Volume 132, Number 5, December 2020
|Number of page(s)||7|
|Published online||30 December 2020|
(Non-equilibrium) thermodynamics of integrable models: The Generalized Gibbs Ensemble description of the classical Neumann model
1 Sorbonne Université, Laboratoire de Physique Théorique et Hautes Energies, CNRS UMR 7589 4 Place Jussieu, 75252 Paris Cedex 05, France
2 Institut Universitaire de France - 1 rue Descartes, 75005 Paris, France
3 Departamento de Física, Universidad de Buenos Aires and IFIBA CONICET - Buenos Aires, Argentina
4 Departamento de Física, Universidad Nacional de La Plata and IFLP CONICET Diag 113 y 64 (1900) La Plata, Argentina
Received: 10 November 2020
Accepted: 10 November 2020
We study the motion of a classical particle subject to anisotropic harmonic forces and constrained to move on the sphere. In the integrable-systems literature this problem is known as the Neumann model. We choose the spring constants in a way that makes the connection with the so-called p = 2 spherical disordered system transparent. We tackle the problem in the N → ∞ 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, constructed with N conserved charges in involution, captures the long-time averages of all relevant observables of the soft model after sudden changes in the parameters (quenches). We reveal the full dynamic phase diagram with four different phases in which the particles' position and momentum are both extended, only the position quasi-condenses or condenses, and both condense. The scaling properties of the fluctuations allow us to establish in which of these cases the strict and soft spherical constraints are equivalent. We thus confirm the validity of the GGE hypothesis for the Neumann model on a large portion of the dynamic phase diagram.
PACS: 05.20.-y – Classical statistical mechanics / 45.50.Jf – Few- and many-body systems / 75.10.Nr – Spin-glass and other random models
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