Geodesic
Algebraic approach to directed stochastic avalanches
Teoretičeskaâ i matematičeskaâ fizika, Tome 169 (2011) no. 3, pp. 444-462 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

Using directed Abelian algebras recently introduced by Alcaraz and Rittenberg, we study a two-dimensional directed stochastic sandpile model analytically. We obtain some exact expressions for the probabilities of possible toppling events involved in the transfer of an arbitrary number of particles to a site in the stationary state. We propose a description of the virtual-time evolution of directed avalanches on two-dimensional lattices. Because the general problem is intractable, we apply the algebraic approach to special cases of directed deterministic avalanches and trivial stochastic avalanches describing simple random walks of two particles. Studying these cases clarifies the role of each particular kind of toppling in the avalanche evolution. In the general case of the quadratic directed algebra, we exactly determine the maximum possible values of the current of particles at any given virtual instant and the occupation number (“height”) of each site at any instant.
Keywords: stochastic process, nonequilibrium stationary state, avalanche dynamics, directed stochastic sandpile, directed Abelian algebra. 
@article{TMF_2011_169_3_a7,
     author = {B. L. Aneva and I. G. Brankov},
     title = {Algebraic approach to directed stochastic avalanches},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {444--462},
     year = {2011},
     volume = {169},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2011_169_3_a7/}
}
TY  - JOUR
AU  - B. L. Aneva
AU  - I. G. Brankov
TI  - Algebraic approach to directed stochastic avalanches
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 2011
SP  - 444
EP  - 462
VL  - 169
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/TMF_2011_169_3_a7/
LA  - ru
ID  - TMF_2011_169_3_a7
ER  - 
%0 Journal Article
%A B. L. Aneva
%A I. G. Brankov
%T Algebraic approach to directed stochastic avalanches
%J Teoretičeskaâ i matematičeskaâ fizika
%D 2011
%P 444-462
%V 169
%N 3
%U http://geodesic.mathdoc.fr/item/TMF_2011_169_3_a7/
%G ru
%F TMF_2011_169_3_a7
B. L. Aneva; I. G. Brankov. Algebraic approach to directed stochastic avalanches. Teoretičeskaâ i matematičeskaâ fizika, Tome 169 (2011) no. 3, pp. 444-462. http://geodesic.mathdoc.fr/item/TMF_2011_169_3_a7/

[1] P. Bak, C. Tang, K. Wiesenfeld, Phys. Rev. Lett., 59:4 (1987), 381–384 | DOI

[2] D. Dhar, Phys. Rev. Lett., 64:14 (1990), 1613–1616 | DOI | MR | Zbl

[3] V. B. Priezzhev, D. V. Kitarev, E. V. Ivashkevich, Phys. Rev. Lett., 76:12 (1996), 2093–2096 | DOI

[4] E. V. Ivashkevich, V. B. Priezzhev, Physica A, 254:1–2 (1998), 97–116, arXiv: cond-mat/9801182 | DOI

[5] D. Dhar, Physica A, 263:1–4 (1999), 4–25, arXiv: cond-mat/9808047 | DOI

[6] D. Dhar, R. Ramaswamy, Phys. Rev. Lett., 63:16 (1989), 1659–1662 | DOI | MR

[7] S. S. Manna, J. Phys. A, 24:7 (1991), L363–L369 | DOI

[8] R. Pastor-Satorras, A. Vespignani, J. Phys. A, 33:3 (2000), L33–L39, arXiv: cond-mat/9907307 | DOI | MR

[9] A. Vazquez, Numerical simulations of directed sandpile models, arXiv: cond-mat/0003420

[10] F. C. Alcaraz, V. Rittenberg, Phys. Rev. E, 78:4 (2008), 041126, 13 pp. | DOI | MR

[11] M. Paczuski, K. Bassler, Phys. Rev. E, 62:4 (2000), 5347–5352, arXiv: cond-mat/0005340 | DOI | MR

[12] M. Kloster, S. Maslov, C. Tang, Phys. Rev. E, 63:2 (2001), 026111, 4 pp., arXiv: cond-mat/0005528 | DOI

[13] N. Zh. Bunzarova, Phys. Rev. E, 82:3 (2008), 031116, 14 pp. | DOI | MR