In the last ten years a multitude of scientific theories moved from the domain of peer reviewed scientific journals into the arena of pop science. This review is an analysis of Self-organized criticality (SOC), a theory which recently made this move. Although it is not necessarily the case for SOC, a theory can, once in the pop science regime, acquire a level of acceptance and momentum that may or may not be warranted by its actual scientific credibility. This is not surprising since the motivations of publishers in the pop science arena are different from those of scientists involved in the peer review process.
For this and other reasons it is not unusual for the authors (and/or subscribers) of these theories to state how their theory will revolutionize science as we know it. A more skeptical viewpoint might say that while it is possible that one of these theories will make some contribution to a few fields of science, it seems unlikely that all of these pop science theories will revolutionize science as we know it. Following an even more pessimistic line of reasoning, one could argue that these pop science theories are nothing more than additions to a long line of failed revolutions put forward by the popular press.
Regardless of one's level of skepticism, it can be entertaining to investigate the scientific merits and foundations of pop science theories. Furthermore, when such a theory does become useful to a particular field, the rewards of an investigation are obvious. In the case of SOC, however, one is compelled to investigate further for two additional reasons. Firstly, the theory makes sense intuitively. Secondly, the signature (power law distributions, see figure 1) of SOC phenomena are ubiquitous in Nature.
To be fair, SOC is well established in the domain of peer reviewed scientific journals. Since the coining of the phrase (Bak, Tang, and Wiesenfeld, 1987) "more than 2,000 papers have been written on the subject, making" this initial paper "the most cited in physics..." (Bak, 1996).
As championed by theoretical physicist Per Bak, SOC is relevant to a large number of naturally occurring phenomena from avalanches, to prices on the commodities market, to the use of the English language in newspapers. Part II of this review offers an overview of the theory. The overview describes SOC in terms of sand pile dynamics. The sand pile is a vehicle which allows an intuitive understanding of the principles of the theory. Simple computer modeling of SOC is then discussed. The explanation of computer models provides a basis for comprehension how SOC is applied to Earth Science. In the last part of section II laboratory results from SOC sand pile experiments are compared to those predicted by the computer models.
In regard to the Earth Sciences, proponents of the theory see multiple manifestations of SOC. The most commonly cited example of SOC in Geology is the Gutenberg-Richter magnitude frequency relationship. In part III of this review, SOC as a model of seismicity and it's subsequent constraints on earthquake rupture physics is discussed.
Finally, part IV summarizes the key points of the review. Furthermore, it speaks to the merits of SOC and what it has brought or can bring to Seismology.
II. THE THEORY OF SELF-ORGANIZED CRITICALITY
A. An explanation of the theory
Self-organized criticality is hypothesized to link the multitude of complex phenomena observed in Nature to simplistic physical laws and / or one underlying process. It is a theory of the internal interactions of large systems. Specifically, it states that large interactive systems will self-organize into a critical state (one governed by a power law, see figure 1). Once in this state small perturbations result in chain reactions, which can affect any number of elements within the system.
A commonly used, intuitive example calls upon the hypothetical generation of a pile of sand. Imagine sand being added one grain at a time to a sandbox. At first, the grains land harmlessly on the stable slope of a proto-sand pile. As more grains are added the slope of the pile increases. Eventually, the slope locally reaches a critical value such that the addition of one more grain results in an "avalanche". The avalanche fills in empty areas of the sandbox. With the addition of still more grains the sandbox will overflow. Sand is thus added and lost from the system. When the count of grains added equals the count of grains lost (on average) then, according to the theory, the sand pile has self-organized to a critical state.
The addition of one more grain may or may not result in an avalanche. Eventually, however, an added grain will cause an avalanche. The key elements of the theory are as follows:
1. The next avalanche can be of any size, (ranging from a single grain to a catastrophic collapse of the sand pile). Moreover, the size distribution of the avalanches will follow a power law. For example, if one were to count the number of avalanches and the number of grains involved in each avalanche over a 24 hour period, one would find that there was 1 avalanche which involved 1,000 grains, 10 avalanches which involved 100 grains, 100 avalanches which involved 10 grains, and so on (see figure 1).
2. Simple physical laws dictate the interactions of individual grains of sand. The specifics of these laws are not important, however, as the system will robustly self-organize to a critical state for a variety of laws. Highly specific physical laws are not necessary in the generation of the power law distribution.
