Thermalization in the one-dimensional Salerno model lattice.

Thudiyangal Mithun,Alan Bishop,Panayotis G Kevrekidis,Aleksandra Maluckov,Avadh Saxena,Bertin Many Manda,Avinash Khare,Charalampos Skokos

doi:10.1103/physreve.103.032211

Abstract

The Salerno model constitutes an intriguing interpolation between the integrable Ablowitz-Ladik (AL) model and the more standard (nonintegrable) discrete nonlinear Schrödinger (DNLS) one. The competition of local on-site nonlinearity and nonlinear dispersion governs the thermalization of this model. Here, we investigate the statistical mechanics of the Salerno one-dimensional lattice model in the nonintegrable case and illustrate the thermalization in the Gibbs regime. As the parameter interpolating between the two limits (from DNLS toward AL) is varied, the region in the space of initial energy and norm densities leading to thermalization expands. The thermalization in the non-Gibbs regime heavily depends on the finite system size; we explore this feature via direct numerical computations for different parametric regimes.

Highlights

Enlightening of the thermalization properties of lattice dynamical models and complex networks is a crucial issue in understanding and exploiting the transport and localization phenomena of relevance to a wide range of physical problems
It is natural to expect that the application of statistical and thermodynamical approaches and related mixing, ergodicity, and energy equipartition concepts to such problems is a topic of substantial ongoing interest
We have used a set of diagnostics such as the probability distribution of amplitudes and finite time averages of local probability density as well as, e.g., Lyapunov characteristic exponents

Summary

INTRODUCTION

Enlightening of the thermalization properties of lattice dynamical models and complex networks is a crucial issue in understanding and exploiting the transport and localization phenomena of relevance to a wide range of physical problems. Our study is partially motivated by recent findings regarding the statistics of different discrete nonlinear physical systems [12, 19] in which, as a core mechanism, the relaxation of nonlinear localized excitations and related ergodization is considered. We adopt a grand-canonical description of SM, decomposing the parameter space of energy and norm densities (corresponding to the conserved quantities of the energy and total norm, respectively) into Gibbs and non-Gibbs regimes In the former, we expect “regular” thermalization. The non-ergodicity may be a feature of the thermodynamic limit of infinite lattices Motivated by this array of recent developments and challenging observations, we numerically investigate the thermalization in the SM with respect to both the Gibbs and non-Gibbs regimes.

MODEL DESCRIPTION

STATISTICAL MECHANICS OF THE SALERNO NETWORK

Gibbs regime

Non-Gibbs regime

CONCLUSIONS & FUTURE CHALLENGES