Nonequilibrium Statistical Mechanics of Generalized Spin Van Der Waals Model.

We have derived the autocorrelation function for the x-component of the total spin operator for a constant coupling spin-exchange interaction model, i.e., the spin van der Waals model. The autocorrelation function is Gaussian both in the high- and low-temperature XY-like regions. It is oscillatory in the low-temperature Ising-like region, but Gaussian in the high-temperature region. Critical slowdown appears only in the Ising-like regime. We have obtained exact expressions for the susceptibility and fluctuation for this model. We find that for T > T(,c), the susceptibility and fluctuation are the same whether the system is XY-like or Ising-like. For T < T(,c), they are the same in the XY-like regime, but they are different in the Ising-like regime. The bounds on these two quantities are consistent with the Falk and Bruch's Bounds. By applying the method of recurrence relations developed by Lee, we have derived the relaxation function for this model. As T (--->) T(,c) for the Ising-like system, the observed critical slowdown is manifested by a collapse of Hilbert space from infinite dimensions to finite dimensions. The dimensionality of Hilbert space for the XY-like system, however, remains infinite as T passes through T(,c). We have also generalized the spin van der Waals model to include four-spin interaction with the spin values 1/2 and 1. We find that the generalized spin van der Waals model exhibits a first order and a second order phase transitions. The autocorrelation function for the generalized spin-1 van er Waals model has been derived for three thermodynamic phases. It is Gaussian in the paramagnetic phase, and it is oscillatory in the ferroquadrupolar phase. However, in the ferromagnetic phase, the autocorrelation function is complicated oscillatory if it is in the Ising-like regime. On the other hand, the autocorrelation function is Gaussian if it is in the XY-like regime. We find that the autocorrelation function for the generalized spin- 1/2 van er Waals model is rather simpler than the spin-1 case, owing to the absence of quadrupolar ordering.