On Thermalization in A Nonlinear Variant of the Discrete NLS Equation

TL;DR

This work investigates thermalization and localization in a genuinely nonlinear lattice toy model derived from the 2D defocusing NLS, featuring conserved norm $A$, energy $H$, and a tunable nonlinear dispersion parameter $D$. Using transfer-integral thermodynamics to define Gibbs regions (valid for $D<0.5$) and extensive long-time simulations, it maps an $(a,h)$ phase diagram distinguishing ergodic, nonergodic, Gibbs, and non-Gibbs regimes. Diagnostics based on finite-time averages $q(T)$, excursion-time PDFs with tail exponent $\gamma$, and compacton-energy analyses reveal an ergodic-to-nonergodic crossover and a transition from one-site to two-site localization as $D$ crosses 1, with ergodic dynamics also found outside the Gibbs framework. The results show that stronger nonlinear coupling promotes faster approach to $q(T) \sim 1/T$ and expands localization tendencies, offering new insights into thermalization and nonstandard statistics in genuinely nonlinear lattices.

Abstract

We study the thermalization properties of a fully nonlinear lattice model originally derived from the two-dimensional cubic defocusing nonlinear Schrödinger 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.

Figures (6)

• Figure 1: Phase diagram for the toy model with (a) $D=0.25$ and (b) $D=2$. In (a), the temperature curves for $\beta=0$ and $\beta=\infty$ are given by $h=a^2/2$ (blue line, TIO blue circles) and $h=a^2/8$ (black line, TIO magenta circles), respectively. In both panels, red circles indicate initial states characterized by a decay exponent $\gamma = 2$ in the probability distribution of excursion times, marking the ergodic–nonergodic crossover. The orange dashed line shows the maximum energy achievable with a constant norm distribution, $h_{\text{max}}(A_\ell = a) = \tfrac{a^2}{4}(1+2D)$. The black solid line, $h_{\text{min}}(A_\ell = a) = \tfrac{a^2}{4}(1-2D)$, is the corresponding minimum for the initial states with a homogeneous norm and coincides with the ground state $h(\beta=\infty)$ for $D=0.25$ in (a). In (b), this $h_{\text{min}}(A_\ell = a)$ line is not the true ground state, as heterogeneous norm distributions allow energies that become unbounded from below in the thermodynamic limit for $D>0.5$. While the TIO is only applicable in the cyan region of panel (a), each of the different regimes and their region of applicability are discussed in the text.
• Figure 2: Finite time averages $q(T)$ for $a=0.25$ with (a) $D=0.25$ for $h=0.02, 0.03125, 0.05, 0.07, 0.1$, and (b) $D=2$ for $h=0.02, 0.03125, 0.1, 1, 2$.
• Figure 3: The probability density function of the initial states for $a=0.25$ with (a) $D=0.25$, (b) $D=2$ for different energy densities $h$. The black dashed lines in both panels refer to a power law decay with an exponent $\gamma=2$ ($\sim 1/{\tau_+}^2$), to point out the results that has $\gamma\leq 2$ (nonergodic), and results that has $\gamma> 2$ (ergodic).
• Figure 4: Increasing energy density reduces chaotic mixing and leads to long-lived localization at fixed $D=0.25$ and $a=0.25$. Local norm densities $a_\ell(t)$ are color coded in time for (a) $h=0.02$ ($\beta>0$), (b) $h=0.03125$ ($\beta=0$), and (c) $h=0.07$ ($\beta<0$). Part (d) shows a zoomed view of (c), where one-site localization persists for long times.
• Figure 5: High-energy localization on two and three sites for $D=2$. Local norm densities $a_\ell$ are color-coded in time for $a=0.25, h=1$. We present the whole lattice in top figure. We zoom to a 3-site, in-phase localization in the middle panel. We zoom to the dominant 2 site staggered localization in bottom. The initial state is prepared according to Eq.(\ref{['eq:compact']}) setting $a=0.25, x=19, c=21$.