- Email: [email protected]
Self-organization of magnetic domain patterns
Physica A 185 (1992) 3-10 North-Holland
Self-organization of magnetic domain patterns Per Bak a and Henrik Flyvbjerg b aDepartment of Physics, Brookhaven National Laboratory, Upton, NY 11973, USA bThe Niels Bohr Institute, Blegdamsvej 17, DK-2100 Copenhagen O, Denmark We model the self-organizing dynamics of two-dimensional magnetic domain patterns with magnetic bubble traps. Avalanches of topological rearrangements and domain destruction are simulated numerically. Asymptotic forms for distributions of avalanche sizes and lifetimes are found analytically. They are power laws with exponential damping factors. Experimental results are shown to obey these asymptotic forms, indicating that the experimentally observed self-organized state is sub-critical.
1. Introduction In r e c e n t e x p e r i m e n t s with d o m a i n g r o w t h in m a g n e t i c garnet film it was o b s e r v e d that the system could self-organize into barely stable configurations l~ke the o n e s h o w n in fig. 1 [1-3]. In such a configuration small increments in tile external field trigger avalanche-like processes o f topological rearrangem e n t s and d o m a i n destruction spanning 2 orders o f m a g n i t u d e in size and lifetime, the size being the n u m b e r o f d o m a i n s disappearing in the avalanche. T h e e x p e r i m e n t a l distributions o f avalanche sizes and lifetimes a p p e a r to
tD
i
~5
Fig. 1. Digitized photograph of a magnetic domain pattern. Fig. 8b from ref. [2]. 0378-4371/92/$05.00 ~) 1992 - Elsevier Science Publishers B.V. All rights reserved
4
P. Bak, H. Flyvbjerg / Self-organization of magnetic domain patterns
follow power laws, and it was suggested that the system might have organized itself into a critical state, thus constituting an example of self-organized criticality [4]. We have modelled the dynamical properties of these domain patterns and reanalyzed the experimental results of ref. [3] in terms of our model. A brief description of the model and the analysis of experimental data is given below. A more detailed account will appear elsewhere [5].
2. Kinematics and dynamics With reference to fig. 1 and similar patterns, the topological number n or topology of a domain is the number of walls forming its boundary. We only consider domain patterns with positive domain wall tension. Then the kinematics and dynamics of a domain pattern closely resemble that of an ideal soap froth in 2 dimensions; see ref. [5] for details. For a certain range of external field strength, however, Westervelt et al. found that domains with n = 5 may shrink only to a certain small size, and then remain static [1-3]. These pentagonal bubble traps do not exist in soap froths, where all domains with topology n ~< 5 shrink and eventually vanish. Since pentagons do not necessarily disappear from the pattern, the pattern may end up in a static configuration, from which all domains of 4 or less neighbours have vanished, and all domains with 5 neighbours have become traps. Such a configuration is static, because no domains can shrink, hence no domains can grow, due to conservation of area. Individual avalanches of domain destruction and topological rearrangements are separated in time by these static configurations. In order to model the dynamics of these avalanches, we make three assumptions, observe a topological conservation law, and make an approximation:
Assumption 1. Any domain with topology n < 5 shrinks, and disappears from the pattern of domains, when its area vanishes. Assumption 2. Any domain with n = 5 shrinks until its area reaches trap size, w h e r e u p o n it remains in the pattern with static area. Assumption 3. Except that two or more bubble traps cannot be neighbours; such a configuration is unstable, and if it is created by the dynamics, the neighbour traps disappear together from the pattern. Conservation law. When a domain with n neighbours vanishes, its topological deficit 6 - n is transferred to its neighbours, so that the average number of neighbours to a domain remains 6. See fig. 2.
P. Bak, H. Flyvbjerg / Self-organization of magnetic domain patterns
(a)
5
(t~)
Fig. 2. A domain with n neighbours disappears, and its neighbours lose a total of 6 - n neighbours. (a) n = 3, (b) n = 5. Notice that two neighbours each lose a neighbour, while one gains a neighbour.
