Communications in Mathematical Physics

, Volume 242, Issue 1–2, pp 81–135 | Cite as

Self-Averaging of Wigner Transforms in Random Media

  • Guillaume BalEmail author
  • Tomasz Komorowski
  • Lenya  Ryzhik


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.


Mathematical Model Error Estimate Acoustic Wave Limit Theorem Wave Length 
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  1. 1.
    Bal, G., Papanicolaou, G., Ryzhik, L.: Radiative transport limit for the random Schroedinger equation. Nonlinearity 15, 513–529 (2002)CrossRefMathSciNetzbMATHGoogle Scholar
  2. 2.
    Bal, G., Papanicolaou, G., Ryzhik, L.: Self-averaging in time reversal for the parabolic wave equation. Stochastics and Dynamics 2, 507–531 (2002)CrossRefzbMATHGoogle Scholar
  3. 3.
    Bal, G., Ryzhik, L.: Time reversal for classical waves in random media. Comptes Rendus de l’Acad. Sci, Serié I/Math 333, 1041–1046 (2001)Google Scholar
  4. 4.
    Bal, G., Ryzhik, L.: Time reversal and refocusing in random media. SIAM J. Appl. Math. 63, 1475–1498 (2003)CrossRefGoogle Scholar
  5. 5.
    Bambusi, D., Graffi, S., Paul, T.: Long time semiclassical approximation of quantum flows: A proof of the Ehrenfest time. Asymptot. Anal. 21, 149–160 (1999)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Bardos, C., Fink, M.: Mathematical foundations of the time reversal mirror. Asympt. Anal. 29, 157–182 (2002)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Billingsley, P.: Convergence of Probability Measures. New York: Wiley, 1968Google Scholar
  8. 8.
    Blomgren, P., Papanicolaou, G., Zhao, H.: Super-Resolution in Time-Reversal Acoustics. J. Acoust. Soc. Am. 111, 230–248 (2002)CrossRefGoogle Scholar
  9. 9.
    Bouzouina, A., Robert, D.: Uniform semiclassical estimates for the propagation of quantum observables. Duke Math. J. 111, 223–252 (2002)CrossRefMathSciNetzbMATHGoogle Scholar
  10. 10.
    Erdös, L., Yau, H.T.: Linear Boltzmann equation as the weak coupling limit of a random Schrödinger Equation. Commun. Pure Appl. Math. 53, 667–735 (2000)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Clouet, J.F., Fouque, J.P.: A time-reversal method for an acoustical pulse propagating in randomly layered media. Wave Motion 25, 361–368 (1997)CrossRefzbMATHGoogle Scholar
  12. 12.
    Fink, M., Prada, C.: Acoustic time-reversal mirrors. Inverse Problems 17, R1–R38 (2001)Google Scholar
  13. 13.
    Fouque, J.P., Sølna, K.: SIAM J. for Multiscale Modeling and Simulation 1, 239–259 (2003)CrossRefGoogle Scholar
  14. 14.
    Gérard, P., Markowich, P.A., Mauser, N.J., Poupaud, F.: Homogenization limits and Wigner transforms. Comm. Pure Appl. Math. 50, 323–380 (1997)CrossRefMathSciNetGoogle Scholar
  15. 15.
    Kesten, H., Papanicolaou, G.: A limit theorem for stochastic acceleration. Commun. Math. Phys. 78, 19–63 (1980)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Kuperman, W., Hodgkiss, W., Song, H., Akal, T., Ferla, C., Jackson, D.: Phase-conjugation in the ocean. J. Acoust. Soc. Am. 102, 1–16 (1997)Google Scholar
  17. 17.
    Lions, P.-L., Paul, T.: Sur les mesures de Wigner. Rev. Mat. Iberoamericana 9, 553–618 (1993)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Poupaud, F., Vasseur, A.: Classical and quantum transport in random media. J. Math. Pure et Appl. 82, 711–748 (2003)CrossRefGoogle Scholar
  19. 19.
    Papanicolaou, G., Ryzhik, L., Sølna, K.: Statistical stability in time reversal. To appear in SIAM J. Appl. Math., 2003Google Scholar
  20. 20.
    Ryzhik, L., Papanicolaou, G., Keller, J.B.: Transport equations for elastic and other waves in random media. Wave Motion 24, 327–370 (1996)CrossRefMathSciNetzbMATHGoogle Scholar
  21. 21.
    Sølna, K.: Focusing of time-reversed reflections. Waves in Random Media 12, 365–385 (2002)CrossRefGoogle Scholar
  22. 22.
    Spohn, H.: Derivation of the transport equation for electrons moving through random impurities. J. Stat. Phys. 17, 385–412 (1977)zbMATHGoogle Scholar
  23. 23.
    Strook, D., Varadhan, S.R.S.: Multidimensional Diffusion Processes. Berlin-Heidelberg-New York: Springer-Verlag, 1979Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Guillaume Bal
    • 1
    Email author
  • Tomasz Komorowski
    • 2
  • Lenya  Ryzhik
    • 3
  1. 1.Department of Applied Physics & Applied MathematicsColumbia UniversityNew YorkUSA
  2. 2.Institute of MathematicsUMCSLublinPoland
  3. 3.Department of MathematicsUniversity of ChicagoChicagoUSA

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