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(Ogata,, 1999 ; Helmstetter and Sornette,, 2002 Interestingly, the evolution model of Bak and Sneppen, ( 1993 (another paradigm of SOC) can be interpreted to reproduce the statistics of earthquakes from this (slow) time scale (Ito,, 1995 This perspective opened a whole new line in statistical seismology, but this is a ... |
(Bak et al.,, 2002 Corral, 2004a, Corral, 2004b, Conclusions We started this chapter showing some remarkable statistical properties of earthquake occurrence, and ended up mingling with infinite-dimensional sandpiles models for self-organized criticality. In between, we learnt a few things about branching processes. Now... |
First, besides any model, we can say a few things just by looking at the data: earthquakes and other natural hazards follow a power-law distribution of sizes, in some cases with an exponential cutoff due to finite-size effects (the Earth is finite, after all!). For the particular values of the exponents found, this imp... |
(Mantegna and Stanley,, 1999 ; Newman,, 2005 one wonders what the points in common with these systems and natural hazards can be. |
Regarding Otsuka’s rupture model, we showed how, by using a fairly simple stochastic cascade setup for the local dynamics of fault patches and the mathematical formalism for branching processes, one can reproduce the global statistical properties of real earthquake occurrences (and other natural hazards). This is quite... |
But Otsuka’s model is a particular case of the Galton-Watson branching process. So, first, we presented in an easy way the main results already known for such processes (main results in relation to our interests). We explained how the machinery of probability generating functions allows to find a formula for the activi... |
In this regard, we showed how, by using a simple feedback mechanism, one can turn the critical point into an attractor of the model. A global condition, related with boundary dissipation, acts on the probability of activation, in such a way that when this probability is low, it increases, and vice versa when it is high... |
It is worth mentioning that going beyond the mean-field limit and turn to lattice (more realistic) systems makes things terribly complicated, and the researcher has to rely more and more on computer simulations and losses the guide of exact, or at least approximated analytical treatments. But this makes the mathematica... |
As a final point, we have to recognize that criticality and self-organized criticality are not the only ways to generate power-law distributions. In fact, much simpler processes that yield power laws exist, as reviewed in Sornette, ( 2004 ); Mitzenmacher, ( 2004 ); Newman, ( 2005 A well known mechanism that escapes fro... |
Appendix Properties of power-law distributions Some facts about the power-law distribution are remarkable. Let us consider the probability density [MATH] defined between [MATH] and [MATH] We may first calculate its mean, i.e., the expected value of [MATH] , given by |
[EQUATION] It is easy to check that, when [MATH] (i.e. [MATH] ), this integral becomes infinite, so, mathematicians would state the expected value of the energy does not exist, whereas physicists would say that that value is infinite. We take the second option, which is more informative as we are aware of what we are d... |
Although previously we interpreted as good news the fact that most earthquakes are of small size and only very few of them are devastating, the situation is certainly not so favorable. The reason is that the rare big events, despite their scarcity, are the ones responsible for the dissipation of energy in the system. F... |
[EQUATION] no matter how big is [MATH] (see next subsection for details). A second peculiar property of power laws is scale invariance. Let us introduce the concept of scale transformation, considering an arbitrary function that we call [MATH] The idea of a scale transformation is to look at the function |
[MATH] at a different scale, as for instance, using a mathematical microscope. We can have a view of the function at the scale of meters (if [MATH] and [MATH] were distances) and try to see how it looks at the scale of centimeters. This is performed through a scale transformation, denoted by an operator [MATH] acting o... |
[EQUATION] where [MATH] and [MATH] are two constants called scale parameters, performing a linear transformation on [MATH] and [MATH] In the case of the meters-centimeters example, |
[MATH] In general, almost every function changes under a scale transformation; the exception can be found looking for the function or functions that verify the following condition, |
[EQUATION] It is trivial to check that a solution is given by the power-law function [EQUATION] with [MATH] given by [EQUATION] in other words, a power law with exponent |
[MATH] does not change under a scale transformation if the scale factors are related through [EQUATION] Figure 12 shows how indeed this is the case, with [MATH] [MATH] , and [MATH] Note that the constant of proportionality in equation ( 91 ), contained in the symbol |
[MATH] , does not play any role here. More importantly, it can also be demonstrated that not only the power law is a solution, but it is the only solution valid for all values of [MATH] |
