On Thermalization in A Nonlinear Variant of the Discrete NLS Equation

TL;DR

This paper investigates how a nonlinear lattice model derived from the 2D cubic defocusing NLS equation exhibits both ergodic and nonergodic regimes, revealing complex thermalization and localization phenomena influenced by system parameters.

Contribution

It provides the first detailed analysis of thermalization, ergodicity, and localization in a nonlinear lattice model beyond standard Gibbsian regimes, highlighting parameter-dependent behaviors.

Findings

Ergodic regimes exist outside traditional Gibbsian parameters.

Localization patterns depend on the nonlinear dispersion parameter D.

Increasing D accelerates the transition toward 1/T scaling in temperature.

Abstract

We study the thermalization properties of a fully nonlinear lattice model originally derived from the two-dimensional cubic defocusing nonlinear Schr\"odinger equation (NLS) using analytical and numerical methods. Our analysis reveals both ergodic and nonergodic regimes; importantly, we find broad parameter ranges where the dynamics is ergodic even though it lies outside the Gibbsian parameter regime (for both $D=0.25$ and $D=2$), and a higher-energy range where ergodicity breaks down. We observe that in a certain range of parameters, the system requires non-standard statistical descriptions, indicating a breakdown of conventional thermalization. We examine the influence of the nonlinear dispersion parameter $D$ on the system's behavior, showing that increasing $D$ enhances fluctuations and speeds up the crossover of $q(T)$ toward the $\sim 1/T$ scaling. By analyzing excursion times, probability density functions, and localization patterns, we characterize transitions between ergodic and nonergodic behavior. In long-time numerical simulations within the non-ergodic regime for $D>1$, stable localization over two sites is observed, while $D<1$ favors single-site localization in the high energy density regimes. Our results provide insights into the interplay between thermalization, localization, and non-standard statistical behavior in genuinely nonlinear systems.