The random neighbour approximation. We neglect neighbour correlations. We keep track of individual domains, and whenever one of them disappears and its neighbours as a consequence should have their topologies changed, we temporarily appoint randomly chosen domains to be those neighbours, and change their topology. A n y domain is appointed neighbour with probability proportional to its own n u m b e r of neighbours. The first two assumptions hold for curvature driven domain growth in 2D, except for the possibility that pentagons can stop shrinking. We expect the third assumption to catch essential features of the actual mechanism of destabilization of pentagonal bubble traps described in refs. [1-3]. It is certainly in accord with photos in refs. [1-3], none of which have pentagonal bubble traps next to each other. The topological conservation law is a consequence of having only three-vertices in the domain boundary network. The random neighbour approximation is motivated by the experimental observation that correlations between nearest neighbour domains are small, and effectively vanish for more distant neighbours. It has led to quite accurate results for curvature driven domain growth in 2 dimensions [6].
3. Simulation We have simulated the dynamics described in the previous section. We considered whether this dynamics drives the system towards a definite distribution of topologies, i.e. will the system self-organize, and, if so, to which type of state? Will the avalanches remain limited in size in the limit of an infinite ensemble of domains, i.e. is the stationary state sub-critical? Or will, possibly after a transient time, avalanches as large as the ensemble occur for any ensemble size, i.e. is the self-organized state critical [4]? We found that after an initial transient time the system has self-organized to a steady state with avalanches causing negligible fluctuations in the limit of
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P. Bak, H. Flyvbjerg / Self-organizationof magnetic domain patterns
7
infinite ensemble. The sub-critical nature of the self-organized state is illustrated by simulation results for the distribution of avalanche sizes, D(s), shown in fig. 3a. The data shown in this figure were produced by equilibrating an initial 106 domains through 106 avalanches, and binning the sizes of the following 106 avalanches to obtain D(s). In the figure, we see that for 1 0 < s < 1 0 0 , D(s) closely follows a straight line with slope - 3 / 2 , i.e. D(s) ~ s -3/2. For large values of s, however, D(s) falls off faster than this power law. The following section describes how.
4. Analytical results The random neighbour approximation permits us to write down a recursion relation for D(s). A calculation based on this recursion relation reveals that
D(s)ocs-3/2exp(-S/So)
for s----~~ ;
(1)
see ref. [5] for details. In eq. (1) the power 3/2 on s is a fixed number equal to the mean field exponent for D(s) for self-organized critical sand-pile models [7, 8]. This equality is not so surprising, since the mean field description of sand-pile models is a random neighbour approximation to leading order in 1/d, d being the dimension of space. In eq. (1) we also notice the exponential damping factor exp(-S/So). Only for So = ~ is D(s) without a characteristic scale and the self-organized state critical. While the data plotted in fig. 3a may appear to obey a power law distribution, the same data plotted in fig. 3b a s s3/2D(s) against s on semi-log paper clearly exhibit the exponential damping factor. The straight line passing through the data points is the exponential factor in eq. (1). The value for s o was determined in the simulation. The overall amplitude in eq. (1) depends on the detailed properties of the branching process, and was fitted to the data points in fig. 3b. The branching process is critical, if s o = ~, which is the case, when the average number ( b ) of shrinking pentagons created by one shrinking pentagon reaching trap size, is exactly equal to one. Then the process just goes on, with no inherent scale in it. If ( b ) > 1 the average number of shrinking pentagons increases with time, and the process is super-critical and run-away. If ( b ) < 1, the process is sub-critical and terminates after a finite number of branchings. Simulations gave the result ( b ) = 0.926 . . . . i.e. the process we have simulated is not critical.
(2)
8
P. Bak, H. Flyvbjerg / Self-organization of magnetic domain patterns
Since there is no assumption or mechanism in our model that assures ( b ) = 1, it is maybe not so surprising that it is not critical. On the other hand, self-organized critical systems exist [4], so we are induced to reflect on what it is that makes a model with self-organization critical. In the case considered here, a conservation law for the average number of shrinking pentagons would clearly do the job. But there is no good reason the model should contain such a conservation law. In the limit of an infinite ensemble, the avalanche is described as a random branching process with constant branching ratios [5]. The theory for such processes gives us the asymptotic behaviour for the distribution of lifetimes [9]: D ( T ) oc T -2 e x p ( - T / T o ) .
