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Effects of randomness and spatially dependent relaxation on sandpile models
Physica A 253 (1998) 307–314
E ects of randomness and spatially dependent relaxation on sandpile models Pui-Man Lam a; ∗ , Isiaka Akanbi a , David E. Newman b
a Department b Oak
of Physics, Southern University, Baton Rouge, Louisiana 70813, USA Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA Received 22 November 1997
Abstract We investigate two types of randomness in the relaxation of sandpile models when the slope at some point becomes over critical. In one type of randomness, the number of particles nf , falling to its nearest neighbors in the resulting relaxation, is not constant but random, even though an equal number fall in each direction. We nd that this kind of randomness does not change the universality class of the models. Another type of randomness is introduced by having all nf particles to fall in one single direction, but with the direction chosen randomly. We nd that c 1998 Elsevier this type of randomness has a strong e ect on the universality of the models. Science B.V. All rights reserved
1. Introduction The phenomenon of self-organized criticality (SOC) [ 1 –3] is characterized by spontaneous and dynamical generation of spatial and temporal scale invarance in extended nonequilibrium systems. Spatial scaling invariance is characterized by fractal geometry, while temporal scale invariance is manifested by 1=! power spectrum. Most of the models studied so far are cellular automata or sandpile models, which, starting from an arbitrary initial state, evolve automatically into a critical state characterized by power-law correlations in both spatial and temporal scales. In most sandpile models, the slope at a point (x; y) is described by an integer variable z(x; y). If z(x; y) exceeds a critical value zc , it is updated synchronously as follows: z(x; y) → z(x; y) − nf ; z(x ± 1; y) → z(x ± 1; y) + nf =4 ; z(x; y ± 1) → z(x; y ± 1) + nf =4 : ∗
Corresponding author.
c 1998 Elsevier Science B.V. All rights reserved 0378-4371/98/$19.00 Copyright PII S 0 3 7 8 - 4 3 7 1 ( 9 7 ) 0 0 6 7 9 - 1
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Here we have taken a square lattice and nf 6zc is an integer divisible by four. In the original BTW model, nf is a constant at every point of the lattice. In this paper we want to investigate the e ect of having nf random on di erent sites of the lattice. For instance for zc = 8 we can let nf be either 4 or 8 at random. We nd that this kind of randomness does not noticeably a ect the universality of the model. Besides the above updating rule, we also investigate the following updating rule. When z(x; y) exceeds a critical value zc it is updated according to the rule: z(x; y) → z(x; y) − nf (x; y) ; z(x0 ; y0 ) → z(x0 ; y0 ) + nf (x; y) ; where (x0 ; y0 ) is a randomly chosen nearest neighbor of (x; y) and nf (x; y) can be chosen either constant or random. In this case, instead distributing nf grains equally in the four directions, all the nf grains fall in a single, randomly chosen direction. We nd that this type of randomness has a strong e ect on the universality class of the model.
2. BTW model with random nf In this model, when the slope z(x; y) at a point (x; y) exceeds a critical value zc , its value is updated according to: z(x; y) → z(x; y) − nf (x; y) ; z(x ± 1; y) → z(x ± 1; y) + nf (x; y)=4 ; z(x; y ± 1) → z(x; y ± 1) + nf (x; y)=4 : Here we have chosen zc = 8 and nf (x; y) is a random variable whose value can be either 4 or 8. The system is an N × N square lattice with open boundary conditions on all four sides. Sand grains are added one at a time only when the slope at every site is below critical. Once the slope at a single site is above critical, we stop adding sand and wait for the system to relax through avalanching. We resume adding sand when all sites are again below critical. So there is only one avalance, if any, at all times. We repeat this process until a steady state is reached in which the average number of grains added is equal to the average number of grains leaving the system through the boundaries. After that we can start measuring the distributions of the sizes of the avalanche D(S) and the duration of the avalanches D(T ). These distributions should have power law forms over a large range of the variables S and T , of the forms D(S) ∼ S − S ; D(T ) ∼T − T . Notice that our de nition of D(T ) is di erent from that of BTW. In BTW, their D(T ) is weighted by the average response S=T , but our de nition of D(S) is the same as theirs. For BTW S ∼ 1:0 in two dimensions. Fig. 1 shows log–log plots of our results for D(S) vs. S for both the cases nf = 4 and nf taking random values 4 and 8. The data are obtained for an 80 × 80 square
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Fig. 1. log–log plots of the distribution of avalanche size D(S) vs. S for the BTW model on square lattice for constant and random nf , with the same number nf =4 particles falling in each of the four directions during relaxation of over critical sites.
