MODEL MULTILINEAL PATTERN FORMATION: A COMPUTER EXPERIMENT
Gadomski Adam, Cyrankiewicz Michał
Institute of Mathematics and Physics
University of Technology and Agriculture
85-796 Bydgoszcz, al. Kaliskiego 7, Poland
Received:
Received 19 July, 2001
DOI: 10.12921/cmst.2001.07.02.75-90
OAI: oai:lib.psnc.pl:523
Abstract:
Multilineal pattern formation in the square lattice is studied. It turns out that lognormality both in space (size) and time appears to be a main signature thereof. A diffusive character of the emerging patterns has been emphasized, and some elucidation, supporting an understanding of the model phenomenon in question is provided. Excursions towards dynamics of the multiline as well as a relation to known kinetic or dynamic models have been offered as well.
Key words:
discrete space, evolution, multilineal patterns, nonlinearity
References:
[1] H. J. Herrmann, Phys. Rep. 136 (1986) 153; T. Vicsek, Fractal Growth Phenomena, World
Scientific, Singapore, 1992.
[2] F. Family, T. Vicsek, eds., Dynamics of Fractal Surface, World Scientific, Singapore, 1991.
[3] F. Family, D. P. Landau, eds., Kinetics of Aggregation and Gelation, North Holland, Amsterdam, 1984.
[4] T. Halpin-Healy, Y.-C. Zhang, Phys. Rep. 254 (1995) 215.
[5] D. W. Heermann, Podstawy Symidacji Komputerowych w Fizyce, WNT, Warszawa, 1997.
[6] J. J. Binney, N. J. Dowrick, A. J. Fisher, M. E. J. Newman, Zjawiska Krytyczne. Wstęp do teorii
grupy renormalizacyjnej, WN-PWN, Warszawa, 1998.
[7] M. Eden, Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, edited by J. Neyman, University of California Press, Berkeley, 1961.
[8] L. Schimansky-Geier, T. Pöschel, eds., Stochastic Dynamics, Springer-Verlag, Berlin, 1997.
[9] T. A. Witten, Jr., L. M. Sander, Phys. Rev. Lett. 47 (1981) 1400.
[10] C. Moore, Material presented at the NATO ASI “Complexity from Microscopic to Macroscopic Scales: Coherence and Large Deviations”, 17-27 April 2001, Geilo, Norway.
[11] H. Gould, F. Family, H. E. Stanley, Phys. Rev. Lett. 50 (1983) 689; A. Christou, R. B. Stinchcombe, J. Phys. A: Math. Gen. 19 (1986) L454.
[12] A. Gadomski, M. Schönhof, K. Bończak, in Stochastic Processes in Physics, Chemistry and
Biology, edited by J. A. Freund and T. Pöschel, Lecture Notes in Physics No. 557 (Springer-Verlag, Berlin, 2000), pp. 496-506.
[13] A. Gadomski, Fractals, Vol. 9 (2001) in press.
[14] B. Berg, D. Foerster, Phys. Lett. 106B (1981) 323.
[15] M. Niemiec, A. Gadomski, J. Łuczka, Acta Phys. Pol. B 32 (2001) 1513.
[16] D. Helbing et al., in Traffic and Granular Flow’99, edited by D. Helbing, H. J. Herrmann, M.
Schreckenberg, D. E. Wolf (Springer-Verlag, Berlin, 2000), pp. 193-204.
[17] D. Levine, P. J. Steinhard, Phys. Rev. B 34 (1986) 596.
[18] M. Cyrankiewicz, Diploma Thesis, University of Technology and Agriculture, Bydgoszcz, 2001.
[19] J. Lyklema, Fundamentals of Interface and Colloid Science. Vol. I, Academic Press, London, 1991.
[20] D. Stauffer, D. P. Landau, Phys. Rev. B 39 (1989) 9650; D. Stauffer, N. Jan, Canad. J. Phys. 66
(1988) 187; R. H. Swendsen, Phys. Rev. B 15 (1977) 689.
[21] M. Cieplak, M. Suan Li, Fractals 2 (1994) 481.
[22] W. Przygocki, A. Włochowicz, Fizyka Polimerów, WN-PWN, Warszawa, 2000, chaps 1-4.
