SUPERCRITICAL BIFURCATION-THEORY AND STABILITY OF POINT ATTRACTORS

A novel approach is employed for studying a paramagnetic/ferromagnetic phase transition. Here, the Fokker-Planck transport equation is used to describe the time dependence of the spin distribution function (ord er parameter) for the XY model in mean-field theory. The evolution of the phase-space trajectories from initial nonequilibrium states is obt ained. This is then applied in explaining the otherwise well-known beh avior of the XY model using a fully dynamical-systems approach. The at tractors of this infinite-dimensional dynamical system are determined and their stability for any value of a system parameter delta, that pl ays the role of the absolute temperature, is obtained. A supercritical bifurcation occurs for delta=1/2, and this bifurcation corresponds to the paramagnetic/ferromagnetic phase transition. For an arbitrary ini tial spin density, a unique equilibrium magnetization is shown as bein g due to a continuum of fixed points existing in the temperature range 0<delta<1/2. These fixed points attract all the phase-space trajector ies, except those lying on the stable manifold of a trivial fixed poin t. The trivial fixed point at the origin is stable if delta>1/2, other wise it is a saddle point. These two types of fixed points determine t he limit behavior of the dynamical system and therefore also the equil ibrium state of the XY model in the approximation used here.