Dynamics of the spin-1 Ising Blume-Emery-Griffiths model by the path probability method

The dynamic behavior of the spin-1 Ising Blume-Emery-Griffiths model Hamilt onian with bilinear and biquadratic nearest-neighbor exchange interactions and a single-ion potential is studied by using the path probability method of Kikuchi. First the equilibrium behavior of the model is given briefly in order to understand the dynamic behavior. Then, the path probability metho d is applied to the model and the set of nonlinear differential equations, which is also called the dynamic or rate equations, is obtained. The dynami c equations are solved by using the Runge-Kutta method in order to study th e relaxation of order parameters. The relaxation of the order parameters ar e investigated for the system which undergoes the first- and second-order p hase transitions, especially near and far from the transition temperatures. From this investigation, the "flatness" property of metastable states and the "overshooting" phenomenon are seen explicitly. On the other hand, the s olutions of the dynamic equations are also expressed by means of a flow dia gram for temperatures near and far from the transition temperatures. The st able, metastable and unstable solutions are shown in the flow diagrams, exp licitly and the role of the unstable points, as separators between the stab le and metastable points, is described. The dynamic behavior of the model i s also studied by using the kinetic equations based on the Zwanzig-Nakajima projection operator formalism and the variational principle. Finally, it i s found that one can investigate the dynamic behavior of the system by the path probability method more comprehensively than via the kinetic equations based on the Zwanzig-Nakajima projection operator formalism and the variat ional principle. (C) 2000 American Institute of Physics. [S0021-9606(00)507 14-4].