3. Avalanches are not strictly periodic. In other words, although there may be 10 avalanches involving 100 grains in a 24 hour period, each of these avalanches did not occur every 2.400 hours.
4. The surface of the sand pile will have a fractal dimension.
Of course, the notion of Nature as a sand pile seems excessively simple. Likewise, it seems excessively simple to use the notion of an avalanche to model phenomena as complex as prices on the commodities market and the use of the English language in newspapers.
What makes SOC so intriguing, however, is that it may actually be able to model these natural phenomena. The essential link is that these phenomena maintain power law distributions in what can be considered very noisy conditions, and SOC computer models successfully generate power law distributions for a variety of conditions as well.
B. Models on the computer
The underlying algorithm for SOC models is relatively simple to understand. In essence, the algorithm keeps track of numbers associated with points on a grid. Numbers on the grid can increase, decrease or stay the same. If a number on the grid gets too big then the algorithm decreases that number, and subsequently increases numbers elsewhere.
For example, consider a two-dimensional grid. Each point z(x,y) in the grid has a number n associated with it. The number is a counter for the point z(x,y). In terms of the sand pile, it can be thought of as the average slope of the sand pile at that point (on the grid).
In the course of running the program, the number n at a random point within the grid is increased by one unit as follows:
z(x,y) -> z(x,y)+1.
In other words, the average slope of the sand pile at that point is increased by one unit.
If this increase causes the value of n to be too big then there will be a redistribution of n as follows:
z(x,y) -> z(x,y)-4
z(x+/-1,y) -> z(x+/-1,y)+1
z(x,y+/-1) -> z(x,y+/-1)+1.
In other words, the slope of the sand pile at that point becomes too steep and grains of sand roll down the sand pile to nearby grid points.
If this redistribution causes n to be too big at one of the nearby grid points then the process continues. In other words, if the sand rolls to a point that was already poised for an avalanche then the avalanche continues.
If the redistribution does not cause n to be too big at one of the nearby points then the process stops. Another point z(x,y) is then chosen at random, its n is increased by one, and the process continues indefinitely.
One can imagine that for each point in the grid there is a domain D, which is the set of all points whose n was changed as a result of the initial perturbation. Figure 2 plots D for several different single perturbations that resulted in an avalanche. Each domain D has a finite size s, which is the aerial extent of an avalanche. Figure 3 plots D as a function of s. The figure clearly shows a power law distribution, implying the system has self-organized to a critical state.
C. Models in the laboratory
In the computer models discussed, and in the hypothetical sand pile, the system self-organized to a critical state. Critical state is a term common to thermodynamics. It refers to the critical point in, for example, the pressure-temperature phase space of a liquid-gas system.
Figure 4 is a hypothetical liquid-gas phase diagram. If an experimenter places a sample in the pressure-temperature space of region 1, the sample is in the gas phase. Likewise, if the sample is placed in region 2 the sample is in the liquid phase. If the sample is moved from region to region there is a phase transformation. If the sample is moved following the dashed arrow the phase change will occur abruptly at a specific temperature and pressure.
If the sample is moved following the solid arrow the phase change will occur continuously. In this continuos phase change there will be gas bubbles in equilibrium with the liquid. The gas bubbles will have a preferred size depending on the temperature and pressure.
If, however, the experimenter places the sample at the critical point (labeled 3), the gas bubbles show scale invariance. In other words, if the experimenter computes a gas bubble size distribution, the distribution will follow a power law.
This, however, is not an example of SOC because the system did not self-organize to the critical point. The system's pressure and temperature had to be carefully tuned so that it was exactly at the critical point. Thus, to be an example of SOC phenomena sand pile experiments in the laboratory must generate a power law distribution without careful tuning by the experimenter.
Nagel (1992) performed experiments with sand piles in three sided sand boxes, and rotating, semicylindrical drums filled with sand (see figure 5). The sand boxes had sand added to them at a constant rate and in random locations. The drum was rotated at a constant angular velocity. Beneath each apparatus a capacitor (sensitive to the motion of a single grain of sand) measured the over flow (size of avalanches) from the apparatus. The rotating drum was used to circumvent problems associated with spatially non-random addition of sand, and temporal variations in the rate of sand added to the sand pile.
