Abstract
We consider the application of fractal concepts to polymer statistics and to anomalous transport in randomly porous media. It is found that answers to interesting physics questions can be expressed in terms of several new fractal dimensions (in addition to “the” fractal dimensiond f ): (1)d BB f , the fractal dimension of the backbone, arises in connection with electric current flow, (2)d red, the fractal dimension of the singly connected bonds in the backbone, arises in connection with its equivalence to the thermal scaling power, (3)d E, the fractal dimension of the of the elastic backbone, (4)d u, the fractal dimension of the unscreened perimeter, arises in connection with the viscosity singularity at the gelation threshold, (5)d min the fractal dimension of the minimum path (or “chemical distance”) between two sites, arises in co-nnection with the Aharony-Stauffer conjecture, (6)dw, the fractal dimension of a random walk, (7)d G, the fractal dimension of growth sites that arise as a random walk creates a cluster. Relations among these fractal dimensions are discussed, some of which can be proved and others of which are conjectures whose validity has been established only in certain limiting cases.
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References
H. E. Stanley,Introduction to Phase Transitions and Critical Phenomena, 2nd ed. (Oxford University Press, New York, 1985). See also H. E. Stanley,Fractal Concepts in Statistical Physics (scheduled for completion December 1984).
B. Mandelbrot,Les objects fractals (Flammarion, Paris, 1975) [English versions:Fractals: Form, Chance and Dimension (W. H. Freeman, San Francisco, 1977);The Fractal Geometary of Nature (W. H. Freeman, San Francisco, 1977)].
(a) V. Privman, F. Family, and A. Margolina,J. Phys. A, to be published, (b) I. Majid and H. E. Stanley,J. Phys. A, to be published.
D. Stauffer,Phys. Rep. 54:1 (1979).
F. Family and D. P. Landau, eds.,Kinetics of Aggregation and Gelation (North-Holland, Amsterdam, 1984).
A. Kapitulnik and G. Deutscher,Phys. Rev. Lett. 43:1444 (1982).
R. F. Voss, R. B. Laibowitz, and E. I. Allessandrini,Phys. Rev. Lett. 49:1441 (1982).
P. G. de Gennes,La Recherche 7:919 (1976).
R. Zallen,The Physics of Amorphous Solids (John Wiley and Sons, New York, 1983).
A. Ritzenberg and R. J. Cohen,Phys. Rev. B 30, in press.
H. E. Stanley, inDisordered Systems and Localization, C. Castellani, C. DiCastro and L. Peliti, eds. (Springer, Berlin, 1981), p. 59.
B. Jouhier, C. Allain, B. Gauthier-Manuel, and E. Guyon, inPercolation Structures and Processes, G. Deutscher, R. Zallen, and J. Adler, eds. (Adam Hilger, Bristol, 1983), p. 167.
G. S. Grest and M. H. Cohen, inPercolation Structures and Processes, G. Deutscher, R. Zallen, and J. Adler, eds. (Adam Hilger, Bristol, 1983), p. 187.
L. S. Schulman and P. E. Seiden, inPercolation Structures and Processes, G. Deutscher, R. Zallen, and J. Adler, eds. (Adam Hilger, Bristol, 1983), p. 251.
G. Deutscher, A. Kapitulnik, and M. Rappaport, inPercolation Structures and Processes, G. Deutscher, R. Zallen, and J. Adler, eds. (Adam Hilger, Bristol, 1983), p. 207.
D. W. Schaefer and K. D. Keefer,Phys. Rev. Lett., in press.
D. W. Schaefer, J. E. Martin, P. Wiltzius, and D. S. Cannell,Phys. Rev. Lett. 52:2371 (1984).
D. W. Schaefer, inKinetics of Aggregation and Gelation (North-Holland, Amsterdam, 1984).
C. D. Mitescu and J. Roussenq, inPercolation Structures and Processes, G. Deutscher, R. Zallen, and J. Adler, eds. (Adam Hilger, Bristol, 1983).