(positive real) if [MATH] and [MATH] are related by equation ( 93 (Takayasu,, 1989 ; Newman,, 2005 ; Christensen and Moloney,, 2005 ; Corral,, 2008 In summary, the condition of scale invariance demands that |
[EQUATION] and then, the only solution is the power law. One can verify that other solutions, as [MATH] only work for special values of [MATH] and [MATH] |
Scale invariance is in fact the symmetry associated to scale transformations, in an analogous way as rotational invariance is the symmetry corresponding to rotations. If scale invariance is fulfilled, no characteristic scale can be defined for the variable [MATH] in the same way as if there is rotational invariance in ... |
There is, nevertheless, an important point to be taken into account here. If [MATH] represents a probability density (as it is the case for the energy radiated by earthquakes), then, [MATH] cannot be a power law for all [MATH] because it could not be normalized (its integral from 0 to [MATH] |
would diverge). We have already mentioned that it is necessary to introduce a lower cutoff [MATH] in order to avoid this fact. Also, sometimes the power law cannot be extended to infinity, for physical reasons. So, complete scale invariance is not possible for probability distributions, and one can have only a restrict... |
Scale invariance in the energy of earthquakes has some counter-intuitive consequences. Imagine that you arrive at a new country, and you are worried about earthquakes, and ask the people there the following question: |
how big are typically earthquakes here? Despite the innocence of such a simple question, due to scale invariance no characteristic scale for the energy can be defined and the question has no possible answer. |
Dissipation of energy in the largest scales Let us consider a (continuous) power-law distribution, defined, for simplicity, between 1 and [MATH] with probability density, |
[EQUATION] We are going to see that, for a given [MATH] there exist values of [MATH] such that the contribution to the expected value of |
[MATH] from an interval [MATH] is always smaller than the contribution from [MATH] no matter how big [MATH] is. The contribution of an interval [MATH] |
to the mean value of [MATH] is [EQUATION] Therefore, [EQUATION] and [EQUATION] In order that the last integral is larger than the previous one it is enough that |
[EQUATION] So, [MATH] and this implies that [EQUATION] For [MATH] , the (sufficient) condition becomes [MATH] In the case of earthquake radiated energy, |
[MATH] and equation ( 100 ) is fulfilled. Though, slightly larger values of [MATH] violate the condition; nevertheless, there is nothing special in taking [MATH] |
(it is not a magical number!) and we have that equation ( 100 ) is fulfilled for a larger [MATH] For [MATH] equation ( 100 ) would imply [MATH] but this is not an acceptable exponent for a power-law distribution (normalization would not be fulfilled). |
Rigorous proof of extinction probability Besides graphical arguments (see Fig. ), we want to provide a rigorous proof for the computation of the extinction probability in the Galton-Watson process, given by |
[EQUATION] where [MATH] is properly defined only if the limit exists. To see that this is always the case, we note that [MATH] Hence, [MATH] and |
[MATH] , so [MATH] or, in words, [MATH] is a non-decreasing sequence. As [MATH] , we conclude that [MATH] is bounded and has a limit. To continue our proof, let us treat separately the two cases [MATH] [MATH] . Hence, |
case [MATH] As [MATH] is non-convex for [MATH] , it always lies above any straight line tangent to it (Spivak,, 1967 In particular, we consider the line tangent to [MATH] at the point [MATH] , and |
[EQUATION] Hence [MATH] for [MATH] . Also, it is straightforward to see that [MATH] [EQUATION] and of course [MATH] . So we have that [MATH] with [MATH] Summarizing, [MATH] is a fixed point of [MATH] in the interval [MATH] , but [MATH] (strictly) in [MATH] It is clear that the only option left is [MATH] |
case [MATH] We will start showing that [MATH] in this case. First, as already said, [MATH] is a non-decreasing sequence. Second, as [MATH] is continuous and [MATH] , we have that [MATH] for [MATH] for some [MATH] . So, [MATH] for all [MATH] (because it would then decrease). This means that the only way for [MATH] to ha... |
Now we will see that the equation [MATH] has a unique solution in the interval [MATH] . There must be at least one solution because [MATH] , and [MATH] in [MATH] (here we are using Bolzano’s theorem for [MATH] ). To see that this solution is unique, suppose there are two solutions, [MATH] . As we also have [MATH] , by ... |
So, if [MATH] but [MATH] , then [MATH] must be the unique solution of [MATH] in [MATH] For the sake of rigor, we must point out that some “pathological” cases would need a separate treatment, such as [MATH] , but those are almost never of actual interest. |
Catalan numbers The Catalan numbers owe their name not to a Mediterranean region but to the French-Belgian mathematician from the 19th century Eugène Charles Catalan. “His” numbers count a large variety of objects (Stanley,, 1999 in particular, the rooted trees that arise in the study of branching process when the numb... |
