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Corrections to scaling and probability distribution of avalanches for the stochastic Zhang sandpile model
ARTICLE IN PRESS
Physica A 363 (2006) 299–306 www.elsevier.com/locate/physa
Corrections to scaling and probability distribution of avalanches for the stochastic Zhang sandpile model Duan-Ming Zhanga, Yan-Ping Yina,, Gui-Jun Pana,b, Fan Suna a
Department of Physics, Huazhong University of Science and Technology, Wuhan 430074, China b Faculty of Physics and Electronic Technology , Hubei University, Whan 430062, China Received 12 May 2005 Available online 9 September 2005
Abstract We study the distributions of dissipative and nondissipative avalanches separately in the stochastic Zhang (SP-Z) sandpile in two dimension. We find that dissipative and nondissipative avalanches obey simple power laws and do not have the logarithmic correction, while the avalanche distributions in the Abelian Manna model should include a logarithmic correction. We use the moment analysis to determine the numerical critical exponents of dissipative and nondissipative avalanches, respectively, and find that they are different from the corresponding values in the Abelian Manna model. All these indicate that the stochastic Zhang model and the Abelian Manna model belong to distinct universality classes, which imply that the Abelian symmetry breaking changes the scaling behavior of the avalanches in the case of the stochastic sandpile model. r 2005 Elsevier B.V. All rights reserved. Keywords: Self-organized criticality; Sandpile model; Dissipative; Nondissipative; Moment analysis
1. Introduction Since its introduction in 1987, the sandpile model has been considered as the prototype of a self-organized critical (SOC) system [1]. The identification of the universality classes is the most important problem in the field of SOC. It still remains an unclear problem that whether the Abelian symmetry breaking changes the scaling behavior of the avalanches in the stochastic sandpile model. Recent studies showed that the nonAbelian stochastic directed sandpile model (NA-SDM) and the Abelian stochastic directed sandpile model (A-SDM) belong to the same universality class under the condition of parallel update [2]. However, our previous work on directed stochastic models presented that the NA-SDM and the A-SDM do not belong to the same universality class [3]. It seems to imply that the Abelian symmetry breaking changes the scaling behavior of the avalanches in the directed stochastic sandpile model. In the case of stochastic sandpile model, does the Abelian symmetry breaking change the scaling behavior of the avalanches? In order to get a deeper
Corresponding author. Tel./fax: +86 027 87543882.
E-mail address: Aegeanfi[email protected] (Y.-P. Yin). 0378-4371/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2005.08.032
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understanding of this problem, we reconsider the distributions of dissipative and nondissipative avalanches in the stochastic Zhang model. Drossel performed that in two-dimensional BTW sandpile, dissipative avalanche distributions follow clean power laws [4]. Distributions of nondissipative avalanches must also follow power laws in the infinite-size limit, but are subject to much stronger corrections to scaling. Dickman and Campelo showed that in one and two-dimensional (2D) Manna’s sandpile, avalanche distributions in general do not follow simple power laws, but rather include a logarithmic correction [5]. Thus , it would be interesting to find whether the logarithmic correction appears in other models exhibiting SOC. After the Zhang model was introduced by Zhang [6] in 1989, many numerical works were made on this model [7–9]. In this paper we study the dissipative and nondissipative avalanches separately in the twodimensional stochastic Zhang (SP-Z) model. We find that the distributions of dissipative and nondissipative avalanches of the stochastic Zhang model exhibit a standard FSS behavior, and follow pure power laws that they do not have the logarithmic correction to scaling. It seems to imply that the logarithmic correction presented by Ref. [5] is dependent on the inherent mechanism of the Abelian Manna model. We also determine the critical exponents of dissipative and nondissipative avalanches by using the moment analysis [10–14]. Our principal result is that the scaling behavior of dissipative and nondissipative avalanches in the stochastic Zhang sandpile is different from that in the Abelian Manna model. It indicates that the stochastic Zhang model and the Abelian Manna model belong to different universality classes. 