- Email: [email protected]
Effect of anisotropy on the self-organized critical states of Abelian sandpile models
Physica A 266 (1999) 358–361
E ect of anisotropy on the self-organized critical states of Abelian sandpile models Tomoko Tsuchiya ∗ , Makoto Katori Department of Physics, Faculty of Science and Engineering, Chuo University, Kasuga, Bunkyo-ku, Tokyo 112-8551, Japan
Abstract The directed Abelian sandpile models are deÿned on a square lattice by introducing a parameter, c, representing the degree of anisotropy in the avalanche processes, in which c = 1 is for the isotropic case. We calculate the expected number of the topplings per added particle, hT i, which depends on the lattice size L as Lx for large L. Our exact solution gives that x = 1 when any anisotropy is included in the system, while x = 2 in the isotropic case. These results allow us to introduce a new critical exponent, Â, deÿned by ≡ limL→∞ hT i=L with c 6= 1 as ∼ |c − 1|− c 1999 Elsevier Science for |c − 1| 1. From the explicit expression of hT i, we obtain  = 1. B.V. All rights reserved. PACS: 64.60.-i; 05.40.+j; 05.60.+w; 46.10.+z Keywords: Abelian sandpile models; Avalanche sizes; Critical exponents; Anisotropy
1. Introduction Bak et al. proposed the attractive concept of the self-organized criticality (SOC), in which a system spontaneously develops into a critical state characterized by power-law correlations and lack of any scale [1–4]. They have introduced stochastic cellular automata called the sandpile models whose time-evolution produces the SOC [1,2]. Among many models exhibiting the SOC, the Abelian sandpile model (ASM) is special for theorists since it is tractable by analytic calculations. This model can be deÿned on an arbitrary ÿnite set of N sites. Each variable at the site x; z(x), represents the height of the sand column at the position, which takes positive integers. In a stable state, all z(x) are equal to or less than a critical value, zc , i.e. z(x) ∈ {1; 2; : : : ; zc } for any x. ∗
Corresponding author.
c 1999 Elsevier Science B.V. All rights reserved. 0378-4371/99/$ – see front matter PII: S 0 3 7 8 - 4 3 7 1 ( 9 8 ) 0 0 6 1 6 - 5
T. Tsuchiya, M. Katori / Physica A 266 (1999) 358–361
359
The time evolution consists of the following two processes. (i) Adding a particle at a randomly chosen site x: z(x) → z(x) + 1. (ii) The toppling rule is deÿned in terms of an N × N matrix, . If any z(x) ¿ zc , then z(y) → z(y) − (x; y), where (x; y) is the (x; y)-element of matrix. When a toppling occurs at a boundary site, then some particles leave the system. This dissipation plays a key role to make the system relax to a stable state. Dhar studied the ASM and gave the following exact formulas for the SOC state [5]. • There are two classes of conÿgurations, recurrent and transient. Only recurrent conÿgurations are allowed to occur in the SOC state and their occurrence probabilities are equal to each other. The number of allowed conÿgurations is given as NR = Det. • The expected number of topplings at site y due to the avalanche caused by adding a particle at site x can be given by, G(x; y) = [−1 ](x; y), where −1 is the inverse matrix of . Then the expected number of topplings per added particle, hT i, is given as hT i =
1 X X −1 [ ](x; y) : N x y
(1)
For the isotropic nearest-neighbor ASM on an L × L square lattice several quantities characterizing the SOC state have been obtained: the fractional number of sites having a given height [6], the height correlations [7] and the exponents characterizing avalanches [8]. In particular, it is shown that hT i varies as Lx for large L and x = 2 [5]. Dhar and Ramaswamy obtained some critical exponents especially deÿned for the completely directed ASM on the Ld−1 × ∞ lattices [9]. Kadano et al. [10] and Kamakura et al. [11] numerically studied the two-dimensional anisotropic ASM and conjectured x = 1. In this paper, we deÿne the directed Abelian sandpile model (DASM) on an L × L square lattice by introducing a parameter representing the degree of anisotropy, c, where c = 1 corresponds to the isotropic case. We calculate hT i for large L. Using our exact expression, we prove the conjecture by Kadano et al. [10] that when any anisotropy is included in the system, x = 1. Then we can deÿne the limit ≡ limL→∞ hT i=L with c 6= 1 and our exact solution shows that diverges as c → 1 as ∼ |c − 1|− for |c − 1| 1. We refer to  as the anisotropy exponent in the SOC state and determine  = 1.
