Chaos and ergodicity breaking in the discrete nonlinear Schrödinger equation

The discrete nonlinear Schrödinger equation (DNLSE) is known to exhibit distinct dynamical phases. At low energy densities, the system is delocalized and ergodic, but as energy density increases, the system breaks ergodicity due to a coexistence of high-amplitude stable “breather” states with low-amplitude thermal fluctuations. Characterization of these phases is a subject of current research due to both intrinsic mathematical interest and broad experimental relevance of the DNLSE in condensed matter and nonlinear optics. In this talk, we discuss an ensemble sampling approach to study how statistical heterogeneity of phase space orbits emerges from the ergodic phase. We propose the asymptotic variance in the Kolmogorov-Sinai entropy as an order parameter of the transition, and show how nonergodic dynamics onset continuously as energy density increases. Beyond these findings for the DNLSE, our methodology offers a general route to global characterizations of chaotic dynamics in systems where the Lyapunov spectrum computed along individual trajectories is non-representative of the ensemble.