A dynamical order parameter for the transition to nonergodic dynamics in the discrete nonlinear Schr\"odinger equation

TL;DR

This paper introduces a new dynamical order parameter based on the variance of the Kolmogorov-Sinai entropy to precisely identify the transition from ergodic to nonergodic behavior in the discrete nonlinear Schrödinger equation, providing a broadly applicable method.

Contribution

The authors propose a novel ensemble variance of the Kolmogorov-Sinai entropy as a trajectory-independent order parameter for ergodicity breaking in nonlinear lattice systems.

Findings

The order parameter vanishes in the ergodic phase and is finite in the nonergodic phase.

The relaxation time exhibits an essential singularity at the transition point.

The method is broadly applicable to systems with conserved quantities.

Abstract

The discrete nonlinear Schr\"odinger 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.