## Abstract

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.

Original language | English (US) |
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Pages (from-to) | 669-694 |

Number of pages | 26 |

Journal | Journal of Statistical Physics |

Volume | 45 |

Issue number | 3-4 |

DOIs | |

State | Published - Nov 1 1986 |

## All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics