### 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) |
---|---|

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 |

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### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Journal of Statistical Physics*,

*45*(3-4), 669-694. https://doi.org/10.1007/BF01021090

}

*Journal of Statistical Physics*, vol. 45, no. 3-4, pp. 669-694. https://doi.org/10.1007/BF01021090

**One-dimensional XY model : Ergodic properties and hydrodynamic limit.** / Shuhov, A. G.; Soukhov, Iouri M.

Research output: Contribution to journal › Article

TY - JOUR

T1 - One-dimensional XY model

T2 - Ergodic properties and hydrodynamic limit

AU - Shuhov, A. G.

AU - Soukhov, Iouri M.

PY - 1986/11/1

Y1 - 1986/11/1

N2 - 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 Z1. 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.

AB - 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 Z1. 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.

UR - http://www.scopus.com/inward/record.url?scp=34250123666&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=34250123666&partnerID=8YFLogxK

U2 - 10.1007/BF01021090

DO - 10.1007/BF01021090

M3 - Article

AN - SCOPUS:34250123666

VL - 45

SP - 669

EP - 694

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

SN - 0022-4715

IS - 3-4

ER -