Statistical mechanics of a nonlinear discrete system

TL;DR

This paper investigates the statistical mechanics of the discrete nonlinear Schrödinger equation, revealing a phase transition at finite temperature characterized by localized breather excitations, through analytical and numerical methods.

Contribution

It introduces a phase transition in the statistical mechanics of the system and links it to localized excitations, expanding understanding of nonlinear discrete systems.

Findings

Existence of a phase transition at finite temperature.

Discontinuity in the partition function indicates the phase transition.

Formation of breather-like localized excitations at the transition.

Abstract

Statistical mechanics of the discrete nonlinear Schr\"odinger equation is studied by means of analytical and numerical techniques. The lower bound of the Hamiltonian permits the construction of standard Gibbsian equilibrium measures for positive temperatures. Beyond the line of $T=\infty$, we identify a phase transition, through a discontinuity in the partition function. The phase transition is demonstrated to manifest itself in the creation of breather-like localized excitations. Interrelation between the statistical mechanics and the nonlinear dynamics of the system is explored numerically in both regimes.