Abstract
Sand pile models are dynamical systems emphasizing the phenomenon of Self Organized Criticality (SOC). From N stacked grains, iterating evolution rules leads to some critical configuration where a small disturbance has deep consequences on the system, involving numerous steps of grain fall. Physicists L. Kadanoff et al inspire KSPM, a model presenting a sharp SOC behavior, extending the well known Sand Pile Model. In KSPM with parameter D we start from a pile of N stacked grains and apply the rule: \(D\!-\!1\) grains can fall from column i onto the \(D\!-\!1\) adjacent columns to the right if the difference of height between columns i and \(i\!+\!1\) is greater or equal to D. We propose an iterative study of KSPM evolution where one single grain addition is repeated on a heap of sand. The sequence of grain falls following a single grain addition is called an avalanche. From a certain column precisely studied for D = 3, we provide a plain process describing avalanches. We hope that this process is a first stone toward the study of KSPM fixed points structure.
Partially supported by IXXI (Complex System Institute, Lyon) and ANR project Subtile.
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References
Bak, P., Tang, C., Wiesenfeld, K.: Self-organized criticality. Phys. Rev. A 38(1), 364–374 (1988)
Creutz, M.: Cellular automata and self organized criticality. In: Some New Directions in Science on Computers (1996)
Dartois, A., Magnien, C.: Results and conjectures on the sandpile identity on a lattice. In: Morvan, M., Rémila, É. (eds.) Discrete Models for Complex Systems, DMCS 2003. DMTCS Proceedings, Discrete Mathematics and Theoretical Computer Science, vol. AB, pp. 89–102 (2003)
Dhar, D.: Self-organized critical state of sandpile automaton models. Phys. Rev. Lett. 64(14), 1613–1616 (1990)
Durand-Lose, J.O.: Parallel transient time of one-dimensional sand pile. Theor. Comput. Sci. 205(1-2), 183–193 (1998)
Formenti, E., Masson, B., Pisokas, T.: Advances in symmetric sandpiles. Fundam. Inform. 76(1-2), 91–112 (2007)
Gajardo, A., Moreira, A., Goles, E.: Complexity of Langton’s ant. Discrete Applied Mathematics 117(1-3), 41–50 (2002)
Gale, D., Propp, J., Sutherland, S., Troubetzkoy, S.: Further travels with my ant. Mathematical Entertainments Column, Mathematical Intelligencer 17, 48–56 (1995)
Goles, E., Kiwi, M.A.: Games on line graphs and sand piles. Theor. Comput. Sci. 115(2), 321–349 (1993)
Goles, E., Martin, B.: Computational Complexity of Avalanches in the Kadanoff Two-dimensional Sandpile Model. In: TUCS (ed.) Proceedings of JAC 2010 Journées Automates Cellulaires 2010, Turku Finland, pp. 121–132 F.1.1 (December 2010)
Goles, E., Morvan, M., Phan, H.D.: The structure of a linear chip firing game and related models. Theor. Comput. Sci. 270(1-2), 827–841 (2002)
Kadanoff, L.P., Nagel, S.R., Wu, L., Zhou, S.M.: Scaling and universality in avalanches. Phys. Rev. A 39(12), 6524–6537 (1989)
Levine, L., Peres, Y.: Spherical asymptotics for the rotor-router model in z d. Indiana Univ. Math. J., 431–450 (2008)
Moore, C., Nilsson, M.: The computational complexity of sandpiles. Journal of Statistical Physics 96, 205–224 (1999), 10.1023/A:1004524500416
Phan, T.H.D.: Two sided sand piles model and unimodal sequences. ITA 42(3), 631–646 (2008)
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Perrot, K., Rémila, E. (2011). Avalanche Structure in the Kadanoff Sand Pile Model. In: Dediu, AH., Inenaga, S., Martín-Vide, C. (eds) Language and Automata Theory and Applications. LATA 2011. Lecture Notes in Computer Science, vol 6638. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-21254-3_34
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DOI: https://doi.org/10.1007/978-3-642-21254-3_34
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