C.M. Aegerter, K.A. L˝orincz∗, and R.J. Wijngaarden∗ The ubiquitous appearance of power-laws in nature has lead Bak et al. in 1987 to propose that slowly driven non-equilibrium systems self-organize into a critical (SOC) state, which naturally leads to power-law behavior [1].
While in the past 15 years, much progress has been made on the theoretical foundations of a process that naturally leads a system to its critical state [2,3], controlled experi-ments are a rarity in the field [4-6].
From the theoretical point of view, it has been shown that in order to get a proper understanding of SOC, the way in which the critical state is reached is of utmost impor-tance, as this allows the self-organization process to be studied [2,3]. This crucial aspect of SOC has however up to now not been studied experimentally at all. Therefore, we here study experimentally the approach to the critical state in a system which has been shown to obey SOC by stringent criteria, such as finite size scaling and the ful-fillment of scaling relations [7]. Our system consists of a three dimensional pile of rice, where we concentrate in particular on the maximum slope of the pile as the critical state is reached, as well as the temporal scaling behavior of the surface coordinates. Using the theory that extremal dynamics is what underlies SOC, the approach to the crit-ical state can be described analytcrit-ically by what is called the Gap-Equation [8] and its dynamics is determined by the dynamics in the critical state. This is quantified by the fact that the exponent governing the approach of the maximum slope to its critical value,δ, is related to the ex-ponents characterizing the critical state, i.e. the avalanche distribution exponent,τ, the avalanche dimension,Dand the fractal dimension of the active sites,dB. All of these quantities are known experimentally from studies of the critical state, which we have carried out on our rice pile in previous investigations [7]. Thus a combination of the results in the critical state with those from the transient be-havior can lead to a confirmation that extremal dynamics indeed lies at the heart of SOC.
The experiments were carried out on a pile of rice with the surface area of∼1 x 1m2[7]. Long grained rice of di-mensions∼2 x 2 x 7mm3is fed continuously at one edge of the pile in a uniformly distributed line and an image of the pile surface is taken every 30s with a high resolution charge-coupled device (CCD) camera (1596x2048 pixels).
The driving rate corresponds to the dropping of∼1500 grains between two images. This does however qualify as slow driving, as at each point along the line of growth, this only corresponds to about 2 grains being added each time step. Furthermore, one has to compare the number of added grains with that already in the pile, which is of
the order of107−108grains. In order to reconstruct the surface coordinates of the pile, a set of colored lines (red-green-blue) is projected onto the pile perpendicularly. The images are taken at an angle of 45◦ to the projection di-rection leading to a distortion of the lines corresponding to the surface properties, with and accuracy and precision of∼1-2mm[9]. After identification of the different lines from the image, a simple calculation leads to the surface coordinates.
Figure 1: A typical image of the reconstruction of the rice-pile surface. Due to the number of lines projected, and the high reso-lution of the CCD camera, the surface can be reconstructed with an accuracy of∼1−2mm, which is comparable to the size of a grain.
For the experiments on the build-up towards the critical state, we first created a flat pile surface, at an angle far below the critical angle (φ0 ≃ 0.55 compared toφc ≃ 0.8). Each experiment consisted of about 500 images and 9 separate experiments were analyzed [10]. Here, we want to determine a measure for the distance from the critical state (the Gap of Ref. [8]). We therefore determine the maximum local slope of the pile, f(t), as a function of time (see inset of Fig. 2). As the pile gets closer to the critical state, the maximum local angle will approach a critical value,fc. The Gap is then given by the difference G(t) =fc−f(t)of the maximum local slope to its critical value.
When the rice pile approaches the critical state, the max-imum local slope increases towards its critical value. In the context of extremal dynamics, this process can be de-scribed by the Gap-equation, which predicts thatG(t) ∝ t−δ [8]. Furthermore, the value of the exponent charac-terizing the approach to the critical state,δ, is directly re-lated to the avalanche size distribution exponent, τ, the avalanche dimensionD and the fractal dimension of the active sitesdBin the stationary state via
δ= 1−1−dB/D
2−τ . (D.2)
The time dependence of the Gap is shown in Fig. 2, where clearly the approach to the critical state follows a power law over two decades. Experimentally, δis determined
to be 0.8(1), where the biggest contribution of the er-ror comes from the determination offc, which was ob-tained independently from a direct experiment on the max-imum possible slope of a small part of the pile to be fc = 0.92(2)[10]. The value forδobtained in the direct experiment on the transient slopes is in good agreement with the expectation from Eq. D.2 ofδ = 0.75(3)using the values ofτ, DanddBdetermined previously [7]. Thus the approach to the critical state is well described by ex-tremal dynamics.
