Generalised Gibbs Ensemble for spherically constrained harmonic models

TL;DR

This paper analytically constructs the Generalised Gibbs Ensemble for the integrable Soft Neumann Model, revealing how static and dynamic observables align across different phases in the thermodynamic limit.

Contribution

It provides the first analytical calculation of the GGE partition function for the Soft Neumann Model, including static observables and their comparison with long-term dynamics.

Findings

Excellent agreement between static and dynamic averages across phases

Derivation of the GGE partition function in the thermodynamic limit

Insights into the spherical constraint's role in the Neumann model

Abstract

We build and analytically calculate the Generalised Gibbs Ensemble partition function of the integrable Soft Neumann Model. This is the model of a classical particle which is constrained to move, on average over the initial conditions, on an $N$ dimensional sphere, and feels the effect of anisotropic harmonic potentials. We derive all relevant averaged static observables in the (thermodynamic) $N\rightarrow\infty$ limit. We compare them to their long-term dynamic averages finding excellent agreement in all phases of a non-trivial phase diagram determined by the characteristics of the initial conditions and the amount of energy injected or extracted in an instantaneous quench. We discuss the implications of our results for the proper Neumann model in which the spherical constraint is imposed strictly.