(3)
5. Comparison with experiment We do not expect our model to be a quantitatively precise dynamical model for the topological avalanches studied by Westervelt et al. Too many simplifications and idealizations were introduced for that to be possible. But we may expect it to reproduce gross features correctly, like the asymptotic forms given in eq. (1) and eq. (3), power laws with exponential damping factors. In particular, we expect the mean field exponents in s -3/2 and t -2 to be reasonable approximations, since they are consequences of the random neighbour approximation, and that approximation gave quantitatively reasonable results in ref. [61. Figs. 4 and 5 show semi-log plots of the experimental data from ref. [3] with the power-factors divided out. The straight lines in these figures are the results of X2-fits of eq. (1) and eq. (3) to the data points shown as solid circles. The separation of the data into asymptotic data (solid circles) and non-asymptotic data (open circles) was done by varying the point of separation and choosing the separation that maximized the x2-backing. Clearly, the x2-backings obtained for these fits are unrealistically large. We can reduce them by a factor two, and still fit the data with a x2-backing larger than 70%. Comparing the experimental value s o = 29 shown in fig. 4 with our model's value s o = 475, the difference, a factor 16, is surprisingly large in view of our positive experience with the random neighbour approximation. However, as with all approximations, some quantities are approximated better than others, and analytical expressions for s o show that s o is particularly sensitive to errors on ( b ) ; see ref. [5]. A 23% reduction in the relative number of pentagons in the ensemble suffices to reduce s o from 475 to its experimental value 29.
P. Bak, H. Flyvbjerg / Self-organization of magnetic domain patterns
9
0.1
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60
70
80
90
100
110
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Fig. 4. Experimental results from ref. [3] for s3/ZD(s) versus s. T h e straight line is a least-squares fit of the functional form in eq. (1) to the experimental data dotted as filled circles. T h e x2-backing for the fit is 99.8%. It has s 0 - - 2 9 , with s o defined in eq. (1).
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100 120 140 160 1 0 0 2 0 0
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P. Bak, H. Flyvbjerg / Self-organization of magnetic domain patterns
10
6. Discussion and conclusions We have considered the possibility of changing the model presented above in order to reproduce experimental results better. What we need in order to decrease our values for s o and To, is a reduction in the probability that a pentagon shrinking to trap size creates new shrinking pentagons. This we cannot obtain within the random neighbour approximation, since it fixes the relative probabilities with which neighbour domains are chosen. The correlations between topologies of neighbour domains that we have neglected, are known to be negative: domains with lower topological numbers have, on the average, neighbours with higher topological numbers, and vice versa. So the probability that a neighbour to a shrinking pentagon is a hexagon or a pentagonal trap is really smaller than we have assumed with our approximation. This is what we need: it causes less domains to shrink and vanish, hence reduces s o. Irrespective of the details of our model, the sloping straight lines fitting the experimental data in figs. 4 and 5 clearly demonstrate that the data were taken in a sub-critical state.
Acknowledgements We thank Ken Babcock and Robert Westervelt for discussions of their experimental results. PB's research was supported by the Division of Materials Science, US D O E , under contract DE-AC02-76CH00016, HF's research by the Danish Natural Science Research Council under contract 11-8664. H F thanks the Physics D e p a r t m e n t at BNL for hospitality while this work was commenced.
References [1] [2] [3] [4] [5] [6] [7] [8] [9]
K.L. Babcock and R.M. Westervelt, Phys. Rev. A 40 (1989) 2022. K.L. Babcock, R. Seshadri and R.M. Westervelt, Phys. Rev. A 41 (1990) 1952. K.L. Babcock and R.M. Westervelt, Phys. Rev. Lett. 64 (1990) 2168. P. Bak, C. Tang and K. Wiesenfeld, Phys. Rev. Lett. 59 (1987) 381; Phys. Rev. A 38 (1988) 364. P. Bak and H. Flyvbjerg, Phys. Rev. A 45 (1992) 2192. H. Flyvbjerg and C. Jeppesen, Phys. Scr. T 38 (1991) 49. P. AlstrOm, Phys. Rev. A 38 (1988) 4905. D. Dhar and S.N. Majumdar, J. Phys. A 23 (1990) 4333. T.E. Harris, The Theory of Branching Processes (Springer, Berlin, 1963).
