One-dimensional XY model: ergodic properties and hydrodynamic limit
type: Item
instance of: scholarly article
title: One-dimensional XY model: ergodic properties and hydrodynamic limit
zbMATH Open document ID: 0638.46053
DOI: 10.1007/BF01021090
author: A. G. Shukhov
author: Yuri Suhov
published in: Journal of Statistical Physics
publication date: 1986-01-01T00:00:00Z
review text: We prove theorems on convergence to a stationary state in the course of time for the one-dimensional XY model and its generalizations. The key point is the well-known Jordan-Wigner transformation, which maps the XY dynamics onto a group of Bogoliubov transformations on the CAR \(C^*\)- algebra over \(Z^ 1\). The role of stationary states for Bogoliubov transformations is played by quasifree states and for the XY model by their inverse images with respect to the Jordan-Wigner transformation. The hydrodynamic limit for the one-dimensional XY model is also considered. By using the Jordan-Wigner transformation one reduces the problem to that of constructing the hydrodynamic limit for the group of Bogoliubov transformations. As a result, we obtain an independent motion of ``normal modes'', which is described by a hyperbolic linear differential equation of second order. For the XX model this equation reduces to a first-order transfer equation.
Mathematics Subject Classification ID: 46N99
Mathematics Subject Classification ID: 82B10
Mathematics Subject Classification ID: 46L60
zbMATH DE Number: 4040600
zbMATH Keywords: nonequilibrium quantum statistical mechanics
zbMATH Keywords: convergence to a stationary state
zbMATH Keywords: one-dimensional XY model
zbMATH Keywords: Jordan-Wigner transformation
zbMATH Keywords: Bogoliubov transformations on the CAR \(C^ *\)-algebra
zbMATH Keywords: quasifree states
zbMATH Keywords: hydrodynamic limit
zbMATH Keywords: hyperbolic linear differential equation of second order
zbMATH Keywords: first-order transfer equation
MaRDI profile type: Publication
cites work: Relaxation of Local Thermal Deviations from Equilibrium
cites work: Return to equilibrium
cites work: On the dynamics and ergodic properties of the XY model
cites work: On the XY-model on two-sided infinite chain
cites work: Q3913911
cites work: Asymptotic orbits in a free Fermi gas
cites work: Q3752119
cites work: Q4744809
cites work: Unitary equivalence of temperature dynamics for ideal and locally perturbed Fermi-gas
cites work: One-dimensional harmonic lattice caricature of hydrodynamics: Second approximation.
cites work: Gibbs states of a one dimensional quantum lattice
cites work: On uniqueness of KmMS states of one-dimensional quantum lattice systems
cites work: On uniqueness of KMS states of one-dimensional quantum lattice systems
Recommended article: Q3683028
Recommended article: On the dynamics and ergodic properties of the XY model
Recommended article: The one-dimensional Boltzmann gas: The ergodic hypothesis and the phase portrait of small systems
Recommended article: Ergodicity of unlabeled dynamics of Dyson’s model in infinite dimensions
Recommended article: Markovian and non-Markovian dynamics in the one-dimensional transverse-field XY model
Recommended article: Q3689987
Recommended article: Hydrodynamic limit for a spin system on a multidimensional lattice
Recommended article: Non-equilibrium dynamics of an infinite range \(XY\) model in an external field
Recommended article: Hydrodynamic limit of the Kawasaki dynamics on the 1d-lattice with strong, finite-range interaction
description: scientific article; zbMATH DE number 4040600