Recent progress of the theory of dynamical systems and billiards sheds new light on the nonequilibriums statisticaml mechanics. Mixing, weak mixing and continuouss spectruma are associated to relaxation to equilibrium via entropy incrase. The properties of the relaxation time are reflected in the transport properities which could be anomalous both in Sinai billiard with infinite horizona and in the barrier billiard. Numerical simulations are presented to corroborate these properties.