Self-organization of magnetic domain patterns Per Bak a and Henrik Flyvbjerg b aDepartment of Physics, Brookhaven National Laboratory, Upton, NY 11973, USA bThe Niels Bohr Institute, Blegdamsvej 17, DK-2100 Copenhagen O, Denmark We model the self-organizing dynamics of two-dimensional magnetic domain patterns with magnetic bubble traps. Avalanches of topological rearrangements and domain destruction are simulated numerically. Asymptotic forms for distributions of avalanche sizes and lifetimes are found analytically. They are power laws with exponential damping factors. Experimental results are shown to obey these asymptotic forms, indicating that the experimentally observed self-organized state is sub-critical.
1. Introduction In r e c e n t e x p e r i m e n t s with d o m a i n g r o w t h in m a g n e t i c garnet film it was o b s e r v e d that the system could self-organize into barely stable configurations l~ke the o n e s h o w n in fig. 1 [1-3]. In such a configuration small increments in tile external field trigger avalanche-like processes o f topological rearrangem e n t s and d o m a i n destruction spanning 2 orders o f m a g n i t u d e in size and lifetime, the size being the n u m b e r o f d o m a i n s disappearing in the avalanche. T h e e x p e r i m e n t a l distributions o f avalanche sizes and lifetimes a p p e a r to
tD
i
~5
Fig. 1. Digitized photograph of a magnetic domain pattern. Fig. 8b from ref. [2]. 0378-4371/92/$05.00 ~) 1992 - Elsevier Science Publishers B.V. All rights reserved
4
P. Bak, H. Flyvbjerg / Self-organization of magnetic domain patterns
follow power laws, and it was suggested that the system might have organized itself into a critical state, thus constituting an example of self-organized criticality [4]. We have modelled the dynamical properties of these domain patterns and reanalyzed the experimental results of ref. [3] in terms of our model. A brief description of the model and the analysis of experimental data is given below. A more detailed account will appear elsewhere [5].
2. Kinematics and dynamics With reference to fig. 1 and similar patterns, the topological number n or topology of a domain is the number of walls forming its boundary. We only consider domain patterns with positive domain wall tension. Then the kinematics and dynamics of a domain pattern closely resemble that of an ideal soap froth in 2 dimensions; see ref. [5] for details. For a certain range of external field strength, however, Westervelt et al. found that domains with n = 5 may shrink only to a certain small size, and then remain static [1-3]. These pentagonal bubble traps do not exist in soap froths, where all domains with topology n ~< 5 shrink and eventually vanish. Since pentagons do not necessarily disappear from the pattern, the pattern may end up in a static configuration, from which all domains of 4 or less neighbours have vanished, and all domains with 5 neighbours have become traps. Such a configuration is static, because no domains can shrink, hence no domains can grow, due to conservation of area. Individual avalanches of domain destruction and topological rearrangements are separated in time by these static configurations. In order to model the dynamics of these avalanches, we make three assumptions, observe a topological conservation law, and make an approximation:
Assumption 1. Any domain with topology n < 5 shrinks, and disappears from the pattern of domains, when its area vanishes. Assumption 2. Any domain with n = 5 shrinks until its area reaches trap size, w h e r e u p o n it remains in the pattern with static area. Assumption 3. Except that two or more bubble traps cannot be neighbours; such a configuration is unstable, and if it is created by the dynamics, the neighbour traps disappear together from the pattern. Conservation law. When a domain with n neighbours vanishes, its topological deficit 6 - n is transferred to its neighbours, so that the average number of neighbours to a domain remains 6. See fig. 2.
P. Bak, H. Flyvbjerg / Self-organization of magnetic domain patterns
(a)
5
(t~)
Fig. 2. A domain with n neighbours disappears, and its neighbours lose a total of 6 - n neighbours. (a) n = 3, (b) n = 5. Notice that two neighbours each lose a neighbour, while one gains a neighbour.
The random neighbour approximation. We neglect neighbour correlations. We keep track of individual domains, and whenever one of them disappears and its neighbours as a consequence should have their topologies changed, we temporarily appoint randomly chosen domains to be those neighbours, and change their topology. A n y domain is appointed neighbour with probability proportional to its own n u m b e r of neighbours. The first two assumptions hold for curvature driven domain growth in 2D, except for the possibility that pentagons can stop shrinking. We expect the third assumption to catch essential features of the actual mechanism of destabilization of pentagonal bubble traps described in refs. [1-3]. It is certainly in accord with photos in refs. [1-3], none of which have pentagonal bubble traps next to each other. The topological conservation law is a consequence of having only three-vertices in the domain boundary network. The random neighbour approximation is motivated by the experimental observation that correlations between nearest neighbour domains are small, and effectively vanish for more distant neighbours. It has led to quite accurate results for curvature driven domain growth in 2 dimensions [6].
