Quenched dynamics of classical isolated systems: the spherical spin model with two-body random interactions or the Neumann integrable model

LETICIA CUGLIANDOLO; GUSTAVO LOZANO; EMILIO NICOLAS NESSI; MARCO PICCO; ALESSANDRO TARTAGLIA

JOURNAL OF STATISTICAL MECHANICS-THEORY AND EXPERIMENT

IOP PUBLISHING LTD

Lugar: Londres; Año: 2018

1742-5468

We study the Hamiltonian dynamics of the spherical spin model with fully-connected two-body random interactions. In the statistical physics framework, the potential energy is of the so-called p = 2 kind, closely linkedto the O(N) scalar field theory. Most importantly for our setting, the energy conserving dynamics are equivalent to the ones of the Neumann integrable model. We take initial conditions from the Boltzmann equilibrium measure at a temperature that can be above or below the static phase transition, typical of a disordered (paramagnetic) or of an ordered (disguised ferromagnetic) equilibrium phase. We subsequently evolve the configurations with Newton dynamics dictated by a different Hamiltonian, obtained from an instantaneous global rescaling of the elements in the interaction random matrix. In the limit of infinitely many degrees of freedom we identify three dynamical phases depending on the parameters that characterise the initial state and the final Hamiltonian. We next set the analysis of the system with finite number of degrees of freedom in terms of N non-linearly coupled modes. We argue that in the N ! 1 limit the modes decouple at long times. We evaluate the mode temperatures and we relate them to the frequency-dependent effective temperature measured with thefluctuation-dissipation relation in the frequency domain, similarly to what was recently proposed for quantum integrable cases. Finally, we analyse the N - 1 integrals of motion, notably, their scaling with N, and we usethem to show that the system is out of equilibrium in all phases, even for parameters that show an apparent Gibbs-Boltzmann behaviour of the global observables. We elaborate on the role played by these constants ofmotion after the quench and we briefly discuss the possible description of the asymptotic dynamics in terms of a Generalised Gibbs Ensemble.