Abstract
This talk is designed to complement that of D. Stauffer; together both seek to review recent work in percolation theory that has taken place after completion of the two recent reviews by Stauffer (1979) and Essam (1980). Stauffer's talk focusses on new results concerning percolations clusters, while this talk concerns some less well understood topics, including possible applications of percolation to the real world. The organization of this talk is presented in the following outline. After presenting a word of philosophy, we shall describe several topics and exemplify each with a particular system:
John Simon Guggenheim Memorial Fellow., 1980–81.
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
References
R. J. Glauber, J. Math. Phys. 4, 294 (1963).
T. D. Lee and C. N. Yang, Phys. Rev. 87, 410 (1952); the identification of the upper marginal dimensionalityd+ = 6and the first expansions in(d+-d)were carried out in M. E. Fisher, Phys. Rev. Lett. 40, 1610 (1978).
G. Parisi and N. Sourlas, Phys. Rev. Lett. 46, 871 (1981); T. C. Lubensky and A. J. McKane, J. Phys. A 14, L157 (1981).
S. Kirkpatrick, AIP Conf. Proc. 40, 99 (1978).
G. Shlifer, W. Klein, P. J. Reynolds, and H. E. Stanley, J. Phys. A 12, L169 (1979).
The distinction between the Hausdorff-Besicovitch or ‘fractal’ dimensionality D = 2yh-d = γ/ν of the entire system and the fractal dimensionality D† = yh = βδ/ν of the incipient infinite cluster was made clear by remarks of D. Stauffer, G. Harrison, G. Bishop, and W. Klein; it is described in H. E. Stanley, J. Phys. A 10, L211 (1977). The fractal dimension was first related to critical exponents in H. E. Stanley, R. J. Birgeneau, P. J. Reynolds, and J. F. Nicoll, J. Phys. C 9, L553 (1976). Recently, critical properties have been calculated very elegantly for various fractal lattices (see talk by A. Aharony in this conference; also Y. Gefen, B. Mandelbrot, and A. Aharony, Phys. Rev. Lett. 45, 855 (1980). A description of many objects with non-integer Hausdorff-Besicovitch dimensionality may be found in B. B. Mandelbrot, Fractals: Form, Chance, and Dimension (W. H. Freeman, San Francisco, 1977).
R. J. Birgeneau and H. E. Stanley, unpublished.
R. Pike and H. E. Stanley, J. Phys. A 14, L169 (1981).
A. Coniglio, unpublished.
A. Coniglio, Phys. Rev. Lett. 46, 250 (1981).
See, e.g., G. Deutcher, this conference; M. Lagües, R. Ober and C. Taupin, J. Phys. (Paris) 39, L487 (1978).
P. J. Flory, J. Am. Chem. Soc. 63, 3083, 3091, 3096 (1941); W. H. Stockmayer, J. Chem. Phys. 11, 45 (1943).
M. E. Fisher and J. W. Essam, J. Math. Phys. 2, 609 (1961); H. L. Frisch and J. M. Hammersley, J. Soc. Ind. Appl. Math. 11, 894 (1963); D. Stauffer, Chem. Soc. Faraday Trans. 72, 1354 (1976); P. G. de Gennes, J. Phys. (Paris) 36, 1049 (1976).
S. W. Haan and R. Zwanzig, J. Phys. A 10, 1547 (1977); T. Vicsek and J. Kertész, J. Phys. A 14, L31 (1981); E. T. Gawlinski and H. E. Stanley, “Continuum percolation in two dimensions: Monte Carlo tests of scaling and universality for non-interacting discs”, preprint submitted to J. Phys. A Lett; A. Geiger and H. E. Stanley, “Continuum percolation in three dimensions: Molecular dynamics tests of universality for interacting ST2 particles”, preprint.
A. Coniglio, H. E. Stanley and W. Klein, Phys. Rev. Lett. 42, 518 (1979); “Solvent effects on polymer gels: A statistical mechanical model” (preprint submitted to Phys. Rev. B).
T. Tanaka, G. Swislow and I Ohmine, Phys. Rev. Lett. 42, 1556 (1979).
T. W. Barrett, “Site-Bond Correlated-Percolation theory of polymer gelation applied to a polymer with preferential binding at two disparate sites”, preprint (submitted to Phys. Rev. Lett.). The CSK model has also been extended by A. E. Gonzales and S. Muto [J. Chem. Phys. 73, 4668 (1980)] to explain the phenomenon of crossing gelation curves observed by P. Ruiz-Azuara, T. Tanaka, A. Coniglio, H. E. Stanley and W. Klein, “Dependence of the gelation curve on solvent composition: Crossing points”, preprint submitted to J. Chem. Phys.
F. Delyon, B. Souillard, and D. Stauffer, “Percolatioe phase transition without appearance of an infinite network”, preprint submitted to J. Phys. A Lett. See also M. Aizenman, F. Delyon, and B. Souillard, J. Stat. Phys. 23, 267 (1980).
