Quantum dynamics in classical time evolution of correlation functions

The time-dependence of correlation functions under the influence of classical equations of motion is described by an exact evolution equation. For conservative systems thermodynamic equilibrium is a fixed point of these equations. We show that this fixed point is not universally stable, since infinitely many conserved correlation functions obstruct the approach to equilibrium. Equilibrium can therefore be reached at most for suitably averaged quantities or for subsystems, similar to quantum statistics. The classical time evolution of correlation functions shows many dynamical features of quantum mechanics.