CLASSICAL TWO-DIMENSIONAL SANDPILE MODELS
Cite this article as:
Podlazov А. V. CLASSICAL TWO-DIMENSIONAL SANDPILE MODELS. Izvestiya VUZ. Applied Nonlinear Dynamics, 2016, vol. 24, iss. 4, pp. 39-70. DOI: https://doi.org/10.18500/0869-6632-2016-24-4-39-70
I consider sandpile models being open nonlinear systems demonstrating the phenomenon of avalanche-like response to a single disturbance of steady state. I study thoroughly the five most known variants of the two-dimensional rules referred as the models of Dhar–Ramaswamy, Pastor-Satorras–Vespignani, Feder–Feder, Manna and Bak–Tang–Wiesenfeld. The analytical solutions obtained in various ways are known for the first four models and the reasons preventing the construction of a solution are known for the fifth one. The generalization of these results allows to develop a common approach to the theoretical study of self-organized critical phenomena.
Self-organization into the critical state gives rise to the scale-invariant properties. Theirs statistical description can not be generally obtained on the basis of the models’ rules. Intermediate level models mediates between the microlevel of the elements local behavior and the macrolevel of the entire system behavior. The rules of these models are not derived from the rules of original models, but are formulated on the ground of physical intuition, computer simulation results and common understanding of the dynamic processes that hold the system near the critical point.
The collective dynamics of all of the models is reduced at the intermediate level to the processes familiar to mathematical physics, the first of them are asymmetric random walks. On this basis, I propose uniform methods of solution of models. The BTW model is solved for the first time.
All the critical indices are analytically calculated for the models considered. The influence of the rules features of models on their common properties is analysed on theses ground.
The most important for the rules is the aspect whether they are stochastic or deterministic. The former increases the number of avalanche characteristics with different properties, and the later helps large avalanches fit into a finite-size system and results in the system as a whole obtaining dynamic symmetries absent at the level of elements behaviour rules.
DOI: 10.18500/0869-6632-2016-24-4-39-70
Paper reference: Podlazov A.V. Classical two-dimensional sandpile models. Izvestiya VUZ. Applied Nonlinear Dynamics. 2016. Vol. 24, Issue 4. P. 39–70.
1. Bak P. How nature works: The science of self-organized criticality. Springer-Verlag, New York, Inc. 1996.
2. Dhar D., Ramaswamy R. Exactly solved model of self-organized critical phenomena // Phys. Rev. Lett. 1989. Vol. 63, N16. Pp. 1659–1662.
3. Pastor-Satorras R., Vespignani A. Universality classes in directed sandpile models // J. Phys. A. 2000. Vol. 33, N3. Pp. L33–L39.
4. Feder H.J.S., Feder J. Self-organized criticality in a stick-slip process // Phys. Rev. Lett. 1991. Vol. 66, N20. Pp. 2669–2672.
5. Manna S.S. Two-state model of self-organized criticality // J. Phys. A. 1991. Vol. 24, N7. Pp. L363–L639.
6. Bak P., Tang C., Wiesenfeld K. Self-organized criticality: An explanation of 1/f-noise // Phys. Rev. Lett. 1987. Vol. 59, N4. Pp. 381–384.
7. Bak P., Tang C., Wiesenfeld K. Self-organized criticality // Phys. Rev. A. 1988. Vol. 38, N1. Pp. 364–374.
8. Dhar D., Pruessner G., Expert P., Christensen K., Zachariou N. Directed Abelian sandpile with multiple downward neighbors // Phys. Rev. E. 2016. Vol. 93. 042107.
9. Paczuski M., Bassler K.E. Theoretical results for sandpile models of SOC with multiple topplings // Phys. Rev. E. 2000. Vol. 62, Iss. 4. Pp. 5347–5352.
10. Kloster M., Maslov S., Tang C. Exact solution of stochastic directed sandpile model// Phys. Rev. E. 2001. Vol. 63. 026111.
11. Hu C.K., Ivashkevich E.V., Lin C.Y., Priezzhev V.B. Inversion symmetry and exact critical exponents of dissipating waves in the sandpile model // Phys. Rev. Lett. 2000. Vol. 85, N19. Pp. 4048–4051.
12. Ktitarev D.V., L ̈ubeck S., Grassberger P., Priezzhev V.B. Scaling of waves in the Bak–Tang–Wiesenfeld sandpile model // Phys. Rev. E. 2000. Vol. 61, N1. Pp. 81–92.
13. Hughes D., Paczuski M. Large scale structures, symmetry, and universality in sandpiles // Phys. Rev. Lett. 2002. Vol. 88. 054302.
