  We study the spatial Fourier transform of the spin correlation function
G_q(t) in paramagnetic quantum crystals by direct simulation of a 1d lattice of
atoms interacting via a nearest-neighbor Heisenberg exchange Hamiltonian. Since
it is not practical to diagonalize the s=1/2 exchange Hamiltonian for a lattice
which is of sufficient size to study long-wavelength (hydrodynamic)
fluctuations, we instead study the s -> infinity limit and treat each spin as a
vector with a classical equation of motion. The simulations give a detailed
picture of the correlation function G_q(t) and its time derivatives. At high
polarization, there seems to be a hierarchy of frequency scales: the local
exchange frequency, a wavelength-independent relaxation rate 1/tau that
vanishes at large polarization P ->1, and a wavelength-dependent spin-wave
frequency proportional to q^2. This suggests a form for the correlation
function which modifies the spin diffusion coefficients obtained in a moments
calculation by Cowan and Mullin, who used a standard Gaussian ansatz for the
second derivative of the correlation function.