2. Deÿnition of the directed Abelian sandpile models We consider an L × L square lattice. Let c be a positive rational number and choose a positive integer so that c becomes a positive integer. In our model, we assume
360
T. Tsuchiya, M. Katori / Physica A 266 (1999) 358–361
zc = 2(c + 1) and is deÿned as (x; y) = 2(c + 1)
if y = x + e1 or x + e2 ;
(x; y) = −c (x; y) = − (x; y) = 0
if y = x ; (2)
if y = x − e1 or x − e2 ; otherwise
where e1 and e2 are the basis vectors of the square lattice. When c ¿ 1 (c ¡ 1, resp.) the toppling particles prefer toppling to the positive (negative, resp.) directions of e1 and e2 to toppling to the negative (positive, resp.) directions. When c =1, this model is just isotropic and reduced to be the undirected nearest-neighbor ASM. From the toppling rules above mentioned, it can be conÿrmed that the DASM with the parameters (c; ) with c ¡ 1 is the same as the DASM with the parameter (1=c; c). We focus on the cases of c ¿ 1 in the following calculations. The special limits c → 0 and c → ∞ correspond to the completely directed models [9,10]. 3. Calculation of hTi and conclusion To determine hT i, we have to obtain −1 . We need the matrices P and Q satisfying PQ = QP = I and = PQ, where is the diagonal matrix. We found that n2 x2 n1 x1 2 (x1 +x2 )=2 c sin ; (3) sin P(n; x) = L+1 L+1 L+1 n1 x1 2 −(x1 +x2 )=2 n2 x2 c ; sin Q(x; n) = sin L+1 L+1 L+1
0
(n; n ) = n1 ; n01 n2 ; n02 2 c + 1 −
√
n2 n1 + cos ; c cos L+1 L+1
(4)
(5)
where x = (x1 ; x2 ), n = (n1 ; n2 ) and n0 = (n01 ; n02 ). From these expressions and (1), we obtain 2 n2 L L X n1 X sin2 L+1 sin L+1 2c2 √ hT i = 2 n1 n2 2 L (L + 1) c + 1 − c(cos( L+1 ) + cos( L+1 )) n =1 n =1 1
×
2
2 − (−1)n1 (c(L+1)=2 + c−(L+1)=2 ) 2 − (−1)n2 (c(L+1)=2 + c−(L+1)=2 ) √ √ × : n1 n2 {1 + c − 2 c cos( L+1 )}2 {1 + c − 2 c cos( L+1 )}2
(6)
This seems to include the terms which behave as c L =L4 when L1. However, we have performed the double summations in (6) carefully and found that the exponential factor c L is cancelled out. The ÿnal expression is very simple and hT i '
L 3(c − 1)
for L1 :
(7)
T. Tsuchiya, M. Katori / Physica A 266 (1999) 358–361
361
From this we get the critical exponents as x = 1 and  = 1 for any c 6= 1. More details of calculations are reported elsewhere [12]. Since hT i expresses the mean volume of an avalanche, this result implies that the avalanches in the directed cases are one-dimensional, while they are two-dimensional in the undirected cases. We have conÿrmed this result also for the one-dimensional model and expect it for any dimension [12,10]. If this conjecture is true, it may be concluded that the upper critical dimension du is 2 in the c 6= 1 cases, although du = 4 in the c = 1 case. Acknowledgements The authors express their gratitude to K. Honda and M. Shimamura for useful informations of the computer simulation results. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
P. Bak, C. Tang, K. Wiesenfeld, Phys. Rev. Lett. 59 (1987) 381. P. Bak, C. Tang, K. Wiesenfeld, Phys. Rev. A 38 (1988) 364. P. Bak, How Nature Works, Oxford University Press, Oxford, 1997. H.J. Jensen, Self-Organized Criticality, Cambridge University Press, Cambridge, 1998. D. Dhar, Phys. Rev. Lett. 64 (1990) 1613. V.B. Priezzhev, J. Stat. Phys. 74 (1994) 955. S.N. Majumdar, D. Dhar, J. Phys. A 24 (1991) L357. V.B. Priezzhev, D.V. Ktitarev, E.V. Ivashkevich, Phys. Rev. Lett. 76 (1996) 2093. D. Dhar, R. Ramaswamy, Phys. Rev. Lett. 63 (1989) 1659. L.P. Kadano , S.R. Nagel, L. Wu, S.-M. Zhou, Phys. Rev. A 39 (1989) 6524. N. Kamakura, T. Terao, K. Honda, private communication. T. Tsuchiya, M. Katori, J. Phys. A, to appear.
