Sandpiles and diffusion-limited reactions

Joachim Krug

doi:10.1103/PhysRevE.47.730

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Sandpiles and diffusion-limited reactions

Joachim Krug

IBM Research Division, Thomas J. Watson Research Center, Yorktown Heights, New York 10598

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Phys. Rev. E 47, 730 – Published 1 January, 1993

DOI: https://doi.org/10.1103/PhysRevE.47.730

Abstract

We establish the equivalence between a one-dimensional sandpile model recently introduced by Chhabra et al. [Phys. Rev. A (to be published)], and the diffusion-limited reaction A+A→0 with a point source. The equivalence is used to predict the power-law decay of the avalanche size distribution, P(s)∼ $s^{- 3}$ , in agreement with numerical simulations. This result differs from the mean-field treatment of Chhabra et al., demonstrating the importance of fluctuations in this low-dimensional system.

References (14)

P. Bak, C. Tang and K. Wiesenfeld, Phys. Rev. Lett. 59, 381 (1987); Phys. Rev. A 38, 364 (1988).
L. P. Kadanoff, S. R. Nagel, L. Wu and S.-M. Zhou, Phys. Rev. A 39, 6524 (1989).
J. Krug, J. Stat. Phys. 66, 1635 (1992).
D. Dhar, Physica A 186, 82 (1992).
J. M. Carlson, J. T. Chayes, E. R. Grannan and G. H. Swindle, Phys. Rev. Lett. 65, 2547 (1990); Phys. Rev. A 42, 2467 (1990).
J. Krug, J. E. S. Socolar and G. Grinstein, Phys. Rev. A 46, 4479 (1992).
A. B. Chhabra, M. J. Feigenbaum, L. P. Kadanoff, A. J. Kolan, and I. Procaccia, Phys. Rev. A (to be published).
L. P. Kadanoff, A. B. Chhabra, A. J. Kolan, M. J. Feigenbaum and I. Procaccia, Phys. Rev. A 45, 6095 (1992).
A different two-state approximation to the $L^{2}$ model was proposed by Carlson et al. [5].
As in previous studies [3,5,6,7] of the $L^{2}$ model avalanches are considered to be instantaneous, i.e., the addition of a grain is pictured to immediately take the sandpile from one stable configuration to another, with the net loss (or gain) of some number of grains; the dynamics is ``integrated'' between successive additions of sand. In particular, the restriction $S_{i}$ >= 1 in the $L^{3}$ model may be violated during an avalanche, but not in the final stable configuration.
K. Kang and S. Redner, Phys. Rev. A 32, 435 (1985), and references therein.
Z. Cheng, S. Redner and F. Leyvraz, Phys. Rev. Lett. 62, 2321 (1989). These authors actually study the coagulation reaction A+A -> A rather than the annihilation reaction A+A -> 0, but the scaling properties of the two processes are expected to be the same [see, e.g., L. Peliti, J. Phys. A 19, L365 (1985)].
R. Kopelman, J. Stat. Phys. 42, 185 (1986); C. R. Doering and D. ben-Avraham, Phys. Rev. A 38, 3035 (1988).
R. Rammal and G. Toulouse, J. Phys. (Paris) Lett. 44, L13 (1983).

[c1] P. Bak, C. Tang and K. Wiesenfeld, Phys. Rev. Lett. 59, 381 (1987); Phys. Rev. A 38, 364 (1988).

[c2] L. P. Kadanoff, S. R. Nagel, L. Wu and S.-M. Zhou, Phys. Rev. A 39, 6524 (1989).

[c3] J. Krug, J. Stat. Phys. 66, 1635 (1992).

[c4] D. Dhar, Physica A 186, 82 (1992).

[c5] J. M. Carlson, J. T. Chayes, E. R. Grannan and G. H. Swindle, Phys. Rev. Lett. 65, 2547 (1990); Phys. Rev. A 42, 2467 (1990).

[c6] J. Krug, J. E. S. Socolar and G. Grinstein, Phys. Rev. A 46, 4479 (1992).

[c7] A. B. Chhabra, M. J. Feigenbaum, L. P. Kadanoff, A. J. Kolan, and I. Procaccia, Phys. Rev. A (to be published).

[c8] L. P. Kadanoff, A. B. Chhabra, A. J. Kolan, M. J. Feigenbaum and I. Procaccia, Phys. Rev. A 45, 6095 (1992).

[c9] A different two-state approximation to the $L^{2}$ model was proposed by Carlson et al. [5].

[c10] As in previous studies [3,5,6,7] of the $L^{2}$ model avalanches are considered to be instantaneous, i.e., the addition of a grain is pictured to immediately take the sandpile from one stable configuration to another, with the net loss (or gain) of some number of grains; the dynamics is ``integrated'' between successive additions of sand. In particular, the restriction $S_{i}$ >= 1 in the $L^{3}$ model may be violated during an avalanche, but not in the final stable configuration.

[c11] K. Kang and S. Redner, Phys. Rev. A 32, 435 (1985), and references therein.

[c12] Z. Cheng, S. Redner and F. Leyvraz, Phys. Rev. Lett. 62, 2321 (1989). These authors actually study the coagulation reaction A+A -> A rather than the annihilation reaction A+A -> 0, but the scaling properties of the two processes are expected to be the same [see, e.g., L. Peliti, J. Phys. A 19, L365 (1985)].

[c13] R. Kopelman, J. Stat. Phys. 42, 185 (1986); C. R. Doering and D. ben-Avraham, Phys. Rev. A 38, 3035 (1988).

[c14] R. Rammal and G. Toulouse, J. Phys. (Paris) Lett. 44, L13 (1983).

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Sandpiles and diffusion-limited reactions

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