Abstract
We study the steady state of the Abelian sandpile models with stochastic toppling rules. The particle addition operators commute with each other, but in general these operators need not be diagonalizable. We use their Abelian algebra to determine their eigenvalues, and the Jordan block structure. These are then used to determine the probability of different configurations in the steady state. We illustrate this procedure by explicitly determining the numerically exact steady state for a one dimensional example, for systems of size ≤12, and also study the density profile in the steady state.
Similar content being viewed by others
References
Dhar, D.: Theoretical studies of self-organized criticality. Physica A 369, 29 (2006)
Manna, S.S.: Two-state model of self-organized criticality. J. Phys. A, Math. Gen. 24, L363 (1991)
Frette, V., Christensen, K., Mathe-Sorensen, A., Feder, J., Jossang, T., Meakin, P.: Avalanche dynamics in a pile of rice. Nature 379, 49 (1996)
Chessa, A., Vespignani, A., Zapperi, S.: Critical exponents in stochastic sandpile models. Comput. Phys. Commun. 121, 299 (1999)
Bak, P., Tang, C., Wiesenfeld, K.: Self-organized criticality: An explanation of the 1/f noise. Phys. Rev. Lett. 59, 381 (1987)
Bak, P., Tang, C., Wiesenfeld, K.: Self-organized criticality. Phys. Rev. A 38, 364 (1988)
Povolotsky, A.M., Priezzhev, V.B., Hu, C.K.: The asymmetric avalanche process. J. Stat. Phys. 3, 1149 (2003)
Alcaraz F.C., Rittenberg V.: Directed Abelian algebras and their applications to stochastic models. arXiv:0806.1303
Kloster, M., Maslov, S., Tang, C.: Exact solution of a stochastic directed sandpile model. Phys. Rev. E 63, 026111 (2001)
Paczuski, M., Bassler, K.E.: Theoretical results for sandpile models of self-organized criticality with multiple topplings. Phys. Rev. E 62, 5347 (2000)
Dhar, D.: Steady state and relaxation spectrum of the Oslo rice-pile model. Physica A 340, 535 (2004)
Dickman, R., Alva, M., Muñoz, M., Peltola, J., Vespignani, A., Zapperi, S.: Critical behavior of a one-dimensional fixed-energy stochastic sandpile. Phys. Rev. E 64, 056104 (2001)
Stilck, J.F., Dickman, R., Vidigal, R.R.: Series expansion for a stochastic sandpile. J. Phys. A, Math. Gen. 37, 1145 (2004)
Vidigal, R.R., Dickman, R.: Asymptotic behavior of the order parameter in a stochastic sandpile. J. Stat. Phys. 118, 1 (2005)
Diaz-Guilera, A.: Noise and dynamics of self-organized critical phenomena. Phys. Rev. A 45, 8551 (1992)
Vespignani, A., Zapperi, S., Pietronero, L.: Renormalization approach to the self-organized critical behavior of sandpile models. Phys. Rev. E 51, 1711 (1995)
Vespignani, A., Dickman, R., Munoz, M., Zapperi, S.: Driving, conservation, and absorbing states in sandpiles. Phys. Rev. Lett. 81, 5676 (1998)
Ben-Hur, A., Biham, O.: Universality in sandpile models. Phys. Rev. E 53, R1317 (1996)
Lubeck, S., Usadel, K.D.: Numerical determination of the avalanche exponents of the Bak-Tang-Wiesenfeld model. Phys. Rev. E 55, 4095 (1997)
Milshtein, E., Biham, O., Solomon, S.: Universality classes in isotropic, Abelian, and non-Abelian sandpile models. Phys. Rev. E 58, 303 (1998)
Lubeck, S.: Moment analysis of the probability distribution of different sandpile models. Phys. Rev. E 61, 204 (2000)
Menech, M.D., Stella, A.L.: From waves to avalanches: Two different mechanisms of sandpile dynamics. Phys. Rev. E 62, R4528 (2000)
Dickman, R., Campelo, J.M.M.: Avalanche exponents and corrections to scaling for a stochastic sandpile. Phys. Rev. E 67, 066111 (2003)
Satorras, R., Vespignani, A.: Universality classes in directed sandpile models. J. Phys. A 33, L33 (2000)
Biham, O., Milshtein, E., Malcai, O.: Evidence for universality within the class of deterministic and stochastic sandpile models. Phys. Rev. E 63, 061309 (2001)
Bonachela, J.A., Munoz, M.: Confirming and extending the hypothesis of universality in sandpiles. arXiv:0806.4079
Bonachela, J.A., Munoz, M.: How to discriminate easily between directed percolation and Manna scaling. Physica A 384, 89 (2007)
Mohanty, P.K., Dhar, D.: Generic sandpiles have directed percolation exponents. Phys. Rev. Lett. 89, 104303 (2002)
Bonachela, J.A., Ramasco, J.J., Chate, H., Dornic, I., Munoz, M.A.: Sticky grains do not change the universality class of isotropic sandpiles. Phys. Rev. E 74, 050102 (2006)
Mohanty, P.K., Dhar, D.: Critical behavior of sandpile models with sticky grains. Physica A 384, 34 (2007)
Dhar, D.: Some results and a conjecture for Manna’s stochastic sandpile model. Physica A 270, 69 (1999)
Schutz, G.M., Ramaswamy, R., Barma, M.: Pairwise balance and invariant measures for generalized exclusion processes. J. Phys. A, Math. Gen. 29, 837 (1996)
Lubeck, S., Dhar, D.: Continuously varying exponents in sandpile models. J. Stat. Phys. 102, 1 (2001)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Sadhu, T., Dhar, D. Steady State of Stochastic Sandpile Models. J Stat Phys 134, 427–441 (2009). https://doi.org/10.1007/s10955-009-9683-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10955-009-9683-7