Abstract
Lattice statistical models of equilibrium critical phenomena generally obey finite size scaling (FSS) ansatz. However, the critical behavior of the prototypical BTW sandpile model demonstrating self-organized criticality at out of equilibrium is described by a peculiar multiscaling behaviour. FSS hypothesis is verified here on two versions (RSM1 and RSM2) of a rotational sandpile model (RSM) with broken mirror symmetry. In these models, sand grains flow only in the forward direction and in a specific rotational direction from an active site after toppling. The toppling rules are such that RSM1 will have less randomness whereas RSM2 will have more randomness with respect to RSM. RSM1 is expected to be more closer to BTW whereas RSM2 is expected to be more closer to Manna’s stochastic model. Both RSM1 and RSM2 are found to belong to the same universality class of RSM. The scaling functions of RSM1 and RSM2 are also found to obey usual FSS hypothesis at out of equilibrium instead of multiscaling as in BTW.
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References
H.E. Stanley, Introduction to Phase Transitions and Critical Phenomena (Oxford University Press, New York, 1971)
J.L. Cardy, Finite-size Scaling, edited by J.L. Cardy (North Holland, Amsterdam, 1988)
P. Bak, How Nature Works: the Science of Self-Organized Criticality (Copernicus, New York, 1996)
H.J. Jensen, Self-Organized Criticality (Cambridge University Press, Cambridge, 1998)
K. Christensen, N.R. Moloney, Complexity and Criticality (Imperial College Press, London, 2005)
K. Chen, P. Bak, S.P. Obukhov, Phys. Rev. A 43, 625 (1990)
H. Takayasu, H. Inaoka, Phys. Rev. Lett. 68, 966 (1992)
P. Bak, K. Snappen, Phys. Rev. Lett. 71, 4083 (1993)
P. Bak, C. Tang, K. Wiesenfeld, Phys. Rev. Lett. 59, 381 (1987)
C. Tebaldi, M. De Menech, A.L. Stella, Phys. Rev. Lett. 83, 3952 (1999)
S. Lübeck, Phys. Rev. E 61, 204 (2000)
Spiral self-avoiding walk: H.J.W. Blöte, H.J. Hilhorst, J. Phys. A 17, L111 (1984)
Spiral lattice animal: T.C. Li, Z.C. Zhou, J. Phys. A 18, 67 (1985)
Spiral percolation: S.B. Santra, I. Bose, J. Phys. A 25, 1105 (1992)
S.B. Santra, S.R. Chanu, D. Deb, Phys. Rev. E 75, 041122 (2007)
S.S. Manna, J. Phys. A: Math. Gen. 24, L363(1991)
S.S. Manna, Physica A 179, 249 (1991)
P. Grassberger, S.S. Manna, J. Phys. (France) 51, 1077 (1990)
S.S. Manna, J. Stat. Phys. 63, 653 (1991)
D. Dhar, R. Ramaswamy, Phys. Rev. Lett. 63, 1659 (1989)
A. Ben-Hur, O. Biham, Phys. Rev. E 53, R1317 (1996)
S. Lübeck, K.D. Usadel, Phys. Rev. E 55, 4095 (1997)
S.S. Manna, A.L. Stella, Physica A 316, 135 (2002)
S.B. Santra, W.A. Seitz, Int. J. Mod. Phys. C 11, 1357 (2000)
A. Chessa, H.E. Stanley, A. Vespignani, S. Zapperi, Phys. Rev. E 59, R12 (1999)
Toppling waves are defined as the number of toppling during the propagation of an avalanche starting from a critical site O without toppling O more than once. Each toppling of O creates a new toppling wave
D.V. Ktitarev, S. Lübeck, P. Grassberger, V.B. Priezzhev, Phys. Rev. E 61, 81 (2000)
M. De Menech, A.L. Stella, Phys. Rev. E 62, R4528 (2000)
M. Paczuski, S. Boettcher, Phys. Rev. E 56, R3745 (1997)
R. Karmakar, S.S. Manna, A.L. Stella, Phys. Rev. Lett. 94, 088002 (2005)
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Ahmed, J., Santra, S. Finite size scaling in BTW like sandpile models. Eur. Phys. J. B 76, 13–20 (2010). https://doi.org/10.1140/epjb/e2010-00198-x
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DOI: https://doi.org/10.1140/epjb/e2010-00198-x