as the root (corresponding to the zero generation of the associated branching process) plus the remaining [MATH] nodes, these latter can be distributed as a varying number of nodes associated to the first branch, [MATH] |
and the rest to the second branch, [MATH] respectively. Therefore, the number of trees [MATH] of size [MATH] fulfills, [EQUATION] |
where [MATH] is taken equal to one, as there is only one way in which a branch can have no elements. Note that from here we obtain |
[EQUATION] and so on this simple formula generates all Catalan numbers. The curious reader can check Figure 13 , where all possible rooted trees with no more than two branches per node, of size up to 4, are shown. |
If we want a closed expression for these numbers, we may define a generating function [EQUATION] One can obtain an expression for [MATH] just using the properties of the Catalan numbers (Wilf,, 1994 First, let us calculate |
[EQUATION] As we know that [MATH] , we end up with a quadratic equation for [MATH] , namely, [EQUATION] which allows us to isolate [MATH] |
[EQUATION] One of both functions (depending on the [MATH] sign) is then the generating function of the Catalan numbers. We are going to recover these numbers from its generating function. First, one needs the Taylor expansion of [MATH] around [MATH] which is |
[EQUATION] where, remember, [MATH] , and so, [EQUATION] Then, substituting in [MATH] , one can realize that only the minus sign can correspond to a generating function, and |
[EQUATION] from where we obtain a first expression for the Catalan numbers, [EQUATION] A more comfortable formula can be obtained using that |
[EQUATION] and then one finds, [EQUATION] the standard expression for the Catalan numbers, now valid for all [MATH] Normalization and non-normalization of the total size distribution |
We are going to illustrate how the total size probability distribution, [MATH] is only normalized in the subcritical and critical cases. We use the binomial distribution for the distribution of the number of offsprings, with [MATH] and [MATH] From the main text, we know that |
[EQUATION] It can be checked, using the generating function of the Catalan numbers, that this expression is normalized for [MATH] but not for [MATH] In order to see this, let us first consider the generating function of the Catalan numbers, derived in the previous subsection of the Appendix, |
[EQUATION] Then, introducing [MATH] [EQUATION] and using the expression for [MATH] [EQUATION] We can distinguish two cases, first, [MATH] , for which, |
[EQUATION] and for the opposite case, [MATH] [EQUATION] Therefore, [EQUATION] Remembering the results for the extinction probability for the binomial distribution, |
[EQUATION] which obviously is not normalized for [MATH] We could also have arrived to the same result using, not the generating function of the Catalan numbers, but the generating function [MATH] of the size [MATH] |
Stirling’s Approximation Usually, Stirling’s formula is demonstrated by means of the Euler-Maclaurin formula. However, if one knows some elementary properties of the gamma distribution, Stirling’s formula arises almost spontaneously, by means of a probabilistic trick. |
Remember that the factorial is associated to the gamma function, [MATH] , which is defined as [EQUATION] for [MATH] (Abramowitz and Stegun,, 1965 This allows to introduce the gamma distribution (Durrett,, 2010 with probability density given by |
[EQUATION] for [MATH] (and zero otherwise), and with mean [MATH] and variance [MATH] It turns out that the gamma distribution arises as a sum of a number [MATH] of independent exponential random variables, each with density [MATH] |
(this can be easily demonstrated through successive convolutions of the exponentials, see Durrett, ( 2010 ). But using the central limit theorem, the gamma distribution will converge, in the limit [MATH] to a normal distribution (see Fig. 14 ), with mean [MATH] |
and standard deviation [MATH] (in this case the notation is different to the rest of the chapter). Then, it will be possible to transform the gamma function into a Gaussian integral. Indeed, |
[EQUATION] The key point is to find the value of [MATH] for which both functions overlap. This happens around the mean or the mode of both distributions, corresponding, respectively, to [MATH] and [MATH] Substituting both values in |
[EQUATION] we get [EQUATION] and therefore, looking for the normal probability density inside the integral, [EQUATION] The value of [MATH] is obtained from [MATH] |
(for independent random variables the variance of a sum is the sum of variances, which is one for each exponential distribution in our sum). Substituting, and replacing the lower integration limit by [MATH] due to the fact that the standard deviation [MATH] is much smaller than the mean [MATH] one obtains, |
[EQUATION] valid, remember, in the limit [MATH] This proof has some parts in common with the more elaborated one of Khan, ( 1974 |
and less resemblance with that of van den Berg, ( 1995 Acknowledgements. We would like to dedicate this work to the colorful scientist Per Bak, in the 25 years of his invention of self-organized criticality and in the 10th anniversary of his untimely death. The chapter originates, in part, from a lecture that one of th... |
