Abstract
We consider diffusive systems with static disorder, such as Lorentz gases, lattice percolation, ants in a labyrinth, termite problems, random resistor networks, etc. In the case of diluted randomness we can apply the methods of kinetic theory to obtain systematic expansions of dc and ac transport properties in powers of the impurity concentrationc. The method is applied to a hopping model on ad-dimensional cubic lattice having two types of bonds with conductivityσ andσ 0=1, with concentrationsc and 1−c, respectively. For the square lattice we explicitly calculate the diffusion coefficientD(c,σ) as a function ofc, to O(c2) terms included for different ratios of the bond conductivityσ. The probability of return at long times is given byP 0(t)≈ [4πD(c,σ)t]−d/2, which is determined by the diffusion coefficient of the disordered system.
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References
E. H. Hauge, inTransport Phenomena, G. Kirczenov and J. Marro, eds. (Lecture Notes in Physics, No. 31, Springer-Verlag, Berlin, 1974), p. 338; J. R. Dorfman and H. van Beijeren, inStatistical Mechanics, Part B, Time Dependent Processes, B. J. Berne, ed. (Plenum Press, New York, 1977).
Th. M. Nieuwenhuizen, P. F. J. v. Velthoven, and M. H. Ernst,Phys. Rev. Lett., (1986).
D. Frenkel, preprint.
M. H. Ernst, P. F. J. van Velthoven, and Th. M. Nieuwenhuizen,J. Phys. A, Math. Gen. 19 (1986).
S. Kirkpatrick,Rev. Mod. Phys. 45:574 (1973); inIll-Condensed Matter, B. Balian, K. Maynard, and G. Toulouse, eds. (North-Holland, Amsterdam, 1979).
B. P. Watson and P. J. Leath,Phys. Rev. B 9:4893 (1974);12:498(E) (1975).
T. Odagaki, M. Lax, and A. Puri,Phys. Rev. B 28:2755 (1983); T. Odagaki,Phys. Rev. B 33:544 (1986).
E. J. Garboczi and M. F. Thorpe,Phys. Rev. B 33:3289 (1986).
A. B. Harris and S. Kirkpatrick,Phys. Rev. B 16:542 (1977).
J. B. T. M. Roerdink and K. E. Shuler,J. Stat. Phys. 41:581 (1985); J. B. T. M. Roerdink,Physica 132A (1985).
D. C. Hong, H. E. Stanley, A. Coniglio, and A. Bunde,Phys. Rev. B 33:4564 (1986).
D. Bedeaux, K. Lakatos-Lindenberg, and K. E. Shuler,J. Math. Phys. 12:2116 (1971).
J. P. Straley,Phys. Rev. B 15:5733 (1977).
T. Morita and T. Horiguchi,J. Math. Phys. 12:981 (1971).
S. Alexander, J. Bernasconi, W. R. Schneider, and R. Orbach,Rev. Mod. Phys. 53:175 (1981).
P. J. H. Denteneer and M. H. Ernst,Phys. Rev. B 29:1755 (1984).
I. S. Gradshteyn and I. M. Ryzhik,Table of Integrals, Series and Products (Academic Press, New York, 1965).
M. Abramowitz and I. A. Stegun,Handbook of Mathematical Functions (Dover, New York, 1970).
K. Golden and G. Papanicolaou,Commun. Math. Phys. 90:473 (1983).
G. L. Montet,Phys. Rev. B 7:650 (1973).
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Ernst, M.H., van Velthoven, P.F.J. Random walks on cubic lattices with bond disorder. J Stat Phys 45, 1001–1030 (1986). https://doi.org/10.1007/BF01020586
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DOI: https://doi.org/10.1007/BF01020586