Geodesic
Transmission of waves and particles through long random barriers
Teoretičeskaâ i matematičeskaâ fizika, Tome 68 (1986) no. 3, pp. 433-448

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The logarithmic damping rate of the coeffficient of transmission, averaged over the scatterer configurations, of a long one-dimensional barrier is expanded in powers of the scatterer concentration, and this expansion is analyzed. It is shown that the damping rate is analytic at low concentrations and for nonresonant scattering in both the case of completely randomly distributed scatterers as well as when there are statistical correlations in their distribution. The technique of the proof is analogous to the technique employed with the Kirkwood–Salsburg correlation equations of statistical physics. 
@article{TMF_1986_68_3_a10,
     author = {A. V. Marchenko and L. A. Pastur},
     title = {Transmission of waves and particles through long random barriers},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {433--448},
     publisher = {mathdoc},
     volume = {68},
     number = {3},
     year = {1986},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_1986_68_3_a10/}
}
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A. V. Marchenko; L. A. Pastur. Transmission of waves and particles through long random barriers. Teoretičeskaâ i matematičeskaâ fizika, Tome 68 (1986) no. 3, pp. 433-448. http://geodesic.mathdoc.fr/item/TMF_1986_68_3_a10/