This experiment tested for the presence of a power law in the temporal occurrence of avalanches. Nagel did not find a power law distribution. Instead, he found that avalanches occur periodically in time, suggesting an oscillation in the system. The periodic oscillation is derived from variations in the slope of the sand pile. The pile "does not collapse until the slope reaches an upper value [[theta]] m. When the slope becomes greater than this maximum angle of stability the pile is unstable and a global avalanche occurs. The avalanche brings the slope to a smaller value called the angle of repose" (Nagel, 1992).
Experiments by Nagel and others at the University of Chicago are criticized, however, because the techniques only record avalanches that slide out of the experimental apparatus. To circumvent this problem, similar experiments were conducted at the University of Michigan (Bretz, et. al., 1992). These experiments video taped landslides. The video tape was digitized and the size of the landslides subsequently counted. The results showed that some systems did exhibit a power law relationship while others did not.
The experiments by Bretz, et. al. bring up a problem with sand piles as examples of SOC phenomena. Specifically, it is questionable whether or not adjusting parameters such as grain size, cohesion, and rates of addition are in fact a form of tuning the experiments. In other words, if the experiments are sensitive to these parameters, and the experimenter must not violate certain conditions in performing the experiment, is the system really self-organizing to the critical state?
Defenders of SOC, however, are not concerned with whether or not laboratory physicists are tuning their sand piles to produce power law distributions. They point to the fact that these power laws exist on their own in Nature, where it is impossible to claim that the observer is tuning the results. Specifically, they point to seismicity plots; the Gutenberg-Richter relationship.
III. SELF-ORGANIZED CRITICALITY AND SEISMOLOGY
The Gutenberg-Richter relationship refers to the power law distribution between frequency and magnitude for earthquakes. For example, consider the entire global occurrence of earthquakes. The Gutenberg-Richter relationship states that, on average, there is one magnitude '8' every year, 10 magnitude '7's, 100 magnitude '6's, etc. (see figure 6).
Defenders of SOC point to this power law distribution as one of the best examples of the "finger print" of an SOC phenomena. They do so for two reasons. Firstly, earthquakes follow this power law distribution despite the abundance of noise in the natural environment. Secondly, it is simple to modify computer sand pile simulations to mimic fault plane interactions and earthquakes.
Consider a two dimensional array of blocks. Each block is connected by springs to its nearest neighbor, as well as to a driving plate (see figure 7). This system is analogous to a "sand pile" in that the local addition of slope to the sand pile:
z(x,y) -> z(x,y)+1
is like adding stress to the "fault plane". The addition of stress (sand) eventually causes a block to fail and an earthquake (avalanche) occurs:
z(x,y) -> z(x,y)-4
z(x+/-1,y) -> z(x+/-1,y)+1
z(x,y+/-1) -> z(x,y+/-1)+1.
If the subsequent redistribution of stress or "stress drop" causes a nearby block to exceed the critical stress value, then it too will fail. Thus, earthquakes can range in size from the failure of a single block to failures which encompass nearly the entire system. As with computer models of sand piles, this simple set of rules produces computer-earthquakes which follow a power law distribution.
The idea of using spring-slider block models to generate synthetic seismicity has a long history in Seismology. Burridge and Knopoff (1967) made similar simulations both in the laboratory and on the computer. These simulations generated power law distributions as well. In this sense, simple SOC models bring nothing new to Seismology.
Lomnitz-Adler (1993) did, however, extend the Burridge-Knopoff model. His work took the interactions of z(x,y) as shown above and modified them to more closely simulate current, competing theories of earthquake rupture physics. There are currently two competing models used to describe earthquake rupture physics. The first, typically described as the Heaton pulse model (Heaton, 1992), states that earthquakes have a rupture front which is closely followed by a healing front. The rupture front breaks the rock, and displacement occurs directly behind the rupture front. Displacement at a particular point ceases when a healing front (closely following the rupture front) passes the point. The healing front prevents a complete stress drop on the fault plane. The Heaton pulse model is thus also known as a partial stress drop model.
The second model used to describe earthquake rupture physics is that of the classical crack. The crack model has no healing front. Displacement initiates with the passage of a rupture front and continues until the earthquake itself stops.
