and Computational Methods

doi: 10.18698/2309-3684-2014-3-89110

We propose a full solution for Manna model, two-dimensional conservative sand pile model with isotropic rules of grains redistribution on average. We determined the general properties indices of avalanches distribution (size, area, perimeter, duration, the multiplicity of topplings) for the model both analytically and numerically. The solution is based on spatio-temporal decomposition of avalanches described in terms of toppling layers and waves and on division of the motion of grains into directed and undirected types. The former of the two is treated as the dynamics of active particles with some physical properties described.

[1] Bak P., Tang C., Wiesenfeld K. Self-organized criticality. Phys. Rev. A., 1988, vol. 38, no. 1, pp. 364−374.

[2] Bak P. Workings of Nature: Theory of Selforganized Criticality. Synergetics: From Past to Future. Moscow, Librokom Publ., 2013, no. 66, 276 p.

[3] Manna S.S. Two-state model of self-organized criticality. J. Phys. A: Math. Gen., 1991, vol. 24, pp. L363–L639.

[4] Milshtein E., Biham O., Solomon S. Universality classes in isotropic, Abelian, and non-Abelian sandpile models. Phys. Rev. E., 1998, vol. 58, no. 1, pp. 303–310.

[5] Zhang Y.-C. Scaling theory of self-organized criticality. Phys. Rev. Lett., 1989, vol. 63, no. 5, pp. 470−473.

[6] Ben-Hur A., Biham O. Universality in sandpile models. Phys. Rev. E., 1996, vol. 53, no. 2, pp. 1317–1320.

[7] Malinetskiy G.G., Podlazov A.V. Vestnik MGTU im. N.E. Baumana. Seriya Estestvennye Nauki — Herald of the Bauman Moscow State Technical University. Series: Natural Sciences, 2012, no. 2, pp. 119–128.

[8] Pietronero L., Vespignani A, Zapperi S. Renormalization scheme for selforganized criticality in sandpile models. Phys. Rev. Lett., 1994, vol. 72, no. 11, pp. 1690−1693.

[9] Vespignani A., Zapperi S., Pietronero L. Renormalization approach to the selforganized critical behavior of sandpile models. Phys. Rev. E., 1995, vol. 51, no. 3, pp. 1711−1724.

[10] Díaz-Guilera A. Dynamic renormalization group approach to self-organized critical phenomena. Europhys. Lett., 1994, vol. 26, no. 3, p. 177.

[11] Corral Á., Díaz-Guilera A. Symmetries and fixed point stability of stochastic differential equations modeling self-organized criticality. Phys. Rev. E., 1997, vol. 55, no. 3, pp. 2434−2445.

[12] Dhar D., Ramaswamy R. Exactly solved model of self-organized critical phenomena. Phys. Rev. Lett., 1989, vol. 63, no. 16, pp. 1659−1662.

[13] Pastor-Satorras R., Vespignani A. Universality classes in directed sandpile models. J. Phys. A: Math. Gen, 2000, no. 33, pp. L33–L39.

[14] Paczuski M., Bassler K.E. Theoretical results for sandpile models of SOC with multiple topplings. Phys. Rev. E., 2000, vol. 62, no. 4, pp. 5347−5352.

[15] Kloster M., Maslov S., Tang C. Exact solution of stochastic directed sandpile model. Phys. Rev. E., 2001, vol. 63, no. 2, pp. 026111.

[16] Feder H.J.S., Feder J. Self-organized criticality in a stick-slip process. Phys. Rev. Lett., 1991, vol. 66, no. 20, pp. 2669−2672.

[17] Kadanoff L.P., Nagel S.R., Wu L., Zhou S. Scaling and universality in avalanches. Phys. Rev. A., 1989, vol. 39, no. 12, pp. 6524−6537.

[18] Podlazov A.V. Izvestiya Vuzov. Prikladnaya Nelineynaya Dinamika. Proceedings of the universities: Applied Nonlinear Dynamics, 2012, vol. 20, no. 6, pp. 25−46.

[19] Lübeck S., Usadel K.D. Bak-Tang-Wiesenfeld sandpile model around upper critical dimension. Phys. Rev. E., 1997, vol. 56, no. 5, pp. 5138−5143.

[20] Chessa A., Vespignani A., Zapperi S. Critical exponents in stochastic sandpile models. Comput. Phys. Commun, 1999, vol. 121–122, pp. 299–302.

[21] Lübeck S. Moment analysis of the probability distributions of different sandpile models. Phys. Rev. E., 2000, vol. 61, no. 1, pp. 204−209.

[22] Lübeck S., Usadel K.D. Numerical determination of the avalanche exponents of the Bak-Tang-Wiesenfeld model. Phys. Rev. E., 1997, vol. 55, no. 4, pp. 4095−4099.

[23] Ivashkevich E.V., Ktitarev D.V., Priezzhev V.B. Waves of topplings in an Abelian sandpile. Physica A., 1994, vol. 209, pp. 347−360.

Podlazov A. Two-dimensional self-organized critical Мanna model. Маthematical Modeling and Coтputational Methods, 2014, №3 (3), pp. 89-110

Количество скачиваний: 619