Self-Averaging of Wigner Transforms in Random Media

Abstract

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.

The stochastic acceleration problem in two dimensions

“…The overall strategy of the proof of Theorem 2.1 is similar to that in [1,4,5]. Briefly, it can be summarized as follows.…”

“…This was done by multiplying the right side of (2.5) by a cut-off function Θ(t, X(t), V (t); π) that depends both on the current position (X(t), V (t)) and on the past trajectory π. More precisely, one introduces a time mesh t (1) k = 1/p 1 -the function Θ is equal to zero on the time interval [t (1) k , t (1) k+1 ] if the momentum V (t) is far from V (t (1) k ). This keeps momenta aligned on the time scale p −1 1 and propels the particle forward.…”

“…This keeps momenta aligned on the time scale p −1 1 and propels the particle forward. The second cut-off requires that if X(t), t ∈ [t (1) k , t (1) k+1 ] is close to X(s) with s ≤ t (1) k−1 then Θ = 0 and dynamics is trivial. The latter assumption ensures that "no information is gained" at self-intersections.…”

“…The non-transversality condition of self-intersections is formalized as follows. First, we set-up the mesh t (1) k on which we require the momentum alignment, as in [1,4,5]. In addition, we set-up a finer mesh t (2) k = k/p 2 with p 2 ≫ p 1 and impose that Θ(t) = 0 if at time t the position X(t) is close to X(t (2) k ) with t (2)…”

“…be the set of times spent by the trajectory during the time interval [t (1) k , t (1) k+2 ] in a narrow tube around the past and in a direction transversal to the tube. The following proposition gives an upper bound on the measure of this set.…”