Abstract
Aspects of transport in a highly multiple-scattering environment are investigated by examining random walkers moving in media having anisotropic angular scattering cross sections (turn-angle distributions). A general expression is obtained for the mean square displacement 〈x2〉 of a random walker executing ann-step walk in an infinite homogeneous material, and results are used to predict scaling relations for the probabilityγ(ρ) that a walker returns to the planar surface of a semi-infinite medium at a distanceρ from the point of its insertion.
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References
F. G. Fernald,Appl. Opt. 23:652 (1984).
J. D. Klett,Appl. Opt. 24:1638 (1985).
H. R. Gordon, R. C. Smith, and J. R. V. Zaneveld,Proc. SPIE 489:2 (1984).
M. A. Blizard,Proc. SPIE 637:2 (1986).
B. White, P. Sheng, M. Postel, and G. Papanicolaou,Phys. Rev. Lett. 63:2228 (1989).
J. Chang and Y. C. Yortsos,SPE Form. Eval. 1990 (March): 31; R. A. Beier, SPE paper 20582, presented at the 65th Annual Technical Conference of the Society of Petroleum Engineers, New Orleans, Louisiana (1990).
M. Stern,Nature 254:56 (1975).
R. Bonner and R. Nossal,Appl. Opt. 20:2097 (1981); R. Bonner and R. Nossal, inLaser-Doppler Blood Flowmetry, A. P. Shepherd, Jr., and P. Å. Öberg, eds. (Kluwer Academic, Boston, 1990), Chapter 2.
D. T. Delpy, M. Cope, P. van de Zee, S. Arridge, S. Wray, and J. Wyatt,Phys. Med. Biol. 33:1433 (1988); B. Chance, J. S. Leigh, H. Miyake, D. S. Smith, S. Nioka, R. Greenfeld, M. Finander, K. Kaufmann, W. Levy, M. Yound, P. Cohen, H. Yoshioka, and R. Boretsky,Proc. Natl. Acad. Sci. USA 85:4971 (1988).
J. M. Schmitt, G. X. Zhou, E. C. Walker, and R. T. Wall,J. Opt. Soc. Am. A 14:2141 (1990).
R. Bonner, R. Nossal, S. Havlin, and G. H. Weiss,J. Opt. Soc. Am. A 4:423 (1987).
R. Nossal, J. Kiefer, G. H. Weiss, R. Bonner, H. Taitelbaum, and S. Havlin,Appl. Opt. 27:3382 (1988); G. H. Weiss, R. Nossal, and R. F. Bonner,J. Mod. Opt. 36:349 (1989).
R. Nossal, R. F. Bonner, and G. H. Weiss,Appl. Opt. 28:2238 (1989).
R. Nossal and R. F. Bonner,SPIE Proc. 1431:21 (1991).
A. H. Gandjbakhche, R. Nossal, and R. F. Bonner,Appl. Opt., submitted.
J. Frenkel,Kinetic Theory of Liquids (Dover, New York, 1955), p. 464.
R. Nossal and G. H. Weiss,J. Stat. Phys. 10:245 (1974).
W. J. Taylor,J. Chem. Phys. 16:257 (1948).
L. G. Henyey and J. L. Greenstein,Ap. J. 93:70 (1941).
M. Patterson, B. Chance, and B. C. Wilson,Appl. Opt. 28:2331 (1989).
R. Nossal and J. M. Schmitt,Proc. SPIE 1430:37 (1991).
Y. Kuga and A. Ishimaru,J. Opt. Soc. Am. A 1:831 (1984).
M. P. van Albada and A. Lagendijk,Phys. Rev. Lett. 55:2692 (1985); P. E. Wolf and G. Maret,Phys. Rev. Lett. 55:2696 (1985).
M. Kac,Probability and Related Topics in Physical Sciences (Interscience, New York, 1959), p. 35.
A. H. Gandjbakhche, Thesis, Université de Paris VII (1989); A. H. Gandjbakhche, P. Mills, and P. Snabre,Appl. Opt., submitted.
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Gandjbakhche, A.H., Bonner, R.F. & Nossal, R. Scaling relationships for anisotropic random walks. J Stat Phys 69, 35–53 (1992). https://doi.org/10.1007/BF01053781
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DOI: https://doi.org/10.1007/BF01053781