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Stochastic transport in disordered systems

J. Chem. Phys. 78, 6849 (1983); doi:10.1063/1.444631

Issue Date: 1 June 1983

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Muhammad Sahimi, Barry D. Hughes, L. E. Scriven, and H. T. Davis
Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, Minnesota 55455
We develop a theory of stochastic transport in disordered media, starting from a linear master equation with random transition rates. A Green function formalism is employed to reduce the basic equation to a form suitable for the construction of a class of effective medium approximations (EMAs). The lowest order EMA, developed in detail here, corresponds to recent approximations proposed by Odagaki and Lax [Phys. Rev. B 24, 5284 (1981], Summerfield [Solid State Commun. 39, 401 (1981)], and Webman [Phys. Rev. Lett. 47, 1496 (1981)]. It yields an effective transition rate Wm which can be identified as the memory kernel of a generalized master equation, and used to define an associated continuous-time random walk on a uniform lattice. The long-time behavior of the mean-squared displacement arising from an initially localized state can be found from Wm, as can diffusion constants in any case where the long-time behavior of the system is diffusive. Detailed calculations are presented for seven lattice systems in one, two, and three dimensions, and for a variety of probability density functions f(w) for the transitions rates. For percolation-type densities, i.e., those with only a fraction p<1 of the bonds transmitting, the EMA predicts three distinct kinds of behavior: localization, ``fractal'' transport with slower than linear growth of the mean-squared displacement, and diffusion in the cases p<pc, p=pc, p>~pc, respectively, where pc is the bond percolation threshold of the lattice. Depending on the form of f(w) near w=0, critical exponents may take values independent of f(w) (``universality'') or heavily dependent on f(w) (``nonuniversality''). The Journal of Chemical Physics is copyrighted by The American Institute of Physics.
History: Received 7 June 1982; accepted 9 September 1982
Permalink: http://link.aip.org/link/?JCPSA6/78/6849/1
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KEYWORDS and PACS

Keywords
PACS
  • 72.90.+y
    Electronic transport in condensed matter Other topics in electronic transport in condensed matter
  • 66.90.+r
    Transport properties of condensed matter (nonelectronic) Other topics in nonelectronic transport properties
  • YEAR: 1983

PUBLICATION DATA

ISSN:
0021-9606 (print)   1089-7690 (online)
Publisher:
AIP is a member of CrossRef AIP

REFERENCES (73)
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    64. in M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965), the right-hand side of (A24) defines K(k2), a most unfortunate convention.
  64. M. L. Glasser and I. J. Zucker, Proc. Natl. Acad. Sci. USA 74, 1800 (1977).
  65. See, for example, Ref. 60.
  66. A discussion of lattice Green functions for the hexagonal and triangular lattices, arising from a random walk problem, has recently been given by F. S. Henyey and V. Seshadri, J. Chem. Phys. 76, 5530 (1982).
  67. M. F. Sykes and J. W. Essam, J. Math. Phys. 5, 1117 (1964);
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  69. S. Kirkpatrick, in Ill-Condensed Matter, edited by R. Balian, R. Maynard, and G. Toulouse (North-Holland, Amsterdam, 1979), pp. 321–403.
  70. Quoted from Ref. 43.
  71. It may be noted from Table I that for the simple cubic, body-centered cubic and face-centered cubic lattices, pc[approximate]g(0). This is an example of a more general result: see M. Sahimi, B. D. Hughes, L. E. Scriven, and H. T. Davis, J. Phys. A (in press).
  72. The single-bond EMA predicts that tc = 1, independent of the dimension of the system. Generally accepted values (Refs. 44 and 45) are tc[approximate]1.33 in two dimensions and tc[approximate]1.72 in three dimensions.
  73. Note added in proof: M. J. Stephen and R. Kariotis, Phys. Rev. B 26, 2917 (1982)
    74. have recently obtained the leading correction terms to the EMA for a linear chain using a replica method. K. Kaski, R. A. Tahir-Kheli, and R. J. Elliott, J. Phys. C 15, 209 (1982) have given a CPA analysis of diffusion of a particle in a static but random arrangement of background particles.

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