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Avalanches, Scaling and Coherent Noise

We present a simple model of a dynamical system driven by externally-imposed coherent noise. Although the system never becomes critical in the sense of possessing spatial correlations of arbitrarily long range, it does organize into a stationary state characterized by avalanches with a power-law size distribution. We explain the behavior of the model within a time-averaged approximation, and discuss its potential connection to the dynamics of earthquakes, the Gutenberg-Richter law, and to recent experiments on avalanches in rice piles.


I. INTRODUCTION
There has in the last few years been considerable interest in extended systems which self-organize into a state exhibiting large scale fluctuations and intermittent dynamics. One of the earliest attempts to model systems of this type was made in 1987 by Bak, Tang, and Wiesenfeld, who proposed a simple lattice model for the avalanches produced by depositing grains of sand on an ever-growing sand pile [1]. Despite having only short-range interactions and no tunable parameters, their model organizes itself into a state with long-range spatial correlations and avalanches of size not limited by any finite correlation length. It has been proposed that similar self-organized critical (SOC) behavior could lie behind a wide range of physical phenomena showing 1/f noise and scale-free fluctuation distributions. SOC models have been put forward to describe the dynamics of earthquakes [2], biological evolution [3] and extinction [4], forest fires [5], and many other systems [6]. The common features of these models are that (i) they are all driven very slowly, and (ii) they all have perfect memory, i.e., in the absence of the driving force the model would be entirely stationary.
A distinctive observable consequence of SOC dynamics is that the distribution of fluctuation (or avalanche) sizes takes a power-law form with characteristic exponent τ : The value of τ typically lies in the range 1 < τ ≤ 3 2 , with the value 3 2 corresponding to a critical branching process, appearing if one makes the "random neighbor approximation" in which each site interacts with a randomly-selected small number of other sites [7]. This approximation is equivalent to the limit of infinite dimension, and should give correct results for systems above their critical dimension. In reality however, a number of the systems which are modeled using SOC dynamics in fact display event size distributions with fairly large exponents. Terrestrial earthquakes, for example, appear to follow the Gutenberg-Richter law [9] with τ ≈ 2.0, and the 1D rice pile experiment of Frette and co-workers [10], which has been compared to the sandpile model of Bak, Tang and Wiesenfeld, gives τ = 2.1 ± 0.1.

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A number of models have been proposed which offer explanations for these higher values of τ . The model discussed by Christensen and Olami [2] is one such, but it achieves its result only at the price of having an entirely deterministic dynamics; if one introduces randomness into the model, the simple scaling behavior is destroyed. The reason is that when the exponent describing the distribution of avalanches' spatial extent becomes larger than 2, the mean avalanche size becomes finite and independent of system size, and the spatial overlap between subsequent avalanches becomes insignificant [8]. In the presence of randomness, this can prevent the system from building up any long-range correlations, and ultimately destroy the critical state. We conjecture that, in this regime, any randomness in the positions of the nucleation centers of the avalanches will destroy self-organization of long range spatial correlations.
In this paper, we present a different explanation to account for systems that have larger values of τ . We demonstrate that power-law event size distributions having τ around 2 or greater, are typical of extended systems with quenched memory if they are driven by coherent noise, and that in such systems they are present even in the absence of any interaction between the different parts of the system. (This is different from the situation in the SOC models, where the system is driven by a local driving force, coupled with interactions between the components of the system.) The simplest model demonstrating the phenomenon is defined as follows. Consider a system of N agents, such as grains on the surface of a sand pile or points of contact in a subterranean fault. With each agent i we associate a threshold for movement x i which can take values falling in some specified range and represents the amount of stress that the agent will withstand before it moves. For convenience, we choose to measure x i on a scale on which 0 ≤ x i < 1. The dynamics of the model then consists of the repetition of two steps: We select a random number or "stress level" η from some distribution p stress (η). All x i below η are also exchanged with new random numbers selected uniformly from the interval [0, 1). The number of agents whose thresholds are changed in this fashion is the size s of the avalanche taking place in this time-step.
The random selection of different values for η at each step may be thought of as imposing external stresses which coherently (in other words, simultaneously) influence all of the weaker agents-those having suitably low thresholds for stress-but leaves unchanged the stronger ones. It seems physically reasonable to assume that smaller stresses should be more common than larger ones, and in the following discussion we make the assumption that p stress (η) is largest at η = 0 and falls off to zero as η becomes large. We denote the typical scale of the falloff by σ. The most interesting regime is when σ ≪ 1 and f ≪ 1,

