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 t. The role of stationary states for Bogoliubov transformations is played by quasifree states and for the XYmodel 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 XXmodel this equation reduces to a firstorder transfer equation.