Statistical mechanics of general discrete nonlinear Schr{\"o}dinger models: Localization transition and its relevance for Klein-Gordon lattices

TL;DR

This paper generalizes the statistical mechanics framework of discrete nonlinear Schrödinger models to higher dimensions and nonlinearities, analyzing localization transitions and their implications for Klein-Gordon lattices with energy localization.

Contribution

It extends previous work to more general models, deriving analytical boundaries of the Gibbsian regime and linking localization phenomena to Klein-Gordon lattices.

Findings

Analytical boundary of the normal Gibbsian regime established.

Localization transition characterized for various model cases.

Conditions for persistent energy localization identified.

Abstract

We extend earlier work [Phys.Rev.Lett. 84, 3740 (2000)] on the statistical mechanics of the cubic one-dimensional discrete nonlinear Schrodinger (DNLS) equation to a more general class of models, including higher dimensionalities and nonlinearities of arbitrary degree. These extensions are physically motivated by the desire to describe situations with an excitation threshold for creation of localized excitations, as well as by recent work suggesting non-cubic DNLS models to describe Bose-Einstein condensates in deep optical lattices, taking into account the effective condensate dimensionality. Considering ensembles of initial conditions with given values of the two conserved quantities, norm and Hamiltonian, we calculate analytically the boundary of the 'normal' Gibbsian regime corresponding to infinite temperature, and perform numerical simulations to illuminate the nature of the localization dynamics outside this regime for various cases. Furthermore, we show quantitatively how this DNLS localization transition manifests itself for small-amplitude oscillations in generic Klein-Gordon lattices of weakly coupled anharmonic oscillators (in which energy is the only conserved quantity), and determine conditions for existence of persistent energy localization over large time scales.