One-dimensional XY model: ergodic properties and hydrodynamic limit
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.
nonequilibrium quantum statistical mechanics
Bogoliubov transformations on the CAR \(C^ *\)-algebra
hyperbolic linear differential equation of second order
scientific article; zbMATH DE number 4040600
scientific article