2. Model and simulations The SP-Z model [9] is defined on a D-dimensional square lattice of linear size L in which we assign a nonnegative continuous variable E i called ‘‘energy’’ on each site. At each time step, an amount of energy d is added to a randomly chosen site j according to E j ! E j þ d. The quantity d is a random variable uniformly distributed in ½0; dmax . In our simulations we consider the fixed value dmax ¼ 0:25. When a site acquires an energy larger than or equal to 1 ðE i X1Þ, it becomes active and topples. An active site i relaxes losing all its energy, E i ! 0 , which is randomly redistributed among its nearest P neighbors. In the practical implementation of this rule, we draw four random numbers i0 , 0pi0 p1, with i0 i0 ¼ 1 and update the nearest neighbors i0 by E i0 ! E i0 þ i0 E i . We study the parallel updating that all active sites release their energy simultaneously. We analyze the dissipative and nondissipative avalanches separately. Dissipative avalanches are those in which some energy leaves the system, while nondissipative avalanches are those in which no energy leaves the system. We reported three systems of different sizes for L ¼ 160, 320, and 640 in two dimension. Our results are based on samples of about 107 avalanches for all the systems. We present the avalanche distributions Ps ðsÞ of avalanche sizes s while ‘‘size’’ means the number of topplings in an avalanche, as shown in Fig. 1. The size distributions of dissipative avalanches are presented in Fig. 1(a), while the size distributions of nondissipative avalanches are shown in Fig. 1(b). The morphology of these two kinds of avalanche distributions generally includes a plateau-like region for small s, a rapidly decaying portion for large s, and a power-law-like interval between these limiting regimes. The power-law interval is increased with the size of the system. The probability distribution in the second and third portion generally follows: Ps ðsÞ ¼ sts f s ðs=sc Þ ,
(1)
where f s is a cutoff function decaying rapidly for large argument. The cutoff function f s must take a constant value for s sc . And sc is the cutoff characteristic size which diverges as sc Lbs when the system size L goes to infinity. We compare distributions of dissipative avalanches with those of nondissipative avalanches for system of L ¼ 640 in Fig. 2. It is illustrated that the two kinds of avalanche distributions obey the power laws but have different slopes of the power-law-like interval, which indicates that they present two distinct scaling behaviors. In other words, whether the avalanche is dissipative or nondissipative does not alter the power-law distributions of avalanches, but it changes the scaling behavior of avalanches. Dickman and Campelo showed that distributions of dissipative and nondissipative avalanches in Abelian Manna’s sandpile do not follow pure power laws [5], but include a logarithmic correction to scaling
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(a)
(b) Fig. 1. The probability distribution plot of (a) dissipative avalanches and (b) nondissipative avalanches in the two-dimensional model for L ¼ 160, 320, and 640.
presented as Ps ðsÞ ¼ sts ðIn sÞg f s ðs=sc Þ .
(2)
The variance of f s fluctuates around a constant value with the logarithmic correction, while it has a strong fluctuation with a pure power-law fit. We study the stochastic Zhang model using the method of Dickman and Campelo. First, we take a preliminary estimate of the fitting linear portion ½x0 ; x1 (here x In s ) in the log–log plot of probability distribution of avalanches (Fig. 1). Second, we adjust parameters t and g to minimize the variance of f s ¼ st Ps ðsÞ=ðIn sÞg . f s should take a constant value ideally. Third, we check for any systematic practice and refine the fitting portion accordingly in practice. We use the same x0 for different system sizes, and increase x1 linearly with In L.
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Fig. 2. The probability plot of dissipative and nondissipative avalanches in the two-dimensional model for L ¼ 640.