Fig. 2. Same as Fig. 1, but for the distribution of avalanche duration D(T ) vs. T .
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lattice over 10 000 di erent runs. We found that because of the randomness in the model, it is essential to average over many di erent runs. We have also checked that a size of 80 × 80 is large enough for our purpose. It is chosen as the best compromise between large system size and the large number of runs. The case of constant nf is exactly the BTW model. The slope of the curve gives S ∼ 1:0, in agreement with BTW. The slope for the case of random nf is about 1.1, not noticeably higher than in the case of constant nf . Fig. 2 shows log–log plots of our result for D(T ) vs. T for the cases of nf = 2 and nf taking random values 4 and 8. For the case of constant nf , the slope of the curve gives T ∼ 1:0 and for both the case of random nf the slope gives T ∼ 1:1. Since our D(T ) is di erent from that of BTW, we cannot compare our value with theirs. Our results shows that randomness in nf does not noticeably change the universality of the model.
3. DR model with random nf For the DR model we do exactly the same thing as in the previous section, except now when the slope z(x; y)¿zc , nf =2 grains will fall in each of the positive x and y directions only, instead of in the positive and negative x and y directions, as in the BTW model, with nf an even number and zc ¿2nf . Fig. 3 shows log–log plots of our
Fig. 3. log–log plots of the distribution of avalanche size D(S) vs. S for the DR model on square lattice for constant and random nf , with the same number nf =2 particles falling in each of the two positive x and y directions during relaxation of over critical sites.
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Fig. 4. Same as Fig. 3, but for the distribution of avalanche duration D(T ) vs. T .
results for the size distribution D(S) vs. S for both the case nf = 4 and the case when nf can take values 2 and 4 randomly. The data are obtained for an 80 × 80 square lattice over 10 000 di erent runs. We see that in both cases, the slopes are about the same, giving S ∼ 1:4. Fig. 4 shows log–log plots of our results for the distribution of the avalance duration D(T ) vs. T , for both the case nf = 4 and the case when nf can take values 2 and 4 randomly. The slopes of both curves gives the same value T ∼ 1:4. The case nf = constant is the DR model. In this case DR had shown that if P(t) is − P the probability that an avalance has a duration greater than t, then with R ∞ P(t) ∼ t , 1− 1 the exact result P = 2 . Our D(T ) is related to P(t) by P(t) ∼ t D(T ) dT ∼ t T . Therefore, T is related to P by T = 1 + P = 32 , not far from our numerical result of 1.4. Our results therefore show that randomness in nf also does not a ect the DR model.
4. BTW model with random direction of relaxation In this model, whenever the slope z(x; y) at a point (x; y) exceeds a critical value zc , the relaxation rule is such that all nf grains will fall in a single, but randomly chosen direction. We choose zc = 4. The number nf of falling sand grains is either chosen as constant, nf = 4 or allowed to take randomly the values 2 and 4. That is, when z(x; y)
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Fig. 5. log–log plots of the distribution of avalanche size D(S) vs. S for the BTW model on square lattice for constant and random nf , with all nf particles falling in a single but randomly chosen direction during relaxation of over critical sites.
exceeds a critical value zc , the updating rule is: z(x; y) → z(x; y) − nf (x; y) ; z(x0 ; y0 ) → z(x0 ; y0 ) + nf (x; y) ; where (x0 ; y0 ) is a randomly chosen nearest neighbor of (x; y) and nf (x; y) can be either constant or random. We have simulated this model on an 80 × 80 square lattice over 10 000 di erent runs. The log–log plots of D(S) vs. S are shown in Fig. 5 for both cases of nf constant and random. The data are obtained for an 80 × 80 square lattice over 10 000 di erent runs. For nf constant, the slope of the curve gives S ∼ 0:62, very di erent from the BTW value. For random nf , the slope gives S ∼ 1:2, similar to the value for the BTW model with random nf , but with grains falling equally in all four directions. The log–log plots of D(T ) are shown in Fig. 6. The slopes of the curves give T ∼ 0:6 for constant nf and T ∼ 1:2 for random nf . Here we see that even though randomness in the direction of the falling sand has a strong e ect on the universalities of the models, giving exponents very di erent from the BTW model, a combination of both random nf and random direction of relaxation brings the model back to the universality class of BTW.