H. Nencka-Ficek,J. Phys. A, to be published.
H. E. Stanley,J. Phys. A 10:L211 (1977), Section 4.
D. R. Hamann,Phys. Today 36(5):25 (1983).
A. Coniglio,J. Phys. A 15:3829 (1982)
A. Coniglio,Phys. Rev. Lett. 46:250 (1981)
R. Pike and H. E. Stanley,J. Phys. A 14:L169 (1981)
D. C. Hong and H. E. Stanley,J. Phys. A 16:L525 (1983)
M. F. Sykes,J. Phys. A 17:1565 (1984).
H. J. Herrmann and H. E. Stanley,Phys. Rev. Lett. 53:1121 (1984).
A. Alexander and R. Orbach,J. Phys. (Paris) Lett. 43:L625 (1982).
D. Ben-Avraham and S. Havlin,J. Phys. A 15:L691 (1982).
Y. Gefen, A. Aharony, and S. Alexander,Phys. Rev. Lett. 50:77 (1983).
R. Rammal and G. Toulouse,J. Phys. (Paris) Lett. 44:L13 (1983).
R. Pandey and D. Stauffer,Phys. Rev. Lett. 51:527 (1983); R. Pandey, D. Stauffer, A. Margolina, and J. G. Zabolitzky,J. Stat. Phys. 34:427 (1984).
P. Meakin and H. E. Stanley,Phys. Rev. Lett. 51:1457 (1983).
F. Leyvraz and H. E. Stanley,Phys. Rev. Lett. 51:2048 (1983).
H. E. Stanley and A. Coniglio,Phys. Rev. B (RC) 29:522 (1984).
A. Aharony and D. Stauffer,Phys. Rev. Lett. 52:2368 (1984).
A. Coniglio and H. E. Stanley,Phys. Rev. Lett. 52:1068 (1984).
P. Meakin, F. Leyvraz, and H. E. Stanley,Phys. Rev. B, to be published.
R. Pike and H. E. Stanley,J. Phys. A 14:L169 (1981).
D. C. Hong and H. E. Stanley,J. Phys. A 16:L525;J. Phys. A 16:L525 (1983).
A. Coniglio,J. Phys. A 15:3829 (1982).
S. Havlin and R. Nossal,J. Phys. A 17:L427 (1984).
I. Majid, D. Ben-Avraham, S. Havlin, and H. E. Stanley,Phys. Rev. B 30:1626 (1984).
Z. Djordjevic, S. Havlin, H. E. Stanley, and G. H. Weiss,Phys. Rev. B 30:478 (1984).
S. Havlin, Z. Djordjevic, I. Majid, H. E. Stanley, and G. H. Weiss,Phys. Rev. Lett. 53:178 (1984).
S. Havlin, B. Trus, and H. E. Stanley,Phys. Rev. Lett. 53:1288 (1984).
J. Vannimenus, J. P. Nadal, and C. Martin,J. Phys. A 17:L351 (1984).
H. J. Herrmann, D. C. Hong, and H. E. Stanley,J. Phys. A 17:L261 (1984).
P. G. de Germes,J. Phys. (Paris) Colloque 41:C3 (1980).
P. Meakin and H. E. Stanley,J. Phys. A 17:L173 (1984).
K. Kang and S. Redner,Phys. Rev. Lett. 52:955 (1984).
C. Allain and B. Gauthier-Manuel,Phys. Rev. Lett., in press.
H. D. Bale and P. W. Schmidt,Phys. Rev. Lett. 53:596 (1984).
H. G. E. Hentschel,Phys. Rev. Lett. 52:212 (1984).
H. G. E. Hentschel and J. Deutch,Phys. Rev. A 29:1609 (1984).
R. B. Laibowitz and Y. Gefen,Phys. Rev. Lett. 53:380 (1984).
J. E. Martin and B, W. Ackerson,Phys. Rev. Lett., in press.
M. Matsushita, M. Sano, Y. Hayakawa, H. Honjo, and Y. Sawada,Phys. Rev. Lett. 53:286 (1984).
M. Muthukumar,Phys. Rev. Lett. 50:839 (1983).
L. Niemeyer, L. Pietronero, H. J. Weismann,Phys. Rev. Lett. 52:1033 (1984).
M. Tokuyama and K. Kawasaki,Phys. Lett. 100A:337 (1984).
D. A. Weitz and M. Oliveira,Phys. Rev. Lett. 52:1433 (1984).
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Supported in part at the Center for Polymer Studies by grants from ONR and NSF.
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Stanley, H.E. Application of fractal concepts to polymer statistics and to anomalous transport in randomly porous media. J Stat Phys 36, 843–860 (1984). https://doi.org/10.1007/BF01012944
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DOI: https://doi.org/10.1007/BF01012944