Lomnitz-Adler modified the Burridge-Knopoff model in order to compare the crack model to the partial stress drop model. He further modified the Burridge-Knopoff model to test other key parameters of earthquake rupture physics including frictional and stress loading relationships. The criteria for the model tests was whether or not the said rupture physics could produce a power law distribution. His results indicate (with a few exceptions) that only partial stress drop models can produce power law distributions.
Using these modifications on the classical Burridge-Knopoff model to constrain the rupture physics of earthquakes can, however, lead to spurious results. The principle criticism of the Burridge-Knopoff (and hence the SOC) models is that they fail to properly model elasticity. In the SOC models the amount of displacement experienced by a fault block is the same whether the earthquake involves one block (magnitude 1) or 100,000,000 blocks (magnitude 8). In an elastic solid, however, the displacement experienced is a function of the size of the earthquake. In practical terms, a magnitude 8 earthquake with a fault area on the order of hundreds of square kilometers may locally produce tens of meters of displacement: it is impossible for a magnitude 2 earthquake, with a fault area on the order of a few square meters, to produce tens of meters of slip.
The theory of self-organized criticality (SOC) seeks to explain how the multitude of large interactive systems observed in Nature develop power law relationships from simple rules of interaction. Since SOC models generate power law distributions, proponents of the theory claim that many natural phenomena can eventually be understood via SOC.
The most commonly used paradigm of, and perhaps the best way to understand the theory is the sand pile. Laboratory measurements of sand pile experiments, however, are not consistent in their generation of power law relationships. To generate power law relationships these experiments often require what could be considered tuning by the experimenter.
The most commonly cited example of a naturally occurring SOC phenomena is the Gutenberg-Richter magnitude frequency relation. Computer models of earthquakes which rely on currently used SOC algorithms are, however, incorrect in that they improperly model elasticity.
This however, does not imply that SOC as a theory is without merits. In terms of Seismology, the excitement the theory has generated in the Physics community has led to an increase in the number of physicists interested in the problem of earthquake rupture physics. Similarly, the revitalization of the Burridge-Knopoff type model, and the vigor with which it is now being studied may very well lead to a increased understanding of the physics of the rupture process. Finally, apart from sand piles and earthquakes the theory may still have substantial merit since it is currently the only means of generating power law relationships from simple rules of interaction.
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Figure 1: An example of a power law distribution. Number of avalanches and number of grains involved in an avalanche are plotted in log-log space. The functional relationship has a constant slope.
Figure2: Aerial extent (domain) for several different avalanches in a SOC model. Each avalanche was triggered by the addition of a single grain. Avalanches have orders of magnitude difference in their sizes (Bak, et. al, 1988).
Figure 3: Log-log plot of the frequency of occurrence of given avalanche size D(s) vs. the size of the avalanche s for 200 avalanches Avalanches exhibit a power law distribution (Bak, et. al, 1988).
Figure 4: Generic liquid-gas phase diagram. When a sample is placed in region 1 it thermodynamically behaves like a gas. The sample in region 2 behaves like a liquid. If a sample follows the dashed arrow it will change from a gas to a liquid abruptly. If a sample follows the solid arrow it will change to a gas continuously, with gas bubbles of a fixed size in equilibrium with the liquid. When pressure and temperature are tuned such that the substance is at point 3, the substance is a mixture of gas and liquid. Gas bubbles within this liquid follow a power law distribution in terms of their size and frequency of occurrence.
Figure 5: Schematic diagrams of sand pile experiments done at the University of Chicago. a. Sand is dropped at a constant rate to random locations in a three sided box. Overflow avalanches cause sand to run through a capacitor which measures the size of the overflow. b. Sand in a drum rotating with constant angular velocity. Overflow avalanches are measured by a capacitor as in a (Nagel, 1992).
Figure 6: Log-log plot of global occurrence of earthquakes from 1977- 1995. Seismic moment (used to determine magnitude) is plotted vs. number of events. The distribution follows a power law relationship (Lay and Wallace, 1995).
Figure 7: Conceptual drawing of a two-dimensional spring-slider block model. Leaf springs (K1) connect a moving plate to an array of smaller sliding blocks. These blocks are in turn connected to their nearest neighbors via coil springs (K2, K3). Sliding blocks also have a frictional contact with a fixed plate (Bak, 1996).