Spectrum of Liouville operator and time evolution of classical systems

In the paper we have shown that the distribution function of classical systems with N interacting point particles will approach a function independent of time if the following three conditions are fulfilled: The Liouville operator in L2(Γ) has an absolutely continuous spectrum and a single eigenvalue λ = 0. Approach to equilibrium is understood as the coarse-grained convergence of a distribution function to a coarse-grained equilibrium function. The set of distribution functions is restricted to real functions fεL2(Γ) with the propertyf ≧ 0 a.e. on Γ. Concluding it is noted that the phase space of systems considered is metrically decomposable because of the property TtQE,V = QE,V and also owing to energy being an integral of Hamilton's equations. Hence, the results of the paper apply even to non-ergodic systems.