Some aspects of phase transitions described by the self consistent current relaxation theory

A mathematical model for a nonlinear equation of motion for correlation functions is considered which describes the essential features of the self consistent current relaxation theory for a system experiencing some interaction with a static random field in addition to some quadratic self interaction. It is shown that the plane spanned by the two coupling parameters is separated into a region of ergodic motion and another region where the motion is nonergodic. At the separation line of the two phases all correlation functions can be discussed in terms of scaling laws. The critical exponents vary along the separation line continuously and they can be evaluated explicitely. The separation line consists of two pieces. Transitions across the first piece are characterized by a polarization catastrophy, by a vanishing of the transport coefficient and by a diverging low frequency spectrum ruled by one critical frequency scale. Transitions across the second piece do not exhibit a polarization divergence but show also a power law decrease to zero of the transport coefficient. The low frequency spectrum is the sum of two diverging parts. Each part is described by a scaling law, but the scaling frequencies and the scaling functions are quite different.