A Toy Model for the Diffusion-Limited Aggregation
Cardinal Stefan Wyszynski University
Faculty of Mathematics and Natural Sciences
ul. Wóycickiego 1/3, PL-01-938 Warsaw, PolandE-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:
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).
We consider the deterministic Vicsek fractal with the aim to understand the multifractal properties of the Diffusion-Limited Aggregation.
Key words:
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).
