Self-averaging of Wigner transforms in random media

We establish the self-averaging properties of the Wigner transform of a mixture of states in the regime when the correlation length of the random medium is much longer than the wave length but much shorter than the propagation distance. The main ingredients in the proof are the error estimates for the semiclassical approximation of the Wigner transform by the solution of the Liouville equations, and the limit theorem for two-particle motion along the characteristics of the Liouville equations. The results are applied to a mathematical model of the time-reversal experiments for the acoustic waves, and self-averaging properties of the re-transmitted wave are proved.