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March  2008, 1(1): 85-100. doi: 10.3934/krm.2008.1.85

Self-averaging of kinetic models for waves in random media

Guillaume Bal 1, and  Olivier Pinaud 2,

1. 

Department of Applied Physics and Applied Mathematics, Columbia University, New York NY, 10027
2. 

Université de Lyon, Université Lyon 1, CNRS, UMR 5208 Institut Camille Jordan/ISTIL, Bâtiment du Doyen Jean Braconnier, 43, blvd du 11 novembre 1918, F - 69622 Villeurbanne Cedex, France

Received  November 2007 Revised  November 2007 Published  February 2008

Kinetic equations are often appropriate to model the energy density of high frequency waves propagating in highly heterogeneous media. The limitations of the kinetic model are quantified by the statistical instability of the wave energy density, i.e., by its sensitivity to changes in the realization of the underlying heterogeneous medium modeled as a random medium. In the simplified Itô-Schrödinger regime of wave propagation, we obtain optimal estimates for the statistical instability of the wave energy density for different configurations of the source terms and the domains over which the energy density is measured. We show that the energy density is asymptotically statistically stable (self-averaging) in many configurations. In the case of highly localized source terms, we obtain an explicit asymptotic expression for the scintillation function in the high frequency limit.
Keywords: Itô-Schrödinger equation, self-averaging, statistical stability, scintillation function., Waves in random media, kinetic models.
Mathematics Subject Classification: 35L05, 60H25, 35Q4.
Citation: Guillaume Bal, Olivier Pinaud. Self-averaging of kinetic models for waves in random media. Kinetic & Related Models, 2008, 1 (1) : 85-100. doi: 10.3934/krm.2008.1.85
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