(Non equilibrium) Thermodynamics of Integrable models: The Generalized Gibbs Ensemble description of the classical Neumann Model

We study a classical integrable (Neumann) model describing the motion of a particle on the sphere, subject to harmonic forces. We tackle the problem in the infinite dimensional limit by introducing a soft version in which the spherical constraint is imposed only on average over initial conditions. We show that the Generalized Gibbs Ensemble captures the long-time averages of the soft model. We reveal the full dynamic phase diagram with extended, quasi-condensed, coordinate-, and coordinate and momentum-condensed phases. The scaling properties of the fluctuations allow us to establish in which cases the strict and soft spherical constraints are equivalent, confirming the validity of the GGE hypothesis for the Neumann model on a large portion of the dynamic phase diagram.