Metastable and unstable states of the Blume-Capel model obtained by the cluster variation method and the path probability method

We investigate the temperature dependence of the order parameters of the Bl ume-Capel model for zero magnetic field in the lowest approximation of the cluster variation method which is identical to the mean-field approximation . Besides the stable branches of the order parameters, we establish the met astable and unstable parts of these curves for the various values of the co upling parameter, alpha = D/J. We also study the dynamics of the model by t he path probability method, since the metastable behavior is a dynamical be havior. From this study, the "flatness" property of metastable states, "ove rshooting" phenomenon and as well as the role of the unstable state are see n explicitly. On the other hand, we also investigate how to obtain the meta stable phases with two long-range order parameters. Finally, we calculate t he phase transitions of the metastable and unstable branches of order param eters besides the stable branches and present the complete phase diagram. ( C) 2001 Elsevier Science B.V. All rights reserved.