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Fig. 6. Same as Fig. 5, but for the distribution of avalanche duration D(T ) vs. T .
5. Conclusions We have investigated the e ect of randomness on sandpile models. Two types of randomness have been studied. One type of randomness is in the number nf of falling sand grains due to relaxation when a site becomes over critical. The number nf can be random even though an equal number, nf =4 sand grains fall in each of the four directions on the square lattice. We nd that for this type of randomness, the universality of the models is not changed. More interesting is the case when nf is constant but a random direction is chosen for each relaxation of over critical sites. In this case all nf grains fall in the randomly chosen direction. Here we nd a new universality class in which S ∼ 0:6, very di erent from the BTW value of 1.0. However with the combination of randomness in both nf and in the direction of relaxation, the model reverts back to that of the BTW universality class. The introduction of randomness in nf and in the direction of relaxtion naturally makes our models more realistic towards real sandpiles. We nd that only the special case of constant nf and random direction of relaxation gives rise to new universality classes. However, it is interesting to see that in this case, even microscopic details such as random direction in the relaxation rather than equal distribution in the four directions can have such a strong e ect on the universality of these nonequilibrium models.
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Acknowledgements One of the authors (PML) would like to thank the Fusion Theory Group of the Oak Ridge National Laboratory for hospitality where part of this work was carried out. He would also like to acknowledge support of the Department of Energy grant DE-FG02-97ER25343.A000 and the Louisiana Education Quality Support Fund under contract number LEQSF(95-98)-RD-A-23. References [1] P. Bak, C. Tang, K. Wiesenfeld, Phys. Rev. Lett. 59 (1987) 381. [2] P. Bak, C. Tang, K. Wiesenfeld, Phys. Rev. A 38 (1988) 364. [3] Henrik Jeldtoft Jensen, Self-Organized Criticality, Cambridge Univ. Press, Cambridge, 1998.
E ects of randomness and spatially dependent relaxation on sandpile models Pui-Man Lam a; ∗ , Isiaka Akanbi a , David E. Newman b
a Department b Oak
of Physics, Southern University, Baton Rouge, Louisiana 70813, USA Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA Received 22 November 1997
Abstract We investigate two types of randomness in the relaxation of sandpile models when the slope at some point becomes over critical. In one type of randomness, the number of particles nf , falling to its nearest neighbors in the resulting relaxation, is not constant but random, even though an equal number fall in each direction. We nd that this kind of randomness does not change the universality class of the models. Another type of randomness is introduced by having all nf particles to fall in one single direction, but with the direction chosen randomly. We nd that c 1998 Elsevier this type of randomness has a strong e ect on the universality of the models. Science B.V. All rights reserved
1. Introduction The phenomenon of self-organized criticality (SOC) [ 1 –3] is characterized by spontaneous and dynamical generation of spatial and temporal scale invarance in extended nonequilibrium systems. Spatial scaling invariance is characterized by fractal geometry, while temporal scale invariance is manifested by 1=! power spectrum. Most of the models studied so far are cellular automata or sandpile models, which, starting from an arbitrary initial state, evolve automatically into a critical state characterized by power-law correlations in both spatial and temporal scales. In most sandpile models, the slope at a point (x; y) is described by an integer variable z(x; y). If z(x; y) exceeds a critical value zc , it is updated synchronously as follows: z(x; y) → z(x; y) − nf ; z(x ± 1; y) → z(x ± 1; y) + nf =4 ; z(x; y ± 1) → z(x; y ± 1) + nf =4 : ∗
Corresponding author.
