  The long-time behavior of the infinite temperature spin correlation functions
describing the free induction decay in nuclear magnetic resonance and
intermediate structure factors in inelastic neutron scattering is considered.
These correlation functions are defined for one-, two- and three-dimensional
infinite lattices of interacting spins both classical and quantum. It is shown
that, even though the characteristic timescale of the long-time decay of the
correlation functions considered is non-Markovian, the generic functional form
of this decay is either simple exponential or exponential multiplied by cosine.
This work contains (i) summary of the existing experimental and numerical
evidence of the above asymptotic behavior; (ii) theoretical explanation of this
behavior; and (iii) semi-empirical analysis of various factors discriminating
between the monotonic and the oscillatory long-time decays. The theory is based
on a fairly strong conjecture that, as a result of chaos generated by the spin
dynamics, a Brownian-like Markovian description can be applied to the long-time
properties of ensemble average quantities on a non-Markovian timescale. The
formalism resulting from that conjecture can be described as ``correlated
diffusion in finite volumes.''
