson junctions, that exists as a prelude to a many-body loca...

Integrable many-body systems are characterized by a complete set of preserved actions. Close to an integrable limit, a non-integrable 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 non-chaotic regions. These structures persist up to the ergodization time TE.