c 1998 Elsevier Science B.V. All rights reserved 0378-4371/98/$19.00 Copyright PII S 0 3 7 8 - 4 3 7 1 ( 9 7 ) 0 0 6 7 9 - 1
308
P.-M. Lam et al. / Physica A 253 (1998) 307–314
Here we have taken a square lattice and nf 6zc is an integer divisible by four. In the original BTW model, nf is a constant at every point of the lattice. In this paper we want to investigate the e ect of having nf random on di erent sites of the lattice. For instance for zc = 8 we can let nf be either 4 or 8 at random. We nd that this kind of randomness does not noticeably a ect the universality of the model. Besides the above updating rule, we also investigate the following updating rule. When z(x; y) exceeds a critical value zc it is updated according to the rule: z(x; y) → z(x; y) − nf (x; y) ; z(x0 ; y0 ) → z(x0 ; y0 ) + nf (x; y) ; where (x0 ; y0 ) is a randomly chosen nearest neighbor of (x; y) and nf (x; y) can be chosen either constant or random. In this case, instead distributing nf grains equally in the four directions, all the nf grains fall in a single, randomly chosen direction. We nd that this type of randomness has a strong e ect on the universality class of the model.
2. BTW model with random nf In this model, when the slope z(x; y) at a point (x; y) exceeds a critical value zc , its value is updated according to: z(x; y) → z(x; y) − nf (x; y) ; z(x ± 1; y) → z(x ± 1; y) + nf (x; y)=4 ; z(x; y ± 1) → z(x; y ± 1) + nf (x; y)=4 : Here we have chosen zc = 8 and nf (x; y) is a random variable whose value can be either 4 or 8. The system is an N × N square lattice with open boundary conditions on all four sides. Sand grains are added one at a time only when the slope at every site is below critical. Once the slope at a single site is above critical, we stop adding sand and wait for the system to relax through avalanching. We resume adding sand when all sites are again below critical. So there is only one avalance, if any, at all times. We repeat this process until a steady state is reached in which the average number of grains added is equal to the average number of grains leaving the system through the boundaries. After that we can start measuring the distributions of the sizes of the avalanche D(S) and the duration of the avalanches D(T ). These distributions should have power law forms over a large range of the variables S and T , of the forms D(S) ∼ S − S ; D(T ) ∼T − T . Notice that our de nition of D(T ) is di erent from that of BTW. In BTW, their D(T ) is weighted by the average response S=T , but our de nition of D(S) is the same as theirs. For BTW S ∼ 1:0 in two dimensions. Fig. 1 shows log–log plots of our results for D(S) vs. S for both the cases nf = 4 and nf taking random values 4 and 8. The data are obtained for an 80 × 80 square
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Fig. 1. log–log plots of the distribution of avalanche size D(S) vs. S for the BTW model on square lattice for constant and random nf , with the same number nf =4 particles falling in each of the four directions during relaxation of over critical sites.
Fig. 2. Same as Fig. 1, but for the distribution of avalanche duration D(T ) vs. T .
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lattice over 10 000 di erent runs. We found that because of the randomness in the model, it is essential to average over many di erent runs. We have also checked that a size of 80 × 80 is large enough for our purpose. It is chosen as the best compromise between large system size and the large number of runs. The case of constant nf is exactly the BTW model. The slope of the curve gives S ∼ 1:0, in agreement with BTW. The slope for the case of random nf is about 1.1, not noticeably higher than in the case of constant nf . Fig. 2 shows log–log plots of our result for D(T ) vs. T for the cases of nf = 2 and nf taking random values 4 and 8. For the case of constant nf , the slope of the curve gives T ∼ 1:0 and for both the case of random nf the slope gives T ∼ 1:1. Since our D(T ) is di erent from that of BTW, we cannot compare our value with theirs. Our results shows that randomness in nf does not noticeably change the universality of the model.
3. DR model with random nf For the DR model we do exactly the same thing as in the previous section, except now when the slope z(x; y)¿zc , nf =2 grains will fall in each of the positive x and y directions only, instead of in the positive and negative x and y directions, as in the BTW model, with nf an even number and zc ¿2nf . Fig. 3 shows log–log plots of our
Fig. 3. log–log plots of the distribution of avalanche size D(S) vs. S for the DR model on square lattice for constant and random nf , with the same number nf =2 particles falling in each of the two positive x and y directions during relaxation of over critical sites.