American Geophysical Union In this regard, we thank Armin Bunde, and also Tom Davis, for making his notes on the Catalan numbers publicly available on the Internet, and Anna Deluca and Gunnar Pruessner, for discussions. Cecília M. Clos provided valuable graphical-design assistance. Funding has come from Spanish project... |
# Source: arxiv 1207.5389 # Title: Percolation Models of Self-Organized Critical Phenomena # Sections: all # Downloaded: 2026-03-03T01:58:26.937499+00:00 |
Percolation Models of Self-Organized Critical Phenomena Abstract In this chapter of the e-book “Self-Organized Criticality Systems” we summarize some theoretical approaches to self-organized criticality (SOC) phenomena that involve percolation as an essential key ingredient. Scaling arguments, random walk models, linea... |
The percolation problem The standard theory of percolation Hammer began with an attempt to make statistical predictions about the possibility for a fluid to filter through a random medium, predictions that could be applied to a variety of physics problems, such as epidemic processes with and without immunization, the u... |
Site and bond percolation Given a periodic lattice, embedded in a [MATH] -dimensional Euclidean space, one can choose between two alternative formulations of the percolation problem: site and bond. The differences between site and bond percolation are actually very subtle and are manifest in a typically lower threshold... |
Percolation critical exponents [MATH] [MATH] , and [MATH] Likewise to traditional critical phenomena, characterized by a scale-free statistics of the spontaneously occurring quantities, the geometry of connected clusters in vicinity of the percolation threshold is self-similar (fractal) Stauffer Feder . As [MATH] , the... |
Random walks on percolating clusters The problem of diffusion on fractals Gennes Straley Gefen has stirred considerable attention in the literature, especially, in terms of the random walk approach. If the random walker (an unbiased “ant”) is put on a connected cluster at percolation, then the distance it travels after... |
[EQUATION] The exponent [MATH] is given by [MATH] . Remark that the dependence here is no longer proportional to the time [MATH] , by contrast with uniform spaces. Thus, diffusion is anomalous . The exponent [MATH] describes topological characteristics of the fractal (such as connectivity, etc.) As such, it shows inter... |
In a basic theory of percolation Stauffer it is shown that [MATH] for connected clusters, implying that [MATH] . One sees that the mean-square displacement in Eq. ( ) grows slower-than-linear with time. This slowing down of the transport occurs as a result of multiple trappings and delays of the diffusing particles in ... |
[EQUATION] for [MATH] . Equation ( ) has implications for the ac conductivity at “anomalous” frequency scales, [MATH] , for which the charge carriers move only on the fractal Gefen PRB01 . The various aspects of anomalous diffusion in fractal systems are summarized in the reviews Havlin UFN Bouchaud |
The spectral fractal dimension and the Alexander-Orbach conjecture A hybrid parameter [MATH] is often referred to as the spectral, or fracton, dimension. It is so called because it represents the density of states for vibrational excitations in fractal networks termed fractons Rammal AO . It also appears in the probabi... |
In the past years there has been much excitement about the Alexander-Orbach (AO) conjecture AO that the spectral fractal dimension is exactly [MATH] for percolation clusters in any ambient dimension [MATH] greater than 1. This conjecture is important as it relates the structural characteristics of the fractal, containe... |
Percolation problem on the Riemann sphere It is both interesting and instructive to demonstrate how the spectral fractal dimension may be obtained for threshold percolation on a plane ( [MATH] ). The main idea here PRE97 is to extend the plane by adding the point at infinity to it, then consider a stereographic project... |
Observe that the percolation cluster spanning the plane implies that its stereographic image covers part of the surface of the sphere, the north pole included. Without loss of generality, we may assume that the south pole at which the sphere touches the plane belongs to the cluster. When considered on the Riemann spher... |
[EQUATION] Numerical solution shows that [MATH] , remarkably close to, although slightly smaller than, 4/3. Rigorously speaking, this result disproves the AO conjecture in [MATH] . Despite being this subtle, the observed deviation from [MATH] is important as it helps avoid the secular terms problem when applying a reno... |
Summary Summarizing, the percolation problem presents a non-trivial problem with scale-free behavior. It is related with phase transition-like phenomena as well as the fundamental topology (via the connectedness issues). The percolation clusters provide a particularly clear example of statistical fractals in the limit ... |
