Complexity of two-dimensional patterns
Inhaltsverzeichnis
Referenz
Andrienko, Y.A., Brilliantov, N.V., Kurths, J.: Complexity of two-dimensional patterns. Eur. Phys. J. B 15(3), 539–546 (2000)
DOI
http://dx.doi.org/10.1007/s100510051157
Abstract
To describe quantitatively the complexity of two-dimensional patterns we introduce a complexity measure based on a mean information gain. Two types of patterns are studied: geometric ornaments and patterns arising in random sequential adsorption of discs on a plane (RSA). For the geometric ornaments analytical expressions for entropy and complexity measures are presented, while for the RSA patterns these are calculated numerically. We compare the information-gain complexity measure with some alternative measures and show advantages of the former one, as applied to two-dimensional structures. Namely, this does not require knowledge of the “maximal” entropy of the pattern, and at the same time sensitively accounts for the inherent correlations in the system.
Extended Abstract
Bibtex
@Article{Andrienko2000,
author="Andrienko, Yu.A. and Brilliantov, N.V. and Kurths, J.",
title="Complexity of two-dimensional patterns",
journal="The European Physical Journal B - Condensed Matter and Complex Systems",
year="2000",
volume="15",
number="3",
pages="539--546",
issn="1434-6036",
doi="10.1007/s100510051157",
url="http://dx.doi.org/10.1007/s100510051157 http://de.evo-art.org/index.php?title=Complexity_of_two-dimensional_patterns "
}
Used References
1. A.N. Kolmogorov, Probl. Inf. Transm. 1, 3 (1965)
2. R.Wackerbauer, A. Witt, H. Atmanspacher, J. Kurths, H. Scheingraber, Chaos, Solitons and Fractals 4, 133 (1994).
3. P. Grassberger, Int. J. Theor. Phys. 25, 907 (1986).
4. A. Neiman, B. Shulgin, V. Anishenko, W. Ebeling, L. Shimansky-Geier, J. Freund, Phys. Rev. Lett. 76, 4299 (1996).
5. J.P. Crutchfield, K. Yong, Phys. Rev. Lett. 63, 105 (1989).
6. C.H. Bennett, in Complexity, Entropy and the Physics of Information, edited by W.H. Zurek (Addison-Wesley, Reading, MA, 1990).
7. D.W. McShea, Biol. Phylos. 6, 303 (1991).
8. H. Atmanspacher, C. Rath, G. Wiedenmann, Physica A 234, 819 (1997).
9. J.S. Shiner, M. Davison, P.T. Landsberg, Phys. Rev. E 59, 1459 (1999); P.T. Landsberg, J.S. Shiner, Phys. Lett. A 245, 228 (1998).
10. P.D. Feldman, J.P. Crutchfield, Phys. Lett. A 238, 244 (1998).
11. R. Lopez-Ruiz, H.L. Mancini, X. Calbet, Phys. Lett. A 209, 321 (1995).
12. G. Chaitin, J. ACM 13, 547 (1996).
13. X.Z. Tang, E.R. Tracy, Chaos 8, 688 (1998).
14. A. Witt, A. Neiman, J. Kurths, Phys. Rev. E 55, 5050 (1996).
15. J.P. Crutchfield, D.P. Feldman, Phys. Rev. E 55, R1239 (1997); S. Lloyd, H. Pagels, Ann. Phys. (NY) 188, 186 (1988).
16. U. Schwarz, A.O. Benz, J. Kurths, A. Witt, Astron. Astrophys. 277, 215 (1993).
17. A. Witt, J. Kurths, F. Krause, K. Fischer, Geophys. Astrophys. Fluid Dyn. 77, 77 (1995).
18. P.T. Landsberg, Phys. Lett. A 102, 171 (1984); P.T. Landsberg, in On Self-Organization, edited by R.K. Misra, D. Maas, E. Zwierlein (Springer-Verlag, Berlin, 1994).
19. J.S. Shiner, in Self-Organization of Complex Structures: from Individual to Collective Dynamics, edited by F. Schweitzer (Gordon and Breach, London, 1996).
20. J. Kurths, A. Voss, P. Saparin, A. Witt, H.J. Kleiner, N.Wessel, Chaos 5, 88 (1995).
21. P.I. Saparin, W. Gowin, J. Kurths, D. Felsenberg, Phys. Rev. E 58, 6449 (1998).
22. C.E. Shannon, Bell Syst. Tech. J. 27, 379 (1948). For a recent review see M.C. Mackey, Rev. Mod. Phys. 61, 981 (1989).
23. Equation (1) is written for the case of discrete states, while the continuum set of states implies integration over the set instead of summation; in what follows we deal only with the discrete states.
24. For simplicity we restrict ourselves to the case when only the color of the upper neighbor is accounted. One can certainly take into account the colors of the other neighbors.
25. M.C. Bartelt, V. Privman, Int. J. Mod. Phys. B 5, 2883 (1991); J.W. Evans, Rev. Mod. Phys. 65, 1281 (1993); J.J. Ramsden, J. Stat. Phys. 79, 491 (1995).
26. G.V. Schulz, Z. Phys. Chem. Abt. B 43, 25 (1939).
27. N.V. Brilliantov, Yu.A. Andrienko, P.L. Krapivsky, J. Kurths, Phys. Rev. Lett. 76, 4058 (1996); N.V. Brilliantov, Yu.A. Andrienko, P.L. Krapivsky, Physica A 239, 267 (1997).
28. N.V. Brilliantov, Yu.A. Andrienko, P.L. Krapivsky, J. Kurths, Phys. Rev. E 58, 3530 (1998).
29. P.L. Krapivsky, J. Stat. Phys. 69, 135 (1992).
30. O. Biham, O. Malcai, D.A. Lidar, D. Avnir, Phys. Rev. E 59, R4713 (1999).
31. B.B. Mandelbrot, The Fractal Geometry of Nature (San Francisco: Freeman, 1982).
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