  We consider the quantum and classical Liouville dynamics of a non-integrable
model of two coupled spins. Initially localised quantum states spread
exponentially to the system dimension when the classical dynamics are chaotic.
The long-time behaviour of the quantum probability distributions and, in
particular, the parameter-dependent rates of relaxation to the equilibrium
state are surprisingly well approximated by the classical Liouville mechanics
even for small quantum numbers. As the accessible classical phase space becomes
predominantly chaotic, the classical and quantum probability equilibrium
configurations approach the microcanonical distribution, although the quantum
equilibrium distributions exhibit characteristic `minimum' fluctuations away
from the microcanonical state. The magnitudes of the quantum-classical
differences arising from the equilibrium quantum fluctuations are studied for
both pure and mixed (dynamically entangled) quantum states. In both cases the
standard deviation of these fluctuations decreases as $(\hbar/{\mathcal
J})^{1/2}$, where ${\mathcal J}$ is a measure of the system size. In
conclusion, under a variety of conditions the differences between quantum and
classical Liouville mechanics are shown to become vanishingly small in the
classical limit (${\mathcal J}/\hbar \to \infty$) of a non-dissipative model
endowed with only a few degrees of freedom.