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Fig. 4. Same as Fig. 3, but for the distribution of avalanche duration D(T ) vs. T .
results for the size distribution D(S) vs. S for both the case nf = 4 and the case when nf can take values 2 and 4 randomly. The data are obtained for an 80 × 80 square lattice over 10 000 di erent runs. We see that in both cases, the slopes are about the same, giving S ∼ 1:4. Fig. 4 shows log–log plots of our results for the distribution of the avalance duration D(T ) vs. T , for both the case nf = 4 and the case when nf can take values 2 and 4 randomly. The slopes of both curves gives the same value T ∼ 1:4. The case nf = constant is the DR model. In this case DR had shown that if P(t) is − P the probability that an avalance has a duration greater than t, then with R ∞ P(t) ∼ t , 1− 1 the exact result P = 2 . Our D(T ) is related to P(t) by P(t) ∼ t D(T ) dT ∼ t T . Therefore, T is related to P by T = 1 + P = 32 , not far from our numerical result of 1.4. Our results therefore show that randomness in nf also does not a ect the DR model.
4. BTW model with random direction of relaxation In this model, whenever the slope z(x; y) at a point (x; y) exceeds a critical value zc , the relaxation rule is such that all nf grains will fall in a single, but randomly chosen direction. We choose zc = 4. The number nf of falling sand grains is either chosen as constant, nf = 4 or allowed to take randomly the values 2 and 4. That is, when z(x; y)
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Fig. 5. log–log plots of the distribution of avalanche size D(S) vs. S for the BTW model on square lattice for constant and random nf , with all nf particles falling in a single but randomly chosen direction during relaxation of over critical sites.
exceeds a critical value zc , the updating rule is: z(x; y) → z(x; y) − nf (x; y) ; z(x0 ; y0 ) → z(x0 ; y0 ) + nf (x; y) ; where (x0 ; y0 ) is a randomly chosen nearest neighbor of (x; y) and nf (x; y) can be either constant or random. We have simulated this model on an 80 × 80 square lattice over 10 000 di erent runs. The log–log plots of D(S) vs. S are shown in Fig. 5 for both cases of nf constant and random. The data are obtained for an 80 × 80 square lattice over 10 000 di erent runs. For nf constant, the slope of the curve gives S ∼ 0:62, very di erent from the BTW value. For random nf , the slope gives S ∼ 1:2, similar to the value for the BTW model with random nf , but with grains falling equally in all four directions. The log–log plots of D(T ) are shown in Fig. 6. The slopes of the curves give T ∼ 0:6 for constant nf and T ∼ 1:2 for random nf . Here we see that even though randomness in the direction of the falling sand has a strong e ect on the universalities of the models, giving exponents very di erent from the BTW model, a combination of both random nf and random direction of relaxation brings the model back to the universality class of BTW.
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Fig. 6. Same as Fig. 5, but for the distribution of avalanche duration D(T ) vs. T .
5. Conclusions We have investigated the e ect of randomness on sandpile models. Two types of randomness have been studied. One type of randomness is in the number nf of falling sand grains due to relaxation when a site becomes over critical. The number nf can be random even though an equal number, nf =4 sand grains fall in each of the four directions on the square lattice. We nd that for this type of randomness, the universality of the models is not changed. More interesting is the case when nf is constant but a random direction is chosen for each relaxation of over critical sites. In this case all nf grains fall in the randomly chosen direction. Here we nd a new universality class in which S ∼ 0:6, very di erent from the BTW value of 1.0. However with the combination of randomness in both nf and in the direction of relaxation, the model reverts back to that of the BTW universality class. The introduction of randomness in nf and in the direction of relaxtion naturally makes our models more realistic towards real sandpiles. We nd that only the special case of constant nf and random direction of relaxation gives rise to new universality classes. However, it is interesting to see that in this case, even microscopic details such as random direction in the relaxation rather than equal distribution in the four directions can have such a strong e ect on the universality of these nonequilibrium models.
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Acknowledgements One of the authors (PML) would like to thank the Fusion Theory Group of the Oak Ridge National Laboratory for hospitality where part of this work was carried out. He would also like to acknowledge support of the Department of Energy grant DE-FG02-97ER25343.A000 and the Louisiana Education Quality Support Fund under contract number LEQSF(95-98)-RD-A-23. References [1] P. Bak, C. Tang, K. Wiesenfeld, Phys. Rev. Lett. 59 (1987) 381. [2] P. Bak, C. Tang, K. Wiesenfeld, Phys. Rev. A 38 (1988) 364. [3] Henrik Jeldtoft Jensen, Self-Organized Criticality, Cambridge Univ. Press, Cambridge, 1998.