Abstract
In this lecture we present an overview of the physics of irreversible fractal growth process, with particular emphasis on a class of models characterized by quenched disorder. These models exhibit self-organization, with critical properties developing spontaneously, without the fine tuning of external parameters. This situation is different from the usual critical phenomena, and requires the introduction of new theoretical methods. Our approach to these problems is based on two concepts, the Fixed Scale Transformation, and the quenched-stochastic transformation, or Run Time Statistics (RTS), which maps a dynamics with quenched disorder into a stochastic process. These methods, combined together, allow us to understand the self-organized nature of models with quenched disorder and to compute analytically their critical exponents. In addition, it is also possible characterize mathematically the origin of the dynamics by avalanches and compare it with the continuous growth of other fractal models. A specific application to Invasion Percolation will be discussed. Some possible relations to glasses will also be mentioned.
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
References
Mandelbrot B. B. (1982): The fractal Geometry of Nature. W. H. Freeman, New York.
Witten T. A., Sander L. M. (1981): Phys. Rev. Lett. 47, 1400.
Niemeyer L., Pietronero L., Wiesmann H. J. (1984): Phys. Rev. Lett. 52, 1033.
Bak P., Sneppen K. (1993): Phys. Rev. Lett. 71, 4083.
Wilkinson D., Willemsen J. F. (1983): J. Phys. A 16, 3365.
Bak P., Tang C., Wiesenfeld K. (1987): Phys. Rev. Lett. 59, 381.
Sneppen K. (1992): Phys. Rev. Lett. 69, 3539.
Family F., Zhang Y. C., Vicsek T. (1986): J. Phys. A 19, L733.
De Arcangelis L., Hansen A., Herrmann H. J., Roux S. (1989): Phys. Rev. B 40, 877.
Hansen A., Hinrichsen E. L., Roux S., Herrmann H. J., De Arcangelis L. (1990): Europhys. Lett. 13, 341.
Cafiero R., Pietronero L., Vespignani A. (1993): Phys. Rev. Lett. 70, 3939.
Erzan A., Pietronero L., Vespignani A. (1995): Rev. Mod. Phys. 67, no. 3.
Pietronero L., Schneider W. R., Stella A. (1990): Phys. Rev. A 42, R7496.
Pietronero L., Schneider W. R. (1990): Physica A 119, 249–267.
Marsili M. (1994): J. Stat. Phys. 77, 733.
Gabrielli A., Marsili M., Cafiero R., Pietronero L. (1996): J. Stat. Phys. 84, 889–893.
Marsili M. (1994): Europhys. Lett. 28, 385.
Paczuski M., Bak P., Maslov S. (1996): Phys. Rev. E 53, 414.
Cafiero R., Gabrielli A., Marsili M., Pietronero L. (1996): Phys. Rev. E 54, 1406.
Vendruscolo M., Marsili M. (1996): Phys. Rev. E 54, R1021.
Maslov S. (1995): Phys.Rev.Lett. 74, 562.
Marsili M., Caldarelli G., Vendruscolo M. (1996): Phys. Rev. E 53, 13.
Cafiero R., Gabrielli A., Marsili M., Torosantucci L., Pietronero L. (1996): in preparation.
Tosatti E., Zannetti M., Pietronero L. (1988): Z. Phys. B 73, 161.
Bouchaud J. P., Comtet A., Georges A., Le Doussal P. (1990): Ann. Phys. 201, 285.
Marinari E., Parisi G. (1993): J. Phys. A Math. Gen. 26, L1149.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1997 Springer-Verlag
About this paper
Cite this paper
Pietronero, L., Cafiero, R., Gabrielli, A. (1997). Fractal growth with quenched disorder. In: Rubí, M., Pérez-Vicente, C. (eds) Complex Behaviour of Glassy Systems. Lecture Notes in Physics, vol 492. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0104845
Download citation
DOI: https://doi.org/10.1007/BFb0104845
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-63069-2
Online ISBN: 978-3-540-69123-5
eBook Packages: Springer Book Archive