Our results are presented in Fig. 3 which shows the plot of f s ¼ st Ps ðsÞ=ðIn sÞg versus In s using data from L ¼ 640. For dissipative avalanches in Fig. 3(a), it is shown that the fluctuating spectrum of f s is about 0.55 with pure power-law fit, while the fluctuating spectrum of f s is also about 0.55 at least using t ¼ 1:580 and g ¼ 0:380. It implies that the logarithmic correction is not necessary for dissipative avalanches. For nondissipative avalanches in Fig. 3(b), it is illustrated that the fluctuating spectrum of f s is about 1.05 with pure power-law fit, while the fluctuating spectrum of f s is approximately 0.95 at least using t ¼ 1:490 and g ¼ 0:500, decreased about 10%. It can be considered that the logarithmic correction does not avail to reduce the variance of f s . So we come to the conclusion that in the stochastic Zhang model the variance of f s has no reduction with logarithmic correction to scaling. In other words, the logarithmic correction to scaling is of no necessity in the stochastic Zhang model, while it is needed in the Abelian Manna model. It can be concluded that for the stochastic Zhang model, the distributions of dissipative and nondissipative avalanches follow simple power laws in the infinite-size limit. Under the finite size scaling (FSS) assumption of Eq. (1), the set of exponents fts ; bs g characterize the critical behavior and completely defines the universality class of the model. Then we analyzed the SP-Z model by determining their critical exponents with the moment analysis method [10–14]. The q-moment of s on a lattice of L is defined as Z hsq iL ¼ sq PðsÞ ds . (3) If FSS hypothesis (Eq. (1)) is valid, at least in the asymptotic limit (s ! 1), we can transform z ¼ s=Lbs and obtain Z (4) hsq iL ¼ Lbs ðqþ1ts Þ sqþts f s ðzÞ dzLbs ðqþ1ts Þ . More generally, we have hsq iL Lss ðqÞ , where the behavior of the exponent ss ðqÞ can be obtained via a regression analysis of Inhsq iL as a function of In L, or via the logarithmic derivative qInhsq iL . @ In L Comparing with Eq. (4) we have ss ðqÞ ¼
(5)
ss ðqÞ ¼ bs ðq þ 1 ts Þ .
(6)
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(a)
(b) Fig. 3. Plot of f s ¼ st Ps ðsÞ=ðIn sÞg versus In s for pure power-law fit and fit with logarithmic correction of (a) dissipative avalanches and (b) nondissipative avalanches using data from L ¼ 640.
So we can determine the cutoff exponent bs from the slope of ss ðqÞ versus q, and the avalanche exponent ts from an extrapolation to the horizontal axis (Eq. (6)). In Fig. 4 we have plotted the exponent ss ðqÞ of dissipative and nondissipative avalanches. By measuring the slope of the linear part of the exponent ss ðqÞ, we obtain the cutoff exponent bs ¼ 1:754 0:004 for dissipative avalanche and bs ¼ 1:545 0:005 for nondissipative avalanche. The extrapolation to ss ðqÞ ¼ 0 yields the exponent ts ¼ 1:225 0:001 for dissipative avalanches and ts ¼ 1:647 0:001 for nondissipative avalanches. They are much different from corresponding values in the Abelian Manna model, which are ts ¼ 0:98ð2Þ for dissipative avalanches and ts ¼ 1:30ð1Þ for nondissipative avalanches. It is indicated that the SP-Z model and the Abelian Manna model belong to distinct universality classes. The finite size scaling of Eq. (1) has to be verified and must be consistent with the numerical exponents obtained from the moment analysis. The FSS picture states that by rescaling q s=Lbs and Pq Lbs ts Pðs; LÞ, the data of different system size L must collapse onto the same universal curve. It is presented in Fig. 5 that we
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(a)
(b) Fig. 4. The plot of exponent ss ðqÞ in the two-dimensional model of (a) dissipative avalanches and (b) nondissipative avalanches. The slopes of the linear portion are bs ¼ 1:754 0:004 in (a) and bs ¼ 1:545 0:005 in (b). The extrapolation to the horizontal axis yields the exponents ts ¼ 1:225 0:001 in (a) and ts ¼ 1:647 0:001 in (b).
obtain very good data collapse for dissipative and nondissipative avalanches. The good data collapse of the curves confirms that the size distributions of dissipative and nondissipative avalanches fulfill the finite-size scaling in the SP-Z model, respectively. 3. Conclusion We study the dissipative and nondissipative avalanches separately in the SP-Z model. To study the necessity of the logarithmic correction to scaling, we presented the plot of cutoff function versus In s for both two kinds of avalanches, which indicates that the logarithmic correction does not avail to reduce the fluctuation of the
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(a)
(b) Fig. 5. The plot of curves of the data collapse analysis for (a) size distributions of dissipative avalanches (the values of critical exponents are ts ¼ 1:225 and bs ¼ 1:754) and (b) nondissipative avalanches (the values of critical exponents are ts ¼ 1:647 and bs ¼ 1:545).