II. RESULTS
We have examined the properties of this model both analytically and numerically. Instead of simulating the model directly, we have developed an algorithm which calculates the threshold distribution and avalanche sizes in a formally exact way for a system with N = ∞. Starting off with a uniform distribution of thresholds, the system evolves towards a statistically stationary state. In this state we record the mean threshold distribution and the frequency distribution of avalanches. The results are shown in Figures 1 and 2 for a simulation using exponentially distributed stresses: As Figure 1 shows, the distribution of avalanche sizes s is flat up to a certain point (whose position varies with σ and f ) and then falls off as a power law according to Eq. (1) with τ ≈ 2.0. This power-law behavior appears to be robust in the regime of small f and σ. If, for example, instead of Eq. (2) we employ a Gaussian stress distribution then, although the average distribution of thresholds ( Figure 2) changes radically, the power-law form of the avalanche distribution remains. Notice, however, that the exponent τ changes slightly as the applied stresses are varied. For the Gaussian distribution, for example, we find τ = 2.2±0.1, as opposed to τ = 1.9 ± 0.1 for the exponential. And for steeper distributions of stresses In order to investigate possible connections with spatially-organized models, we have also implemented our model on a lattice and at each time-step eliminated not only those agents whose thresholds for stress fall below the selected level, but also their neighbors. In all cases we observe a power-law distribution of avalanches with exponent in the vicinity of τ = 2.
In order to understand the appearance of this power law, let us consider the timeaveraged behavior of the model. The statistically stationary state arises as a competition between the two processes comprising the dynamics: the stresses which tend to remove lower thresholds from the distribution and thus shift the weight of the distribution to higher values of x, and the aging, which tends to move weight back down again. The result is that the average threshold distribution p thresh (x) is a highly nonhomogeneous, monotonic increasing function of x which, for small σ, tends to have a plateau as x approaches unity (see Figure 2). By balancing the two competing processes, we can calculate p thresh (x) and hence the avalanche distribution. For concreteness, we perform the calculation here for the exponentially distributed stresses of Eq. (2).
The probability of an agent possessing a threshold x lying below the stress level η at any given time-step (and hence of it moving during this time-step) is The total time-averaged rate at which agents move in the interval between x and x + dx is where the x-independent constant W on the right-hand side is the time-averaged rate at which probability is added to p thresh . Rearranging we have The constant is easily fixed by requiring that p thresh (x) integrate to unity, giving For small f and σ, p thresh (x) rises exponentially from zero and then levels off in a plateau around x = −σ log f . Physically, this arises because agents possessing thresholds above this point are affected only by the aging process, which treats them all equally. Below this level, the stress process is important too, and it preferentially moves those with lower thresholds.
The avalanche size distribution is given by The probability p(s|η) of getting an avalanche of a certain size given a certain stress level, depends on the distribution of thresholds, which will in general vary from one time-step to another. However, if we make the "time averaged approximation" (TAA) whereby one assumes that at each time-step the threshold distribution can be approximated by its time- The avalanche size distribution then becomes where we have used Eqs. (5) and (8). We can calculate the stress level η(s) corresponding to an avalanche of size s from the same two equations, which give for e −1/σ ≪ f ≪ 1 and σ ≪ 1. We can now distinguish a number of different regimes. For small avalanches, such that s ≪ σ, the logarithm on the right-hand side can be expanded giving s + σf ≈ σf e η/σ . Substituting into Eq. (9) This gives a flat avalanche distribution for small s up to about s = σf , and then a powerlaw distribution for larger s with exponent τ = 2. The approximation breaks down when s ≈ σ, giving way to a regime in which e η/σ ∼ e s , and hence the avalanche distribution falls off exponentially with s. The various regimes can clearly be seen in the numerical results presented in Figure 1, and the predicted cross-over points between them agree well with the theory.
When f decreases below e −1/σ , the approximations in Eq. (10) break down and instead it becomes valid to write e η/σ ≈ 1 + se 1/σ . In this regime the theory predicts a breakdown in the scaling, a phenomenon which is also seen in the simulations. Thus the reloading process, whose scale is set by f , must be small but necessarily non-zero if we are to see power-law behavior in the avalanche distribution. Notice however that at precisely f = 0 the theory predicts a return to τ = 2 scaling, which is not seen in the simulations, implying that the TAA breaks down in this regime because the distribution p(s|η) becomes too broad to be well approximated by a δ-function.
The physical principle behind the appearance of a power-law distribution here is the interdependence of the avalanche and threshold distributions; the avalanche distribution is a function of the particular distribution of thresholds at any time, but the threshold distribution is itself produced by the action of the avalanches.

III. CONNECTION WITH OTHER MODELS
There are clear similarities between our model and the sand pile model, in which sites also possess a certain threshold stress that they will withstand without adjusting. Furthermore Model (DFBM) [12], the model presented here does not make a clear distinction between small, finite-sized events, and large ones whose size scales like the size of the system. In the BK model, for instance, the spectrum of event sizes contains two separate parts, one composed of small events which scales as s −2 , and another composed of the big events, which occur quasiperiodically. The BK and DFBM models are not statistically stationary, by contrast with our model whose dynamics rapidly reaches a statistically stationary state. Models such as BK and DFBM also show "foreshock" events in which large avalanches are preceded by smaller ones. Our dynamics does not have foreshocks but does display aftershock events, a phenomenon which we discuss in greater detail in next section.