R. B. Griffiths, Phys. Rev. Lett. 23, 17 (1969).
T. C. Lubensky and A. J. McKane, this Conference (poster session).
A. Coniglio and W. Klein, J. Phys. A 13, 2775 (1980).
H. Müller-Krumbhaar, Phys. Lett. A 50, 2708 (1974).
J. Roussenq, Thése, Université de Provence, 1980.
D. Stauffer, J. Phys. (Paris) 42, L99 (1981).
H. E. Stanley, J. Phys. A 12, L329 (1979). A more detailed description and development of the percolation picture, together with extensive comparison to experimental data, is in H. E. Stanley and J. Teixeira, J. Chem. Phys. 73, 3404 (1980). Detailed Monte Carlo calculations have been carried out by R. L. Blumberg, G. Shlifer and H. E. Stanley, J. Phys. A 13, L147 (1980), while renormalization group calculations have been performed by A. Gonzales and P. J. Reynolds, Phys. Lett. 80A, 357 (1980). Detailed network analysis of computer water is presented in A. Geiger, H. E. Stanley and R. L. Blumberg, “Connectivity studies of liquid water: Hydrogen bond networks and four-coordinated water molecules”, preprint. An elementary summary of the overall research thus far is presented in H. E. Stanley, J. Teixeira, A. Geiger, and R. L. Blumberg, Physics 106A, 260 (1981).
E.g., using no adjustable parameters, one finds agreement between the predictions of the binomial theorem (6), based on the assumption of random bond breaking, and the computer simulations of Geiger (Ref. 25) and also M. Mezei and D. L. Beveridge, J. Chem. Phys. 74, 622 (1981); W. L. Jorgensen, Chem. Phys. Lett. 70, 326 (1980); C. Pengali, M. Rao and B. J. Berne, Mol. Phys. 40, 661 (1980).
L. Bosio, J. Teixeira, and H. E. Stanley, Phys. Rev. Lett. 46, 597 (1981); see also the earlier work of R. W. Hendricks, P. G. Mardon and L. B. Shaffer, J. Chem. Phys. 61, 319 (1974).
G. D'Arrigo, G. Maisano, F. Mallamace, P. Migliardo and F. Wanderlingh, “Raman scattering and structure of normal and supercooled water”, preprint scheduled for J. Chem. Phys. 75 (1981); see also R. Bansil, J. L. Taafe, and J. Wiafe Akenten, “Raman spectroscopy of supercooled water”, preprint submitted to J. Chem. Phys.
In this context the word “continuum” means that the sites are not in two discrete states (as in bichromatic percolation), nor in 5 discrete states (as in polychromatic discrete percolation), but rather in a continuum of possible states. Since the term “polychromatic continuum percolation” is easily confused with the term continuum percolation used to describe systems whose elements are not constrained to the vertices of a lattice, we prefer the simpler and more descriptive designation “rainbow percolation”.
D. Bertolini, M. Cassettari, and G. Salvetti, “The dielectric relaxation time of supercooled water down to-21°C”, preprint.
C. A. Angell, “Supercooled Water”, preprint.
M. H. Brodsky, Solid State Commun. 36, 55 (1980).
Recent references on directed percolation include J. Kertész and T. Vicsek, J. Phys. C 13, L343 (1980); D. Dhar and M. Barma, J. Phys. A 14, Ll (1981); S. P. Obukhov, Physics 101A, 145 (1980); J. L. Cardy and R. L. Suger, J. Phys. A 13, L423 (1980); W. Kinzel and J. Yeomans, preprint; E. Domany and W. Kinzel, preprint; E. Domany and W. Kinzel, preprint; P. J. Reynolds, preprint; S. Redner, preprint; S. Redner and A. C. Brown, preprint.
Recent work on restricted valence percolation includes S. G. Whittington, K. M. Middlemiss, G. M. Torrie and D. S. Gaunt, J. Phys. A 13, 3707 (1980); R. Cherry and C. Domb, J. Phys. A 13, 1325 (1980); D. S. Gaunt, J. L. Martin, G. Ord, G. M. Torrie and S. G. Whittington, J. Phys. A 13, 1791 (1980).
Random-site random-bond percolation is analyzed by series expansions in P. Agrawal, S. Redner, P. J. Reynolds and H. E. Stanley, J. Phys. A 12, 2073 (1979) and by renormalization group in H. Nakanishi and P. J. Reynolds, Phys. Lett. 71A, 252 (1979).
The s-state hierarchy was proposed in R.B.Potts,Proc.Camb.Phil.Soc.48,106 (1952).
The n-vector hierarchy was proposed in H.E.Stanley,Phys.Rev.Lett.20,589 (1968).
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1981 Springer-Verlag
About this paper
Cite this paper
Stanley, H.E. (1981). New directions in percolation, including some possible applications of connectivity concepts to the real world. In: Castellani, C., Di Castro, C., Peliti, L. (eds) Disordered Systems and Localization. Lecture Notes in Physics, vol 149. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0012546
Download citation
DOI: https://doi.org/10.1007/BFb0012546
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-11163-4
Online ISBN: 978-3-540-38636-0
eBook Packages: Springer Book Archive