Generalized microcanonical and Gibbs ensembles in classical and quantum integrable dynamics

Abstract

We prove two statements about the long time dynamics of integrable Hamiltonian systems. In classical mechanics, we prove the microcanonical version of the Generalized Gibbs Ensemble (GGE) by mapping it to a known theorem and then extend it to the limit of infinite number of degrees of freedom. In quantum mechanics, we prove GGE for maximal Hamiltonians - a class of models stemming from a rigorous notion of quantum integrability understood as the existence of conserved charges with prescribed dependence on a system parameter, e.g. Hubbard U, anisotropy in the XXZ model etc. In analogy with classical integrability, the defining property of these models is that they have the maximum number of independent integrals. We contrast their dynamics induced by quenching the parameter to that of random matrix Hamiltonians.

5 pages

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Cited by in corpus (14)

• From Quantum Chaos and Eigenstate Thermalization to Statistical Mechanics and Thermodynamics
• Generalized Gibbs ensemble in integrable lattice models
• Equilibration of Quasi-Integrable Systems
• Quenched dynamics of classical isolated systems: the spherical spin model with two-body random interactions or the Neumann integrable model
• Fluctuation Theorem for Quasi-Integrable Systems
• (Non equilibrium) Thermodynamics of Integrable models: The Generalized Gibbs Ensemble description of the classical Neumann Model
• Macrostate equivalence of two general ensembles and specific relative entropies
• The effect of dressing on thermalization of interacting waves
• Rotationally invariant ensembles of integrable matrices
• Gaussian ensemble for quantum integrable dynamics
• Nonlocality as the source of purely quantum dynamics of BCS superconductors
• Nonequilibrium physics in integrable systems and spin-flip non-invariant conserved quantities
• System-Bath Approach to Rotating Brownian Motion
• Generalised Gibbs Ensemble for spherically constrained harmonic models