A dynamical order parameter for the transition to nonergodic dynamics in the discrete nonlinear Schrödinger equation

TL;DR

The paper introduces a trajectory-independent dynamical order parameter for ergodicity breaking in the DNLSE by sampling many microcanonically equivalent initial conditions and analyzing the ensemble variance of the finite-time Kolmogorov–Sinai entropy; it shows $D_{\infty}(e)$ vanishes in the ergodic phase and stays finite when breathers form, with an essential singularity in the relaxation time at a critical energy $e^*\approx 2.77$, offering a general diagnostic for ergodicity transitions in nonlinear lattice systems with conserved quantities.

Abstract

The discrete nonlinear Schrödinger equation (DNLSE) exhibits a transition from ergodic, delocalized dynamics to a weakly nonergodic regime characterized by breather formation; yet, a precise characterization of this transition has remained elusive. By sampling many microcanonically equivalent initial conditions, we identify the asymptotic ensemble variance of the Kolmogorov-Sinai entropy as a dynamical order parameter that vanishes in the ergodic phase and becomes finite once ergodicity is broken. The relaxation time governing the ensemble convergence of the KS entropy displays an essential singularity at the transition, yielding a sharp boundary between the two dynamical regimes. This framework provides a trajectory-independent method for detecting ergodicity breaking that is broadly applicable to nonlinear lattice systems with conserved quantities.

Figures (5)

• Figure 1: Time evolution of a state $\psi$ under the discrete nonlinear Schrödinger equation, Eq. \ref{['eq:eom']}, shown for (a) delocalized, (b) pseudo-localized, and (c) localized phases. The lattice size $N=40$ and amplitude density $a=1$ are fixed while energy densities are $e = 0.8, \,2.3, \,5.5$ from left to right. Bottom: Real component $\mathrm{Re} \,\psi_j(t)$ of the smaller breather evolution in $c$.
• Figure 2: Phase diagram in $(e, a)$ space. Ground state and infinite temperature isotherm was identified in Rasmussen_2001. Breather localizations are still persistent in the pseudo-localized phase; however, the participation ratio (right, shown for $a=1$) is extensive for all states below the green dashed curve identified by Gradenigo_2021_partitionfunction, which approaches the $T=\infty$ boundary as $N\rightarrow \infty$. Evolutions in each phase are shown in Fig. \ref{['fig:time_evolutions']}. Left: Ensemble averages and standard deviations of the participation ratio $Y(N)$
• Figure 3: Ensemble Lyapunov spectra, Kolmogorov-Sinai entropy and standard deviation measurements. Top: ensemble sampled Kolmogorov-Sinai entropy and standard deviation measurements across the $a=1$ slice of the phase diagram. Vertical black bars denote predictions of $D_\infty$. Bottom: sample Lyapunov spectra. Left image shows the $e=1$ spectrum and the two-parameter fit found by Iubini_2021_chaos. Right image shows the magnitudes of the same $e=1$ spectrum contrasted with the $e=6.7$ spectrum on a log scale.
• Figure 4: Finite-time ensemble variance and infinite time predictions of Kolmogorov-Sinai entropy vs energy density. Top left: standard deviation of the sum of finite time positive Lyapunov exponents across $150$ ensemble-sampled trajectories, after averaging over $100$, $1800$, and $11200$ time-units respectively, with $N=100$ and $a=1$. The fits are exponential curves $D(t)=\exp[b + ce]$, where $b=b(t)$, $c=c(t)$ both satisfy logarithmic growth laws shown on the top right. We observe that $c(t)=c_0+C\log(t)$, $b(t) = b_0-B\log(t)$, with $c_0\approx-0.32$, $b_0\approx2.24$, $C\approx0.38$, $B\approx1.07$. Bottom left: breakdown in functional form of $D(e, t)$ near $e=2.5$. Bottom right: mean and standard deviation of posterior probability estimates of $D_\infty$.
• Figure S1: Lyapunov spectrum convergence timescale $\tau_A$. Ratio of matrix norms of off diagonal elements of the $R$ matrices compared to the diagonal elements falling to small values as Oseledets subspaces are iteratively identified, shrinking the off diagonal elements and containing chaotic stretching to the diagonal elements. For very small $\tau_{GS}$ one can achieve more optimal ratios, however a decreased $\tau_{GS}$ makes smaller integration timesteps of increased importance. We recommend setting $\tau_{GS} \geq 10\Delta t$ for an integration timestep $\Delta t$. The relaxation timescale $\tau_A$ is defined by the onset of the plateau.