IV. DISCUSSION
Next, we would like to examine the potential relationship of our model to processes occurring in real physical systems. First we consider earthquakes. To begin with, we ignore spatial correlations and consider the variables x i to be thresholds for movement at various points along a fault. The coherent stress η is provided by long-wavelength background noise from some external source, such as other distant tremors, or movements in the deeper regions of the earth, and the reloading f is due to slow plastic deformation from tectonic movements of the crust. As we have seen, these elements alone lead directly to a power-law distribution of earthquake sizes very close to the observed Gutenberg-Richter law, without the need to invoke interactions between neighboring parts of the fault. That is not to say that such interactions do not exist, only that they are not necessary to produce the observed power law. (Kagan [13] has presented evidence of a fractal pattern in the spatial distribution of earthquake activity, which is an indication that interactions are a feature of the dynamics. This however need not lead us to conclude that these local interactions are necessary for producing the observed size distribution of events.) Another interesting feature of our model is that it shows clear aftershocks. The mechanism for these is straightforward. When a large avalanche takes place, a significant fraction of the thresholds in the system are replaced with new, uniformly distributed ones. Because of the monotonic increasing form of the threshold distribution, this has the effect of shifting the weight of the distribution downwards, increasing the fraction of agents with low threshold for movement. The result is that subsequent stresses on the system have a larger-than-normal effect, and we see an amplification of the usual level of "background" avalanches in the aftermath of a particularly large event. In Figure 3, we show a section of a time series of avalanches from one of our simulations, which clearly displays this aftershock effect. Notice that if we apply the argument iteratively, we would also expect to see sequences of "after-aftershocks" following each of the aftershocks, a behavior which is indeed evident in Figure 3. We have also measured the average probability of getting an event of significant size in the aftermath of another large one, and found that for small times its distribution goes approximately as t −1 (Figure 4). A similar result is seen in the data from real earthquakes, and is commonly referred to as Omori's law [14].
The t −1 distribution can be understood as follows. A large avalanche will redistribute the thresholds of a large fraction of the agents in the system uniformly across the interval of allowed values (0 ≤ x < 1 in this case). A subsequent stress of magnitude η 1 will remove all those agents with x < η 1 , and produce an aftershock extinction of a certain magnitude.
In order to get another significant aftershock we now need a stress η 2 > η 1 in order to reach those agents which were not affected by the first aftershock. In general, if it took a time t to get the first stress, then it will on average, take as long again to get another of the same magnitude, or an aggregate time of 2t until a stress of size η 2 comes along. Repeating the argument, it will take as long again, or a total time of 4t to get the third aftershock, and so forth. Given this exponential increase in the time intervals between these events, it is not hard to show that the histogram of aftershock events should have a t −1 power-law form, regardless of the precise distribution of stresses applied to the system.
Note that our mechanism is by no means the only way to obtain aftershocks. An alternative mechanism has been proposed by Nakanishi [15]  The results of these experiments have also been modeled by Christensen [16] using a SOC model with interacting elements. Clearly, there are aspects of the dynamics captured by their model which are missing from ours, particularly geometrical effects concerned with the spatial distributions of avalanches and the corresponding transport properties of rice in the pile [17]. However, because the exponent τ is greater than 2, making s independent of system size, we can expect these properties to be independent of the largest avalanche events (though on the other hand, they should now depend strongly on the position and nature of the crossover between the two regimes of the avalanche distribution). We suggest that the reverse is also true, i.e., that the observed large avalanches could appear even in the absence of long range spatial correlations.
One characteristic which does seem to distinguish our model from the SOC alternatives is the existence of aftershock events. It might therefore might be profitable to investigate the existence of aftershock avalanches in the experimental data, in order to make a quantitative distinction between the two classes of dynamics. distributions with exponents close to τ = 3 [18]. One may speculate whether these so-called "fat tails" in the distribution are the natural response to the action of external stresses on the market (of which there are indeed many).

V. CONCLUSION
To summarize, we have demonstrated that coherent noise in large systems typically gives rise to intermittent behavior with an "avalanche" type dynamics characterized by a powerlaw distribution of avalanche sizes with exponent in the vicinity of τ = 2. This value is similar to that seen in a number of real systems, including rice piles and earthquakes, suggesting that these systems may in fact be driven by external noise, rather than self-organizing under 12 the influence of short-range internal interactions. If one allows more elaborate types of stress on the system one can obtain power laws with exponents as high as τ = 3.
We believe that the study of systems driven in this fashion by coherent external noise may offer new interpretations of intermittent dynamics in a variety of extended non-equilibrium systems in terms of a direct interplay between small scale structures and long wavelength fluctuations in the system. Such systems might include not only the ricepiles and earthquakes considered here, but possibly also extended chaotic systems such as economics [18] and turbulence [19]. 13