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Volume 27 (4) 2021, 151–157

A Toy Model for the Diffusion-Limited Aggregation

Wolf Marek

Cardinal Stefan Wyszynski University
Faculty of Mathematics and Natural Sciences
ul. Wóycickiego 1/3, PL-01-938 Warsaw, Poland

E-mail: m.wolf@uksw.edu.pl

Received:

Received: 24 November 2021; revised: 9 December 2021; accepted: 14 December 2021; published online: 27 December 2021

DOI:   10.12921/cmst.2021.0000031

Abstract:

We consider the deterministic Vicsek fractal with the aim to understand the multifractal properties of the Diffusion-Limited Aggregation.

Key words:

Diffusion-Limited Aggregation, multifractality

References:

[1]  T.A.  Witten,  L.M.  Sander,  Diffusion-limited  aggregation, a kinetic critical phenomenon, Physical Review Letters 47, 1400–1403 (1981).

[2] D.S. Grebenkov, D. Beliaev, How anisotropy beats fractality in two-dimensional on-lattice diffusion-limited-aggregation growth, Physical Review E 96, 042159 (2017).

[3] L.A. Turkevich, H. Scher, Occupancy-probability scaling in diffusion-limited aggregation, Physical Review Letters 55, 1026 (1985).

[4] C. Amitrano, P. Meakin, H.E. Stanley, Fractal dimension of the accessible perimeter of diffusion-limited aggregation, Physical Review A 40, 1713 (1989).

[5] C. Amitrano, A. Coniglio, F. di Liberto, Growth probability distribution in kinetic aggregation processes, Physical Review Letters 57, 1016 (1986).

[6] G. Paladin, A. Vulpiani, Anomalous scaling laws in multifractal objects, Physics Reports 156(4), 147–225 (1987).

[7]  S. Schwarzer, J. Lee, A. Bunde, S. Havlin, H.E. Roman, H.E.  Stanley,  Minimum  growth  probability  of  diffusion-limited aggregates, Physical Review Letters 65, 603 (1990).

[8] M. Wolf, Hitting probabilities of diffusion-limited-aggregation clusters, Physical Review A, 43, 5504–5517 (1991).

[9] M. Wolf, Size dependence of the minimum-growth probabilities of typical diffusion-limited-aggregation clusters, Physical Review E 47, 1448–1451 (1993).

[10] T. Vicsek, Fractal models for diffusion controlled aggregation, J Phys. A: Math. and Gen. 16(17), L647 (1983).

[11] R.G. Hohlfeld, N. Cohen, Self-similarity and the geometric requirements for frequency independence in antennae, Fractals 7, 79–84 (1999).

[12] S. Fuqi, G. Hongming, G. Baoxin, Analysis of a Vicsek fractal patch antenna, ICMMT 4th International Conference on Proceedings Microwave and Millimeter Wave Technology (2004).

[13] P. Meakin, R.C. Ball, P. Ramanlal, L.M. Sander, Structure of large two-dimensional square-lattice diffusion-limited aggregates: Approach to asymptotic behavior, Physical Review A  35(12), 5233 (1987).

[14] F. Spitzer, Principles of Random Walk, Graduate Texts in Mathematics, Springer, 2nd ed. (2001).

[15] A.P. Roberts, M.A. Knackstedt, Comment on “Hitting probabilities of diffusion-limited-aggregation clusters”, Physical Review E 48, 4143–4144 (1993).

[16] L. Niemeyer, L. Pietronero, H.J. Wiesmann, Fractal dimension of dielectric breakdown, Physical Review Letters 52, 1033–1036 (1984).

[17]  J. Stoer, R. Bulirsch, Introduction to Numerical Analysis, Texts in Applied Mathematics, Springer New York, 2nd ed. (1993).

[18] A. Ralston, P. Rabinowitz, A First Course in Numerical Analysis, Texts in Applied Mathematics, Dover Publications, 2nd ed. (2001).

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