We show the exact expression of the quantum mechanical time correlation function in the
phase space formulation of quantum mechanics. The trajectory-based dynamics that
conserves the quantum canonical distribution–equilibrium Liouville dynamics (ELD)
proposed in Paper I is then used to approximately evaluate the exact expression. It gives
exact thermal correlation functions (of even nonlinear operators, ie, nonlinear functions of
position or momentum operators) in the classical, high temperature, and harmonic limits …