cutoff function. It can be concluded that in the SP-Z model, the avalanche distributions obey simple power laws in the infinite-size limit, unlike in the Abelian Manna model. We performed the moment analysis of the avalanche sizes and determined the critical exponents ts and bs for dissipative and nondissipative avalanches respectively, which are much different from those values in the Abelian Manna model. It is shown by data collapse analysis that the FSS assumption is satisfied in this model. Our result is that the stochastic Zhang model and the Abelian Manna model belong to different universality classes. Acknowledgements This work is supported by the National Nature Science Foundation of China through Grant No. 50272022.
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References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]
P. Bak, C. Tang, K. Wiesenfeld, Phys. Rev. Lett. 59 (1987) 381. D. Hughes, M. Paczuki, Phys. Rev. Lett. 88 (2002) 054302. D.M. Zhang, G.J. Pan, H.Z. Sun, Y.P. Yin, R. Li, Phys. Lett. A 337 (2005) 285. B. Drossel, Phys. Rev. E 61 (2000) R2168. R. Dickman, J.M.M. Campelo, Phys. Rev. E 67 (2003) 066111. Y.C. Zhang, Phys. Rev. Lett. 63 (1989) 470. S. Lubeck, Phys. Rev. E 56 (1997) 1590. A. Giacometti, Phys. Rev. E 58 (1998) 247. R. Pastor-Satorras, A. Vespignani, Eur. Phys. J. B 18 (2000) 197. M. De Menech, A.L. Stella, C. Tebaldi, Phys. Rev. E 58 (1998) 2677. A. Chessa, H.E. Stanley, A. Vespignani, S. Zapperi, Phys. Rev. E 59 (1999) 12. A. Vazquez, O. Sotolongo-Costa, e-print cond-mat/9811414,1999, unpublished. A. Chessa, A. Vespignani, S. Zapperi, e-print cond-mat/9811365, 1998, unpublished. S. Lubeck, Phys. Rev. E 61 (2000) 204.
Physica A 363 (2006) 299–306 www.elsevier.com/locate/physa
Corrections to scaling and probability distribution of avalanches for the stochastic Zhang sandpile model Duan-Ming Zhanga, Yan-Ping Yina,, Gui-Jun Pana,b, Fan Suna a
Department of Physics, Huazhong University of Science and Technology, Wuhan 430074, China b Faculty of Physics and Electronic Technology , Hubei University, Whan 430062, China Received 12 May 2005 Available online 9 September 2005
Abstract We study the distributions of dissipative and nondissipative avalanches separately in the stochastic Zhang (SP-Z) sandpile in two dimension. We find that dissipative and nondissipative avalanches obey simple power laws and do not have the logarithmic correction, while the avalanche distributions in the Abelian Manna model should include a logarithmic correction. We use the moment analysis to determine the numerical critical exponents of dissipative and nondissipative avalanches, respectively, and find that they are different from the corresponding values in the Abelian Manna model. All these indicate that the stochastic Zhang model and the Abelian Manna model belong to distinct universality classes, which imply that the Abelian symmetry breaking changes the scaling behavior of the avalanches in the case of the stochastic sandpile model. r 2005 Elsevier B.V. All rights reserved. Keywords: Self-organized criticality; Sandpile model; Dissipative; Nondissipative; Moment analysis
1. Introduction Since its introduction in 1987, the sandpile model has been considered as the prototype of a self-organized critical (SOC) system [1]. The identification of the universality classes is the most important problem in the field of SOC. It still remains an unclear problem that whether the Abelian symmetry breaking changes the scaling behavior of the avalanches in the stochastic sandpile model. Recent studies showed that the nonAbelian stochastic directed sandpile model (NA-SDM) and the Abelian stochastic directed sandpile model (A-SDM) belong to the same universality class under the condition of parallel update [2]. However, our previous work on directed stochastic models presented that the NA-SDM and the A-SDM do not belong to the same universality class [3]. It seems to imply that the Abelian symmetry breaking changes the scaling behavior of the avalanches in the directed stochastic sandpile model. In the case of stochastic sandpile model, does the Abelian symmetry breaking change the scaling behavior of the avalanches? In order to get a deeper
Corresponding author. Tel./fax: +86 027 87543882.