14. Podlazov A.V. Two-dimensional self-organized critical sandpile models with anisotropic dynamics of the activity propagation// Izvestiya VUZ. AND. 2012. Vol. 20, N6. Pp. 25–46.
15. Malinetskiy G.G., Podlazov A.V. Comparison of two-dimensional isotropic conservative self-organized critical sandpile models // Engineering Journal: Science and Innovation. 2012. Vol. 4, N4. 167.
16. Podlazov A.V. Solution of two-dimensional self-organized critical Manna model // Izvestiya VUZ. AND. 2013. Vol. 21, N6. Pp. 69–87.
17. Podlazov A.V. Two-dimensional self-organized critical Manna model // Mathematical modeling and computational methods. 2014. Vol. 3, N3. Pp. 89–110.
18. Ivashkevich E.V., Ktitarev D.V., Priezzhev V.B. Critical exponents for boundary avalanches in a 2D Abelian sandpile // J. Phys. A: Math. Gen. 1994. Vol. 27, N16. Pp. L585–L590.
19. Dhar D., Manna S.S. Inverse avalanches in the Abelian sandpile model // Phys. Rev. E. 1994. Vol. 49, N4. Pp. 2684–2687.
20. Ivashkevich E.V., Ktitarev D.V., Priezzhev V.B. Waves of topplings in an Abelian sandpile // Physica A. 1994. Vol. 209, N3–4. Pp. 347–360.
21. Paczuski M., Boettcher S. Avalanches and waves in the Abelian sandpile model // Phys. Rev. E. 1997. Vol. 56, N4. Pp. R3745–R3748.
22. Priezzhev V.B., Ktitarev D.V., Ivashkevich E.V. Formation of avalanches and critical exponents in Abelian sandpile model // Phys. Rev. Lett. 1996. Vol. 76, N12. Pp. 2093–2096.
23. Majumdar S.N., Dhar D. Equivalence between the Abelian sandpile model and the q → 0 limit of the Potts model // Physica A. 1992. Vol. 185, N1–4. Pp. 129–145.
BibTeX
author = {А. V Podlazov},
title = {CLASSICAL TWO-DIMENSIONAL SANDPILE MODELS},
year = {2016},
journal = {Izvestiya VUZ. Applied Nonlinear Dynamics},
volume = {24},number = {4},
url = {https://old-andjournal.sgu.ru/en/articles/classical-two-dimensional-sandpile-models},
address = {Саратов},
language = {russian},
doi = {10.18500/0869-6632-2016-24-4-39-70},pages = {39--70},issn = {0869-6632},
keywords = {Self-organized criticality,sandpile models,scale invariance,power laws,finite-size scaling,meso-level models,random walks,spontaneous anisotropy,dynamical symmetries.},
abstract = { I consider sandpile models being open nonlinear systems demonstrating the phenomenon of avalanche-like response to a single disturbance of steady state. I study thoroughly the five most known variants of the two-dimensional rules referred as the models of Dhar–Ramaswamy, Pastor-Satorras–Vespignani, Feder–Feder, Manna and Bak–Tang–Wiesenfeld. The analytical solutions obtained in various ways are known for the first four models and the reasons preventing the construction of a solution are known for the fifth one. The generalization of these results allows to develop a common approach to the theoretical study of self-organized critical phenomena. Self-organization into the critical state gives rise to the scale-invariant properties. Theirs statistical description can not be generally obtained on the basis of the models’ rules. Intermediate level models mediates between the microlevel of the elements local behavior and the macrolevel of the entire system behavior. The rules of these models are not derived from the rules of original models, but are formulated on the ground of physical intuition, computer simulation results and common understanding of the dynamic processes that hold the system near the critical point. The collective dynamics of all of the models is reduced at the intermediate level to the processes familiar to mathematical physics, the first of them are asymmetric random walks. On this basis, I propose uniform methods of solution of models. The BTW model is solved for the first time. All the critical indices are analytically calculated for the models considered. The influence of the rules features of models on their common properties is analysed on theses ground. The most important for the rules is the aspect whether they are stochastic or deterministic. The former increases the number of avalanche characteristics with different properties, and the later helps large avalanches fit into a finite-size system and results in the system as a whole obtaining dynamic symmetries absent at the level of elements behaviour rules. DOI: 10.18500/0869-6632-2016-24-4-39-70 Paper reference: Podlazov A.V. Classical two-dimensional sandpile models. Izvestiya VUZ. Applied Nonlinear Dynamics. 2016. Vol. 24, Issue 4. P. 39–70. Download full version }}