The generalized Langevin equation (GLE) is a reformulation of the Heisenberg equation of motion, and hence, an exact equation. It is the basis of the memory function approach, a very widely used method for studying dynamics of classical and quantum fluids. The GLE was first derived by Mori in a very formal way. A much simpler and more physically motivated derivation was given by us some years later. In this work we provide perhaps the simplest possible derivation of the GLE. The simplicity of the derivation helps to bring out the subtleties present in this important dynamical relationship.

The time-dependent autocorrelation function has been derived for the x component of the total spin for the S = 1/2 constant-coupling anisotropic Heisenberg model, i.e. , the Van der Waals system. For T & T" the time correlation is shown to be Gaussian for both XY-like and Ising-like regimes of the model. For T & T" the correlation function is still Gaussian if the model is XY-like; but it is oscillatory if the model is Isinghke. Critical slow down appears only with the Ising-like system for this time-correlation function.

The autocorrelation function of a single spin (s(t) s(0)) has been obtained from the timeevolution solutions given in the preceding paper by carrying out the ensemble averages explicitly.Slow decay is found in the transverse component and only at high temperatures ( T & T, ). The exponent x. , where (s"(t)s'(0) ) -t, as t~oo, is found to depend discontinuously on the spin-spin interaction strength R = J, /J:~=2 if R =0,~=3 if 0&R~2 (R&1),~=~if R=1, and~=~if R & 2. Slow decay in this model is attributed to nondimensional effects, e.g. , cooperativity. Physical and mathematical mechanisms of the slow decay are described.

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The transverse components of the susceptibility and fluctuation for the spin van der Waals model are obtained for high-and low-temperature XYand Ising-like regimes. The difference between the susceptibility and fluctuation, which arises because of noncommutativity, is related to certain correlation functions of S,. In the XY-like regime, the difference vanishes since the principal long-range order is (S"). But in the Ising-like regime, the difference persists since the principal long-range order now is (S,). Our results are compared with the bounds of Falk and Bruch on the ratio of the susceptibility to the fluctuation. In the critical region as well as in the high-temperature region the upper and lower bounds merge. In these regions, the two correlation functions become identical in the manner indicated by Falk and Bruch, In the low-temperature region, the bounds do not merge. Here we find that in the Ising-like regime of the model the susceptibility and fluctuation are properly different, but in the XY-like regime the two functions nevertheless become identical. Thus, the susceptibility and fluctuation can evidently be the same without the bounds necessarily merging.