Dynamical behaviors of magnetic systems near the critical points in the nonhydrodynamical regime are considered. For this purpose we construct a kinetic equation to describe the time evolution of critical fluctuations that corresponds in a sense to the Boltzmann equation of dilute gases. A graphical technique is developed to deduce the self-consistent equation for time-correlation functions of critical fluctuations from this kinetic equation, and the properties of the time-correlation functions of spin fluctuations for the Heisenberg isotropic and planar ferro- and isotropic antiferromagnets are discussed. The results are then applied to the high-frequency sound attenuation in magnetic systems.
The dynamical theory of critical fluctuations put forward in a previous work is rephrased and extended so as to be applicable below the transition points as well. The theory is illustrated for the isotropic Heisenberg spin systems where the critical dynamical variables can be chosen in such a way as to reproduce the correct characteristic times governing the dynamics of critical fluctuations which were obtained by the dynamical scaling laws. In this connection the general forms of various equal time correlations of long wave length fluctuations have been obtained with the help of the scaling law equation of state. The theory is useful as a starting point for devising approximate schemes of evaluating time correlation functions of physical quantities of our interest, where at least the correct scaling properties of the characteristic times of the problem are retained. This is illustrated by deriving a new self-consistent equation for the time-displaced spin auto-correlation function, and the truncated moments for describing the part of the spectral line shape that is associated with critical fluctuations are introduced and discussed.