Characteristics of Equilibrated Nonlinear Oscillator Systems

TL;DR

This paper develops a comprehensive statistical mechanics framework for large discrete nonlinear Schrödinger equation systems, calculating equilibrium properties and phase diagrams across all nonlinearity levels.

Contribution

It introduces both numerical and analytical methods to determine Lagrange parameters and maps the equilibrium states of DNLSE systems.

Findings

Mapped temperature and chemical potential distributions.

Generated phase diagrams for various nonlinearity levels.

Provided approximate PDFs for site densities.

Abstract

During the evolution of coupled nonlinear oscillators on a lattice, with dynamics dictated by the discrete nonlinear Schr\"odinger equation (DNLSE systems), two quantities are conserved: system energy (Hamiltonian) and system density (number of particles). If the number of system oscillators is large enough, a significant portion of the array can be considered to be an "open system", in intimate energy and density contact with a "bath" - the rest of the array. Thus, as indicated in previous works, the grand canonical formulation can be exploited in order to determine equilibrium statistical properties of thermalized DNLSE systems. In this work, given the values of the two conserved quantities, we have calculated the necessary values of the two Lagrange parameters (typically designated $\beta,\mu$) associated with the grand canonical partition function in two different ways. One is numerical and the other is analytic, based on a published approximate entropy expression. In addition we have accessed a purposely-derived approximate PDF expression of site-densities. Applying these mathematical tools we have generated maps of temperatures, chemical potentials, and field correlations for DNLSE systems over the entire thermalization zone of the DNLSE phase diagram, subjected to all system-nonlinearity levels. The end result is a rather complete picture, characterizing equilibrated large DNLSE systems.