[23] M. Schónhof, Diploma Thesis, Humboldt University at Berlin, Berlin, 2001, chap. 4.
[24] A. A. Adamson, Physical Chemistry of Surfaces, Interscience Publishers, Inc., New York, 1960.
Multilineal pattern formation in the square lattice is studied. It turns out that lognormality both in space (size) and time appears to be a main signature thereof. A diffusive character of the emerging patterns has been emphasized, and some elucidation, supporting an understanding of the model phenomenon in question is provided. Excursions towards dynamics of the multiline as well as a relation to known kinetic or dynamic models have been offered as well.
Key words:
discrete space, evolution, multilineal patterns, nonlinearity
References:
[1] H. J. Herrmann, Phys. Rep. 136 (1986) 153; T. Vicsek, Fractal Growth Phenomena, World
Scientific, Singapore, 1992.
[2] F. Family, T. Vicsek, eds., Dynamics of Fractal Surface, World Scientific, Singapore, 1991.
[3] F. Family, D. P. Landau, eds., Kinetics of Aggregation and Gelation, North Holland, Amsterdam, 1984.
[4] T. Halpin-Healy, Y.-C. Zhang, Phys. Rep. 254 (1995) 215.
[5] D. W. Heermann, Podstawy Symidacji Komputerowych w Fizyce, WNT, Warszawa, 1997.
[6] J. J. Binney, N. J. Dowrick, A. J. Fisher, M. E. J. Newman, Zjawiska Krytyczne. Wstęp do teorii
grupy renormalizacyjnej, WN-PWN, Warszawa, 1998.
[7] M. Eden, Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, edited by J. Neyman, University of California Press, Berkeley, 1961.
[8] L. Schimansky-Geier, T. Pöschel, eds., Stochastic Dynamics, Springer-Verlag, Berlin, 1997.
[9] T. A. Witten, Jr., L. M. Sander, Phys. Rev. Lett. 47 (1981) 1400.
[10] C. Moore, Material presented at the NATO ASI “Complexity from Microscopic to Macroscopic Scales: Coherence and Large Deviations”, 17-27 April 2001, Geilo, Norway.
[11] H. Gould, F. Family, H. E. Stanley, Phys. Rev. Lett. 50 (1983) 689; A. Christou, R. B. Stinchcombe, J. Phys. A: Math. Gen. 19 (1986) L454.
[12] A. Gadomski, M. Schönhof, K. Bończak, in Stochastic Processes in Physics, Chemistry and
Biology, edited by J. A. Freund and T. Pöschel, Lecture Notes in Physics No. 557 (Springer-Verlag, Berlin, 2000), pp. 496-506.
[13] A. Gadomski, Fractals, Vol. 9 (2001) in press.
[14] B. Berg, D. Foerster, Phys. Lett. 106B (1981) 323.
[15] M. Niemiec, A. Gadomski, J. Łuczka, Acta Phys. Pol. B 32 (2001) 1513.
[16] D. Helbing et al., in Traffic and Granular Flow’99, edited by D. Helbing, H. J. Herrmann, M.
Schreckenberg, D. E. Wolf (Springer-Verlag, Berlin, 2000), pp. 193-204.
[17] D. Levine, P. J. Steinhard, Phys. Rev. B 34 (1986) 596.
[18] M. Cyrankiewicz, Diploma Thesis, University of Technology and Agriculture, Bydgoszcz, 2001.
[19] J. Lyklema, Fundamentals of Interface and Colloid Science. Vol. I, Academic Press, London, 1991.
[20] D. Stauffer, D. P. Landau, Phys. Rev. B 39 (1989) 9650; D. Stauffer, N. Jan, Canad. J. Phys. 66
(1988) 187; R. H. Swendsen, Phys. Rev. B 15 (1977) 689.
[21] M. Cieplak, M. Suan Li, Fractals 2 (1994) 481.
[22] W. Przygocki, A. Włochowicz, Fizyka Polimerów, WN-PWN, Warszawa, 2000, chaps 1-4.
[23] M. Schónhof, Diploma Thesis, Humboldt University at Berlin, Berlin, 2001, chap. 4.
[24] A. A. Adamson, Physical Chemistry of Surfaces, Interscience Publishers, Inc., New York, 1960.