E ect of anisotropy on the self-organized critical states of Abelian sandpile models Tomoko Tsuchiya ∗ , Makoto Katori Department of Physics, Faculty of Science and Engineering, Chuo University, Kasuga, Bunkyo-ku, Tokyo 112-8551, Japan
Abstract The directed Abelian sandpile models are deÿned on a square lattice by introducing a parameter, c, representing the degree of anisotropy in the avalanche processes, in which c = 1 is for the isotropic case. We calculate the expected number of the topplings per added particle, hT i, which depends on the lattice size L as Lx for large L. Our exact solution gives that x = 1 when any anisotropy is included in the system, while x = 2 in the isotropic case. These results allow us to introduce a new critical exponent, Â, deÿned by ≡ limL→∞ hT i=L with c 6= 1 as ∼ |c − 1|− c 1999 Elsevier Science for |c − 1| 1. From the explicit expression of hT i, we obtain  = 1. B.V. All rights reserved. PACS: 64.60.-i; 05.40.+j; 05.60.+w; 46.10.+z Keywords: Abelian sandpile models; Avalanche sizes; Critical exponents; Anisotropy
1. Introduction Bak et al. proposed the attractive concept of the self-organized criticality (SOC), in which a system spontaneously develops into a critical state characterized by power-law correlations and lack of any scale [1–4]. They have introduced stochastic cellular automata called the sandpile models whose time-evolution produces the SOC [1,2]. Among many models exhibiting the SOC, the Abelian sandpile model (ASM) is special for theorists since it is tractable by analytic calculations. This model can be deÿned on an arbitrary ÿnite set of N sites. Each variable at the site x; z(x), represents the height of the sand column at the position, which takes positive integers. In a stable state, all z(x) are equal to or less than a critical value, zc , i.e. z(x) ∈ {1; 2; : : : ; zc } for any x. ∗
Corresponding author.
c 1999 Elsevier Science B.V. All rights reserved. 0378-4371/99/$ – see front matter PII: S 0 3 7 8 - 4 3 7 1 ( 9 8 ) 0 0 6 1 6 - 5
T. Tsuchiya, M. Katori / Physica A 266 (1999) 358–361
359
The time evolution consists of the following two processes. (i) Adding a particle at a randomly chosen site x: z(x) → z(x) + 1. (ii) The toppling rule is deÿned in terms of an N × N matrix, . If any z(x) ¿ zc , then z(y) → z(y) − (x; y), where (x; y) is the (x; y)-element of matrix. When a toppling occurs at a boundary site, then some particles leave the system. This dissipation plays a key role to make the system relax to a stable state. Dhar studied the ASM and gave the following exact formulas for the SOC state [5]. • There are two classes of conÿgurations, recurrent and transient. Only recurrent conÿgurations are allowed to occur in the SOC state and their occurrence probabilities are equal to each other. The number of allowed conÿgurations is given as NR = Det. • The expected number of topplings at site y due to the avalanche caused by adding a particle at site x can be given by, G(x; y) = [−1 ](x; y), where −1 is the inverse matrix of . Then the expected number of topplings per added particle, hT i, is given as hT i =
1 X X −1 [ ](x; y) : N x y
(1)
For the isotropic nearest-neighbor ASM on an L × L square lattice several quantities characterizing the SOC state have been obtained: the fractional number of sites having a given height [6], the height correlations [7] and the exponents characterizing avalanches [8]. In particular, it is shown that hT i varies as Lx for large L and x = 2 [5]. Dhar and Ramaswamy obtained some critical exponents especially deÿned for the completely directed ASM on the Ld−1 × ∞ lattices [9]. Kadano et al. [10] and Kamakura et al. [11] numerically studied the two-dimensional anisotropic ASM and conjectured x = 1. In this paper, we deÿne the directed Abelian sandpile model (DASM) on an L × L square lattice by introducing a parameter representing the degree of anisotropy, c, where c = 1 corresponds to the isotropic case. We calculate hT i for large L. Using our exact expression, we prove the conjecture by Kadano et al. [10] that when any anisotropy is included in the system, x = 1. Then we can deÿne the limit ≡ limL→∞ hT i=L with c 6= 1 and our exact solution shows that diverges as c → 1 as ∼ |c − 1|− for |c − 1| 1. We refer to  as the anisotropy exponent in the SOC state and determine  = 1.