The SOC hypothesis The challenge to understand fractals Mandel and the [MATH] “noise” led Bak, Tang, and Wiesenfeld BTW to introduce the concept of self-organized criticality, or SOC. The claim was that irreversible dynamics of systems with many coupled degrees of freedom (“complex” systems) would naturally generate se... |
Char and Aschwanden Ash11 To qualify as SOC, the system must be open, be coupled with the exterior, and involve many interacting degrees of freedom. In addition, its dynamics must be thresholded and nonlinear, and the driving, or energy injection, rate must be very slow (infinitesimal). An important advance of SOC is t... |
SOC vs percolation Before we proceed with the main topics of this chapter, we would like to address the SOC hypothesis against the percolation problem discussed above. Indeed SOC shares with percolation the implications of threshold behavior and spatial self-similarity. An essential difference is that percolation is a ... |
Another important aspect is that in percolation and other traditional critical phenomena, control parameters must be fine tuned to obtain criticality (thus the name “control”). In SOC phenomena, control parameters make part of the dynamical system instead: their values are defined dynamically as the system self-adjusts... |
The “guiding” mechanisms An important issue concerns the mechanisms that “guide” a system to criticality. These mechanisms are of two types. One type is associated with the application of an extremal principle that the dynamics should obey in order to satisfy the microscopic equations of motion. Examples of this type a... |
[MATH] introduced in physics by Wilkinson and Willemsen [MATH] the dynamics proceed along a path of least resistance under the action of capillary forces. Under the condition that the flow rate is infinitesimal the system finds its critical points that are stable and self-organized Invasion . The second type is associa... |
Going with the random walks: DPRW model The percolation problem when account is taken for a dynamical feedback mechanism offers a suitable platform to build toy-models of self-organized critical phenomena. Early attempts in this direction refer to the “dilution-by-hungry-ants” and the “thermal-fuse” models (with and wi... |
Description of the model We consider a hypercubic [MATH] -dimensional ( [MATH] ) lattice confined between two opposite ( [MATH] )-dimensional hyperplanes, which form a parallel-plate “capacitor” as shown in Fig. 3. The plate on the right-hand-side is earthed. Free charges are built by external forces on the capacitor’s... |
When a hole appears on the infinite cluster it causes an activation event with the following consequence: One of the nearest-neighbor occupied sites, which is a random choice, will deliver its charge content to the hole. The hole which has just received the polarization charge becomes ordinary occupied site, while the ... |
Random-walk hopping process Essentially, the holes interchange their position with the nearest-neighbor occupied sites, and it appears reasonable to model this process as interchange hopping process Dyre . In what follows we assume, following Ref. Gennes , that there is a characteristic microscopic hopping time, which ... |
Dynamical geometry of threshold percolation Overall, one can see that the system responds by chain reactions of random-walk hopping processes when it becomes slightly supercritical and it is quiescent otherwise. Excess free charges dissipating at the earthed plate provide a feedback mechanism by which the system return... |
In the DPRW SOC model, chain reactions of the hopping motion acquire the role of “avalanches” in the traditional sandpiles. In the present analysis, we are interested in obtaining the critical exponents of the DPRW model by means of analytical theory. Numerical simulation of the DPRW dynamics is under way for compariso... |
Linear-response theory Dynamics and orderings Starting from an empty lattice (no potential difference between the plates), by randomly adding occupied sites to it, one builds the fractal geometry of the random, or uncorrelated, percolation, characterized by three percolation critical exponents [MATH] [MATH] , and [MATH... |
Frequency-dependent conductivity and diffusion coefficients Given an input electric driving field [MATH] the polarization response of the system is defined through |
[EQUATION] where the response function [MATH] is identically zero for [MATH] as required by causality. We should stress that nonlocal integration over the space variable is not needed here in view of the local (nearest-neighbor) character of the lattice interactions. In a model in which the assumption of locality is re... |
[EQUATION] where [EQUATION] with [MATH] a constant depending on the dimensionality of the lattice and [MATH] and [MATH] the density and charge of the carriers, respectively. The function [MATH] has the sense of the frequency-dependent diffusion coefficient Lax58 . In the zero-frequency limit, Eq. ( ) reproduces the wel... |
[EQUATION] One sees that [MATH] in the limit [MATH] . The Kubo number is a suitable dimensionless parameter which quantifies how the evolution processes in the lattice compare with the microscopic hopping motions. The divergency of the Kubo number at criticality implies that there is a time scale separation: fast hoppi... |
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