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Podlazov A. V. Solution of two-dimensional self-organized critical Manna model. *Izvestiya VUZ. Applied Nonlinear Dynamics*, 2013, vol. 21, iss. 6, pp. 69-87. DOI: 10.18500/0869-6632-2013-21-6-69-87

# Solution of two-dimensional self-organized critical Manna model

We propose a full solution for Manna model – two-dimensional conservative sandpile model with the rules of grains redistribution isotropic at the average. Indices of the probability distributions of avalanches main characteristics (size, area, perimeter, duration, topplings multiplicity) are determined for this model both from theory and from simulations. The solution bases on the spatiotemporal decomposition of avalanches described in terms of toppling layers and waves. The motion of grains is divided into directed and undirected types. The former is treated as the dynamics of active particles with some physical properties described.

- Bak P, Tang C, Wiesenfeld K. Self-organized criticality. Phys. Rev. A. 1988;38(1):364–374. DOI: 10.1103/physreva.38.364.
- Bak P. How nature works: The theory of self-organized criticality. Synergies: From the past to the future.Moscow: Librokom; 2013. 276 p. (In Russian).
- Manna SS. Two-state model of self-organized criticality. J. Phys. A: Math. Gen. 1991;24(7):363–369. DOI: 10.1088/0305-4470/24/7/009.
- Milshtein E, Biham O, Solomon S. Universality classes in isotropic, Abelian, and non-Abelian sandpile models. Phys. Rev. E. 1998;58(1):303–310. DOI: 10.1103/PhysRevE.58.303.
- Zhang Y-C. Scaling theory of self-organized criticality. Phys. Rev. Lett. 1989;63(5):470–473. DOI: 10.1103/PHYSREVLETT.63.470.
- Ben-Hur A, Biham O. Universality in sandpile models. Phys. Rev. E. 1996;53(2):1317–1320. DOI: 10.1103/physreve.53.r1317.
- Malinetsky GG, Podlazov AV. Comparison of two-dimensional isotropic conserved self-organized-critical sand heap models. Herald of the Bauman Moscow State Technical University. Series Natural Sciences. 2012;2:119.
- Pietronero L, Vespignani A, Zapperi S. Renormalization scheme for self-organized criticality in sandpile models. Phys. Rev. Lett. 1994;72(11):1690–1693. DOI: 10.1103/PhysRevLett.72.1690.
- Vespignani A, Zapperi S, Pietronero L. Renormalization approach to the self- organized critical behavior of sandpile models. Phys. Rev. E. 1995;51(3):1711–1724. DOI: 10.1103/physreve.51.1711.
- Daz-Guilera A. Dynamic renormalization group approach to self-organized critical phenomena. Europhys. Lett. 1994;26(3):177–182. DOI: 10.1209/0295-5075/26/3/004.
- Corral A, Daz-Guilera A. Symmetries and fixed point stability of stochastic differential equations modeling self-organized criticality. Phys. Rev. E. 1997;55(3):2434–2445. DOI: 10.1103/PhysRevE.55.2434.
- Dhar D, Ramaswamy R. Exactly solved model of self-organized critical phenomena. Phys. Rev. Lett. 1989;63(16):1659–1662. DOI: 10.1103/PhysRevLett.63.1659.
- Pastor-Satorras R, Vespignani A. Universality classes in directed sandpile models. J. Phys. A: Math. Gen. 2000;33(3):33–39. DOI: 10.1088/0305-4470/33/3/101.
- Paczuski M, Bassler KE. Theoretical results for sandpile models of SOC with multiple topplings. Phys. Rev. E. 2000;62(4):5347–5352. DOI: 10.1103/physreve.62.5347.
- Kloster M, Maslov S, Tang C. Exact solution of stochastic directed sandpile model. Phys. Rev. E. 2001;63(2):026111. DOI: 10.1103/PhysRevE.63.026111.
- Feder HJS, Feder J. Self-organized criticality in a stick-slip process. Phys. Rev. Lett. 1991;66(20):2669–2672. DOI: 10.1103/PhysRevLett.66.2669.
- Podlazov AV. Two-dimensional self-organized critical sandpile models with anisotropic dynamics of the activity propagation. Izvestiya VUZ. Applied Nonlinear Dynamics. 2012;20(6):25–46. DOI: 10.18500/0869-6632-2012-20-6-25-46.
- Lubeck S, Usadel KD. Bak–Tang–Wiesenfeld sandpile model around upper critical dimension. Phys. Rev. E. 1997;56(5):5138–5143. DOI:10.1103/PhysRevE.56.5138.
- Chessa A, Vespignani A, Zapperi S. Critical exponents in stochastic sandpile models. Comput. Phys. Commun. 1999;121-122:299–302.
- Lubeck S. Moment analysis of the probability distributions of different sandpile models. Phys. Rev. E. 2000;61(1):204–209. DOI: 10.1103/physreve.61.204.
- Lubeck S, Usadel KD. Numerical determination of the avalanche exponents of the Bak–Tang–Wiesenfeld model. Phys. Rev. E. 1997;55(4):4095. DOI: 10.1103/PhysRevE.55.4095.
- Kadanoff LP, Nagel SR, Wu L, Zhou S. Scaling and universality in avalanches. Phys. Rev. A. 1989;39(12):6524–6537. DOI: 10.1103/physreva.39.6524.
- Ivashkevich EV, Ktitarev DV, Priezzhev VB. Waves of topplings in an Abelian sandpile. Physica A. 1994;209(3):347–360. DOI: 10.1016/0378-4371(94)90188-0.

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