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We consider the quantum and classical Liouville dynamics of a non-integrablemodel of two coupled spins. Initially localised quantum states spreadexponentially to the system dimension when the classical dynamics are chaotic.The long-time behaviour of the quantum probability distributions and, inparticular, the parameter-dependent rates of relaxation to the equilibriumstate are surprisingly well approximated by the classical Liouville mechanicseven for small quantum numbers. As the accessible classical phase space becomespredominantly chaotic, the classical and quantum probability equilibriumconfigurations approach the microcanonical distribution, although the quantumequilibrium distributions exhibit characteristic `minimum' fluctuations awayfrom the microcanonical state. The magnitudes of the quantum-classicaldifferences arising from the equilibrium quantum fluctuations are studied forboth pure and mixed (dynamically entangled) quantum states. In both cases thestandard deviation of these fluctuations decreases as $(\hbar/{\mathcalJ})^{1/2}$, where ${\mathcal J}$ is a measure of the system size. Inconclusion, under a variety of conditions the differences between quantum andclassical Liouville mechanics are shown to become vanishingly small in theclassical limit (${\mathcal J}/\hbar \to \infty$) of a non-dissipative modelendowed with only a few degrees of freedom.Comment: 12 pages revtex, 11 figures, submitted to Phys Rev

Quantum-classical correspondence for the equilibrium distributions of two interacting spins