3. Simulation We have simulated the dynamics described in the previous section. We considered whether this dynamics drives the system towards a definite distribution of topologies, i.e. will the system self-organize, and, if so, to which type of state? Will the avalanches remain limited in size in the limit of an infinite ensemble of domains, i.e. is the stationary state sub-critical? Or will, possibly after a transient time, avalanches as large as the ensemble occur for any ensemble size, i.e. is the self-organized state critical [4]? We found that after an initial transient time the system has self-organized to a steady state with avalanches causing negligible fluctuations in the limit of
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P. Bak, H. Flyvbjerg / Self-organizationof magnetic domain patterns
7
infinite ensemble. The sub-critical nature of the self-organized state is illustrated by simulation results for the distribution of avalanche sizes, D(s), shown in fig. 3a. The data shown in this figure were produced by equilibrating an initial 106 domains through 106 avalanches, and binning the sizes of the following 106 avalanches to obtain D(s). In the figure, we see that for 1 0 < s < 1 0 0 , D(s) closely follows a straight line with slope - 3 / 2 , i.e. D(s) ~ s -3/2. For large values of s, however, D(s) falls off faster than this power law. The following section describes how.
4. Analytical results The random neighbour approximation permits us to write down a recursion relation for D(s). A calculation based on this recursion relation reveals that
D(s)ocs-3/2exp(-S/So)
for s----~~ ;
(1)
see ref. [5] for details. In eq. (1) the power 3/2 on s is a fixed number equal to the mean field exponent for D(s) for self-organized critical sand-pile models [7, 8]. This equality is not so surprising, since the mean field description of sand-pile models is a random neighbour approximation to leading order in 1/d, d being the dimension of space. In eq. (1) we also notice the exponential damping factor exp(-S/So). Only for So = ~ is D(s) without a characteristic scale and the self-organized state critical. While the data plotted in fig. 3a may appear to obey a power law distribution, the same data plotted in fig. 3b a s s3/2D(s) against s on semi-log paper clearly exhibit the exponential damping factor. The straight line passing through the data points is the exponential factor in eq. (1). The value for s o was determined in the simulation. The overall amplitude in eq. (1) depends on the detailed properties of the branching process, and was fitted to the data points in fig. 3b. The branching process is critical, if s o = ~, which is the case, when the average number ( b ) of shrinking pentagons created by one shrinking pentagon reaching trap size, is exactly equal to one. Then the process just goes on, with no inherent scale in it. If ( b ) > 1 the average number of shrinking pentagons increases with time, and the process is super-critical and run-away. If ( b ) < 1, the process is sub-critical and terminates after a finite number of branchings. Simulations gave the result ( b ) = 0.926 . . . . i.e. the process we have simulated is not critical.
(2)
8
P. Bak, H. Flyvbjerg / Self-organization of magnetic domain patterns
Since there is no assumption or mechanism in our model that assures ( b ) = 1, it is maybe not so surprising that it is not critical. On the other hand, self-organized critical systems exist [4], so we are induced to reflect on what it is that makes a model with self-organization critical. In the case considered here, a conservation law for the average number of shrinking pentagons would clearly do the job. But there is no good reason the model should contain such a conservation law. In the limit of an infinite ensemble, the avalanche is described as a random branching process with constant branching ratios [5]. The theory for such processes gives us the asymptotic behaviour for the distribution of lifetimes [9]: D ( T ) oc T -2 e x p ( - T / T o ) .