2. Deÿnition of the directed Abelian sandpile models We consider an L × L square lattice. Let c be a positive rational number and choose a positive integer so that c becomes a positive integer. In our model, we assume
360
T. Tsuchiya, M. Katori / Physica A 266 (1999) 358–361
zc = 2(c + 1) and is deÿned as (x; y) = 2(c + 1)
if y = x + e1 or x + e2 ;
(x; y) = −c (x; y) = − (x; y) = 0
if y = x ; (2)
if y = x − e1 or x − e2 ; otherwise
where e1 and e2 are the basis vectors of the square lattice. When c ¿ 1 (c ¡ 1, resp.) the toppling particles prefer toppling to the positive (negative, resp.) directions of e1 and e2 to toppling to the negative (positive, resp.) directions. When c =1, this model is just isotropic and reduced to be the undirected nearest-neighbor ASM. From the toppling rules above mentioned, it can be conÿrmed that the DASM with the parameters (c; ) with c ¡ 1 is the same as the DASM with the parameter (1=c; c). We focus on the cases of c ¿ 1 in the following calculations. The special limits c → 0 and c → ∞ correspond to the completely directed models [9,10]. 3. Calculation of hTi and conclusion To determine hT i, we have to obtain −1 . We need the matrices P and Q satisfying PQ = QP = I and = PQ, where is the diagonal matrix. We found that n2 x2 n1 x1 2 (x1 +x2 )=2 c sin ; (3) sin P(n; x) = L+1 L+1 L+1 n1 x1 2 −(x1 +x2 )=2 n2 x2 c ; sin Q(x; n) = sin L+1 L+1 L+1
0
(n; n ) = n1 ; n01 n2 ; n02 2 c + 1 −
√
n2 n1 + cos ; c cos L+1 L+1
(4)
(5)
where x = (x1 ; x2 ), n = (n1 ; n2 ) and n0 = (n01 ; n02 ). From these expressions and (1), we obtain 2 n2 L L X n1 X sin2 L+1 sin L+1 2c2 √ hT i = 2 n1 n2 2 L (L + 1) c + 1 − c(cos( L+1 ) + cos( L+1 )) n =1 n =1 1
×
2
2 − (−1)n1 (c(L+1)=2 + c−(L+1)=2 ) 2 − (−1)n2 (c(L+1)=2 + c−(L+1)=2 ) √ √ × : n1 n2 {1 + c − 2 c cos( L+1 )}2 {1 + c − 2 c cos( L+1 )}2
(6)
This seems to include the terms which behave as c L =L4 when L1. However, we have performed the double summations in (6) carefully and found that the exponential factor c L is cancelled out. The ÿnal expression is very simple and hT i '
L 3(c − 1)
for L1 :
(7)
T. Tsuchiya, M. Katori / Physica A 266 (1999) 358–361
361
From this we get the critical exponents as x = 1 and  = 1 for any c 6= 1. More details of calculations are reported elsewhere [12]. Since hT i expresses the mean volume of an avalanche, this result implies that the avalanches in the directed cases are one-dimensional, while they are two-dimensional in the undirected cases. We have conÿrmed this result also for the one-dimensional model and expect it for any dimension [12,10]. If this conjecture is true, it may be concluded that the upper critical dimension du is 2 in the c 6= 1 cases, although du = 4 in the c = 1 case. Acknowledgements The authors express their gratitude to K. Honda and M. Shimamura for useful informations of the computer simulation results. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
P. Bak, C. Tang, K. Wiesenfeld, Phys. Rev. Lett. 59 (1987) 381. P. Bak, C. Tang, K. Wiesenfeld, Phys. Rev. A 38 (1988) 364. P. Bak, How Nature Works, Oxford University Press, Oxford, 1997. H.J. Jensen, Self-Organized Criticality, Cambridge University Press, Cambridge, 1998. D. Dhar, Phys. Rev. Lett. 64 (1990) 1613. V.B. Priezzhev, J. Stat. Phys. 74 (1994) 955. S.N. Majumdar, D. Dhar, J. Phys. A 24 (1991) L357. V.B. Priezzhev, D.V. Ktitarev, E.V. Ivashkevich, Phys. Rev. Lett. 76 (1996) 2093. D. Dhar, R. Ramaswamy, Phys. Rev. Lett. 63 (1989) 1659. L.P. Kadano , S.R. Nagel, L. Wu, S.-M. Zhou, Phys. Rev. A 39 (1989) 6524. N. Kamakura, T. Terao, K. Honda, private communication. T. Tsuchiya, M. Katori, J. Phys. A, to appear.