E-mail address: Aegeanfi[email protected] (Y.-P. Yin). 0378-4371/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2005.08.032
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understanding of this problem, we reconsider the distributions of dissipative and nondissipative avalanches in the stochastic Zhang model. Drossel performed that in two-dimensional BTW sandpile, dissipative avalanche distributions follow clean power laws [4]. Distributions of nondissipative avalanches must also follow power laws in the infinite-size limit, but are subject to much stronger corrections to scaling. Dickman and Campelo showed that in one and two-dimensional (2D) Manna’s sandpile, avalanche distributions in general do not follow simple power laws, but rather include a logarithmic correction [5]. Thus , it would be interesting to find whether the logarithmic correction appears in other models exhibiting SOC. After the Zhang model was introduced by Zhang [6] in 1989, many numerical works were made on this model [7–9]. In this paper we study the dissipative and nondissipative avalanches separately in the twodimensional stochastic Zhang (SP-Z) model. We find that the distributions of dissipative and nondissipative avalanches of the stochastic Zhang model exhibit a standard FSS behavior, and follow pure power laws that they do not have the logarithmic correction to scaling. It seems to imply that the logarithmic correction presented by Ref. [5] is dependent on the inherent mechanism of the Abelian Manna model. We also determine the critical exponents of dissipative and nondissipative avalanches by using the moment analysis [10–14]. Our principal result is that the scaling behavior of dissipative and nondissipative avalanches in the stochastic Zhang sandpile is different from that in the Abelian Manna model. It indicates that the stochastic Zhang model and the Abelian Manna model belong to different universality classes. 2. Model and simulations The SP-Z model [9] is defined on a D-dimensional square lattice of linear size L in which we assign a nonnegative continuous variable E i called ‘‘energy’’ on each site. At each time step, an amount of energy d is added to a randomly chosen site j according to E j ! E j þ d. The quantity d is a random variable uniformly distributed in ½0; dmax . In our simulations we consider the fixed value dmax ¼ 0:25. When a site acquires an energy larger than or equal to 1 ðE i X1Þ, it becomes active and topples. An active site i relaxes losing all its energy, E i ! 0 , which is randomly redistributed among its nearest P neighbors. In the practical implementation of this rule, we draw four random numbers i0 , 0pi0 p1, with i0 i0 ¼ 1 and update the nearest neighbors i0 by E i0 ! E i0 þ i0 E i . We study the parallel updating that all active sites release their energy simultaneously. We analyze the dissipative and nondissipative avalanches separately. Dissipative avalanches are those in which some energy leaves the system, while nondissipative avalanches are those in which no energy leaves the system. We reported three systems of different sizes for L ¼ 160, 320, and 640 in two dimension. Our results are based on samples of about 107 avalanches for all the systems. We present the avalanche distributions Ps ðsÞ of avalanche sizes s while ‘‘size’’ means the number of topplings in an avalanche, as shown in Fig. 1. The size distributions of dissipative avalanches are presented in Fig. 1(a), while the size distributions of nondissipative avalanches are shown in Fig. 1(b). The morphology of these two kinds of avalanche distributions generally includes a plateau-like region for small s, a rapidly decaying portion for large s, and a power-law-like interval between these limiting regimes. The power-law interval is increased with the size of the system. The probability distribution in the second and third portion generally follows: Ps ðsÞ ¼ sts f s ðs=sc Þ ,
(1)
where f s is a cutoff function decaying rapidly for large argument. The cutoff function f s must take a constant value for s sc . And sc is the cutoff characteristic size which diverges as sc Lbs when the system size L goes to infinity. We compare distributions of dissipative avalanches with those of nondissipative avalanches for system of L ¼ 640 in Fig. 2. It is illustrated that the two kinds of avalanche distributions obey the power laws but have different slopes of the power-law-like interval, which indicates that they present two distinct scaling behaviors. In other words, whether the avalanche is dissipative or nondissipative does not alter the power-law distributions of avalanches, but it changes the scaling behavior of avalanches. Dickman and Campelo showed that distributions of dissipative and nondissipative avalanches in Abelian Manna’s sandpile do not follow pure power laws [5], but include a logarithmic correction to scaling
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(a)
(b) Fig. 1. The probability distribution plot of (a) dissipative avalanches and (b) nondissipative avalanches in the two-dimensional model for L ¼ 160, 320, and 640.
presented as Ps ðsÞ ¼ sts ðIn sÞg f s ðs=sc Þ .
