SUPERCRITICAL BIFURCATION AT A PHASE-TRANSITION

The time evolution of the spin distribution for the XY model in mean-f ield theory is described by the Fokker-Planck transport equation. This equation is Fourier transformed to generate an infinite-dimensional n onlinear dynamical system. The otherwise well-known behavior of the XY model is then explained using a fully dynamical-systems approach. The evolution of the phase-space trajectories of this dynamical system fr om initial nonequilibrium states is obtained. Regardless of the initia l spin distribution, its attractors (all are point attractors) determi ne the equilibrium magnetic state of the model. The phase transition a t a critical temperature is accompanied by a supercritical bifurcation . Numerical results for the time dependence of the phase-space traject ories and the order parameter are presented.