(3)
5. Comparison with experiment We do not expect our model to be a quantitatively precise dynamical model for the topological avalanches studied by Westervelt et al. Too many simplifications and idealizations were introduced for that to be possible. But we may expect it to reproduce gross features correctly, like the asymptotic forms given in eq. (1) and eq. (3), power laws with exponential damping factors. In particular, we expect the mean field exponents in s -3/2 and t -2 to be reasonable approximations, since they are consequences of the random neighbour approximation, and that approximation gave quantitatively reasonable results in ref. [61. Figs. 4 and 5 show semi-log plots of the experimental data from ref. [3] with the power-factors divided out. The straight lines in these figures are the results of X2-fits of eq. (1) and eq. (3) to the data points shown as solid circles. The separation of the data into asymptotic data (solid circles) and non-asymptotic data (open circles) was done by varying the point of separation and choosing the separation that maximized the x2-backing. Clearly, the x2-backings obtained for these fits are unrealistically large. We can reduce them by a factor two, and still fit the data with a x2-backing larger than 70%. Comparing the experimental value s o = 29 shown in fig. 4 with our model's value s o = 475, the difference, a factor 16, is surprisingly large in view of our positive experience with the random neighbour approximation. However, as with all approximations, some quantities are approximated better than others, and analytical expressions for s o show that s o is particularly sensitive to errors on ( b ) ; see ref. [5]. A 23% reduction in the relative number of pentagons in the ensemble suffices to reduce s o from 475 to its experimental value 29.
P. Bak, H. Flyvbjerg / Self-organization of magnetic domain patterns
9
0.1
0.01 I 0
I
I
1
10
20
30
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I 50
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I
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60
70
80
90
100
110
S
Fig. 4. Experimental results from ref. [3] for s3/ZD(s) versus s. T h e straight line is a least-squares fit of the functional form in eq. (1) to the experimental data dotted as filled circles. T h e x2-backing for the fit is 99.8%. It has s 0 - - 2 9 , with s o defined in eq. (1).
10
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0.I
0
I
I
I
I
20
40
60
O0
I
I
i
i
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100 120 140 160 1 0 0 2 0 0
T Fig. 5. Experimental results from ref. [3] for T2D(T) versus T. The straight line is a least-squares fit of the functional form in eq. (3) to the experimental data plotted as filled circles. T h e x2-backing for the fit is 99.7%, with T o = 133 s.
P. Bak, H. Flyvbjerg / Self-organization of magnetic domain patterns
10
6. Discussion and conclusions We have considered the possibility of changing the model presented above in order to reproduce experimental results better. What we need in order to decrease our values for s o and To, is a reduction in the probability that a pentagon shrinking to trap size creates new shrinking pentagons. This we cannot obtain within the random neighbour approximation, since it fixes the relative probabilities with which neighbour domains are chosen. The correlations between topologies of neighbour domains that we have neglected, are known to be negative: domains with lower topological numbers have, on the average, neighbours with higher topological numbers, and vice versa. So the probability that a neighbour to a shrinking pentagon is a hexagon or a pentagonal trap is really smaller than we have assumed with our approximation. This is what we need: it causes less domains to shrink and vanish, hence reduces s o. Irrespective of the details of our model, the sloping straight lines fitting the experimental data in figs. 4 and 5 clearly demonstrate that the data were taken in a sub-critical state.
Acknowledgements We thank Ken Babcock and Robert Westervelt for discussions of their experimental results. PB's research was supported by the Division of Materials Science, US D O E , under contract DE-AC02-76CH00016, HF's research by the Danish Natural Science Research Council under contract 11-8664. H F thanks the Physics D e p a r t m e n t at BNL for hospitality while this work was commenced.
References [1] [2] [3] [4] [5] [6] [7] [8] [9]
K.L. Babcock and R.M. Westervelt, Phys. Rev. A 40 (1989) 2022. K.L. Babcock, R. Seshadri and R.M. Westervelt, Phys. Rev. A 41 (1990) 1952. K.L. Babcock and R.M. Westervelt, Phys. Rev. Lett. 64 (1990) 2168. P. Bak, C. Tang and K. Wiesenfeld, Phys. Rev. Lett. 59 (1987) 381; Phys. Rev. A 38 (1988) 364. P. Bak and H. Flyvbjerg, Phys. Rev. A 45 (1992) 2192. H. Flyvbjerg and C. Jeppesen, Phys. Scr. T 38 (1991) 49. P. AlstrOm, Phys. Rev. A 38 (1988) 4905. D. Dhar and S.N. Majumdar, J. Phys. A 23 (1990) 4333. T.E. Harris, The Theory of Branching Processes (Springer, Berlin, 1963).