(2)
The variance of f s fluctuates around a constant value with the logarithmic correction, while it has a strong fluctuation with a pure power-law fit. We study the stochastic Zhang model using the method of Dickman and Campelo. First, we take a preliminary estimate of the fitting linear portion ½x0 ; x1 (here x In s ) in the log–log plot of probability distribution of avalanches (Fig. 1). Second, we adjust parameters t and g to minimize the variance of f s ¼ st Ps ðsÞ=ðIn sÞg . f s should take a constant value ideally. Third, we check for any systematic practice and refine the fitting portion accordingly in practice. We use the same x0 for different system sizes, and increase x1 linearly with In L.
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Fig. 2. The probability plot of dissipative and nondissipative avalanches in the two-dimensional model for L ¼ 640.
Our results are presented in Fig. 3 which shows the plot of f s ¼ st Ps ðsÞ=ðIn sÞg versus In s using data from L ¼ 640. For dissipative avalanches in Fig. 3(a), it is shown that the fluctuating spectrum of f s is about 0.55 with pure power-law fit, while the fluctuating spectrum of f s is also about 0.55 at least using t ¼ 1:580 and g ¼ 0:380. It implies that the logarithmic correction is not necessary for dissipative avalanches. For nondissipative avalanches in Fig. 3(b), it is illustrated that the fluctuating spectrum of f s is about 1.05 with pure power-law fit, while the fluctuating spectrum of f s is approximately 0.95 at least using t ¼ 1:490 and g ¼ 0:500, decreased about 10%. It can be considered that the logarithmic correction does not avail to reduce the variance of f s . So we come to the conclusion that in the stochastic Zhang model the variance of f s has no reduction with logarithmic correction to scaling. In other words, the logarithmic correction to scaling is of no necessity in the stochastic Zhang model, while it is needed in the Abelian Manna model. It can be concluded that for the stochastic Zhang model, the distributions of dissipative and nondissipative avalanches follow simple power laws in the infinite-size limit. Under the finite size scaling (FSS) assumption of Eq. (1), the set of exponents fts ; bs g characterize the critical behavior and completely defines the universality class of the model. Then we analyzed the SP-Z model by determining their critical exponents with the moment analysis method [10–14]. The q-moment of s on a lattice of L is defined as Z hsq iL ¼ sq PðsÞ ds . (3) If FSS hypothesis (Eq. (1)) is valid, at least in the asymptotic limit (s ! 1), we can transform z ¼ s=Lbs and obtain Z (4) hsq iL ¼ Lbs ðqþ1ts Þ sqþts f s ðzÞ dzLbs ðqþ1ts Þ . More generally, we have hsq iL Lss ðqÞ , where the behavior of the exponent ss ðqÞ can be obtained via a regression analysis of Inhsq iL as a function of In L, or via the logarithmic derivative qInhsq iL . @ In L Comparing with Eq. (4) we have ss ðqÞ ¼
(5)
ss ðqÞ ¼ bs ðq þ 1 ts Þ .
(6)
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(a)
(b) Fig. 3. Plot of f s ¼ st Ps ðsÞ=ðIn sÞg versus In s for pure power-law fit and fit with logarithmic correction of (a) dissipative avalanches and (b) nondissipative avalanches using data from L ¼ 640.
So we can determine the cutoff exponent bs from the slope of ss ðqÞ versus q, and the avalanche exponent ts from an extrapolation to the horizontal axis (Eq. (6)). In Fig. 4 we have plotted the exponent ss ðqÞ of dissipative and nondissipative avalanches. By measuring the slope of the linear part of the exponent ss ðqÞ, we obtain the cutoff exponent bs ¼ 1:754 0:004 for dissipative avalanche and bs ¼ 1:545 0:005 for nondissipative avalanche. The extrapolation to ss ðqÞ ¼ 0 yields the exponent ts ¼ 1:225 0:001 for dissipative avalanches and ts ¼ 1:647 0:001 for nondissipative avalanches. They are much different from corresponding values in the Abelian Manna model, which are ts ¼ 0:98ð2Þ for dissipative avalanches and ts ¼ 1:30ð1Þ for nondissipative avalanches. It is indicated that the SP-Z model and the Abelian Manna model belong to distinct universality classes. The finite size scaling of Eq. (1) has to be verified and must be consistent with the numerical exponents obtained from the moment analysis. The FSS picture states that by rescaling q s=Lbs and Pq Lbs ts Pðs; LÞ, the data of different system size L must collapse onto the same universal curve. It is presented in Fig. 5 that we
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(a)
(b) Fig. 4. The plot of exponent ss ðqÞ in the two-dimensional model of (a) dissipative avalanches and (b) nondissipative avalanches. The slopes of the linear portion are bs ¼ 1:754 0:004 in (a) and bs ¼ 1:545 0:005 in (b). The extrapolation to the horizontal axis yields the exponents ts ¼ 1:225 0:001 in (a) and ts ¼ 1:647 0:001 in (b).
obtain very good data collapse for dissipative and nondissipative avalanches. The good data collapse of the curves confirms that the size distributions of dissipative and nondissipative avalanches fulfill the finite-size scaling in the SP-Z model, respectively. 3. Conclusion We study the dissipative and nondissipative avalanches separately in the SP-Z model. To study the necessity of the logarithmic correction to scaling, we presented the plot of cutoff function versus In s for both two kinds of avalanches, which indicates that the logarithmic correction does not avail to reduce the fluctuation of the
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(a)
(b) Fig. 5. The plot of curves of the data collapse analysis for (a) size distributions of dissipative avalanches (the values of critical exponents are ts ¼ 1:225 and bs ¼ 1:754) and (b) nondissipative avalanches (the values of critical exponents are ts ¼ 1:647 and bs ¼ 1:545).
cutoff function. It can be concluded that in the SP-Z model, the avalanche distributions obey simple power laws in the infinite-size limit, unlike in the Abelian Manna model. We performed the moment analysis of the avalanche sizes and determined the critical exponents ts and bs for dissipative and nondissipative avalanches respectively, which are much different from those values in the Abelian Manna model. It is shown by data collapse analysis that the FSS assumption is satisfied in this model. Our result is that the stochastic Zhang model and the Abelian Manna model belong to different universality classes. Acknowledgements This work is supported by the National Nature Science Foundation of China through Grant No. 50272022.
ARTICLE IN PRESS D.-M. Zhang et al. / Physica A 363 (2006) 299–306
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References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]
P. Bak, C. Tang, K. Wiesenfeld, Phys. Rev. Lett. 59 (1987) 381. D. Hughes, M. Paczuki, Phys. Rev. Lett. 88 (2002) 054302. D.M. Zhang, G.J. Pan, H.Z. Sun, Y.P. Yin, R. Li, Phys. Lett. A 337 (2005) 285. B. Drossel, Phys. Rev. E 61 (2000) R2168. R. Dickman, J.M.M. Campelo, Phys. Rev. E 67 (2003) 066111. Y.C. Zhang, Phys. Rev. Lett. 63 (1989) 470. S. Lubeck, Phys. Rev. E 56 (1997) 1590. A. Giacometti, Phys. Rev. E 58 (1998) 247. R. Pastor-Satorras, A. Vespignani, Eur. Phys. J. B 18 (2000) 197. M. De Menech, A.L. Stella, C. Tebaldi, Phys. Rev. E 58 (1998) 2677. A. Chessa, H.E. Stanley, A. Vespignani, S. Zapperi, Phys. Rev. E 59 (1999) 12. A. Vazquez, O. Sotolongo-Costa, e-print cond-mat/9811414,1999, unpublished. A. Chessa, A. Vespignani, S. Zapperi, e-print cond-mat/9811365, 1998, unpublished. S. Lubeck, Phys. Rev. E 61 (2000) 204.