Figure 2: The maximum local slope as the critical state is ap-proached is shown directly in the inset. The main figure shows the difference of the maximum slope with its critical value. This Gap asymptotically reaches zero as a power-law, where scal-ing is observed over two decades. The exponent is found to be δ= 0.8(1), which is in good agreement with the expectation for an extremal system using the avalanche distribution exponent in the critical state (see above).
On the other hand, the transient dynamics can be observed in the temporal scaling of the dynamics in the critical state.
Due to the presence of the transient timescale in the dy-namics of the pile, there should accordingly be no generic scaling in the temporal behavior as predicted in [2]. This can be studied directly by determining theqth order corre-lation functions of the temporal fluctuations of the surface coordinates [11] given by
Cq(t) =h|h(τ)−h(t+τ)|qi1/qτ . (D.3) These higher order correlation functions of the surface fluctuations in the critical state are shown in Fig. 3(a), for values ofqranging from 1 to 100. Here, clearly no generic scaling behavior is observed, but the growth exponent, in-dicated by the slope ofCq in the double logarithmic plot, decreases strongly as the momentqincreases. This indi-cates the presence of multi-scaling in the temporal behav-ior of the rice-pile surface, as predicted by Paczuski for general SOC systems [2]. This is due to the presence of
a second time-scale, the scale on which the critical state is reached. This scale also appears in the dynamics of the stationary state and hence leads to multi-scaling. From a
Figure 3: Temporal correlation functions of the rice-pile for dif-ferent momentsqwithq= 1, 2, 3, 5, 10, 20, 50, and 100. The lines shown at increasing height correspond to increasing val-ues of q. (a) Results from the stationary state. The slope in the double-logarithmic plot decreases with increasingq, indicat-ing the presence of multi-scalindicat-ing in time for the rice-pile in the stationary state. (b) Results from the transient state. The double-logarithmic plots are parallel for allq, indicating the presence of generic temporal scaling of the rice-pile in the transient state.
simple argument considering the scaling properties of the qth moment of the surface, Paczuski predicts [2] a gen-eral dependence of the growth exponentβin the stationary state on the momentqas
β(q) =β(q= 1)D+ (q−1)α
qD , (D.4)
whereDis the fractal dimension of the avalanches. Note that in ref. [2], the growth exponent forq = 1is defined
D2. AN EXPERIMENTAL APPROACH TO SELF-ORGANIZED CRITICALITY 57 to be one due to the construction of the model. Therefore,
only theq-dependence of the exponent is derived, and ef-fectively time in the model is related to our experimental time to a certain power (given byβ(q= 1)above) [11].
The fractal dimension of the avalanches in the stationary state of our rice-pile has been determined elsewhere [7] to beD = 1.99(2)from finite size scaling of the avalanche size distributions. Thus with the above determination of the generic roughness exponent, theq-dependence of the growth exponentβcan be obtained without any adjustable parameters. In Fig. 4, the experimental determination of theq-dependence ofβis shown together with the predic-tion of Eq. D.4 for both the stapredic-tionary and the transient states. Here the exponent is determined from a linear fit to the curves in Fig. 3(a) over a time scale of 1 to 60 time steps. As can be seen from the figure, the decrease in
Figure 4: The dependence of the growth exponentβon the mo-ment of the correlation functionq, for both the stationary and the transient state, determined in the region of 1-60 and 1-30 time steps respectively. With increasing moment,βdecreases in the stationary state, indicating multi-scaling. The solid line indicates the expectation for a SOC system in the stationary state (eq. D.4), which is in reasonable agreement with the experimental results.
In the transient state however,βdoes not depend on the moment of the correlation functionq, indicating generic scaling. Again this is expected for a SOC system due to the absence of the crit-ical correlation length in the transient state.
the observedβ is somewhat slower than that predicted by Eq. D.4 based on SOC behavior. However, the overall de-pendence and especially the values ofβobtained at high-qs are in good agreement with the expectation from SOC, whereas a generic critical system, with a constant value of βcan be ruled out. In the transient state on the other hand, the figure clearly indicates a value ofβindependent of the momentq. This is again in good accord with the expecta-tions for a SOC system, where in the transient states only the transient correlation length should be present, hence leading to generic temporal scaling, as is observed in the experiment [11]. The set of correlation functions
corre-sponding to the temporal scaling in the transient state is shown in Fig. 3(b). Here, it can be seen that all of the double-logarithmic curves are parallel, indicating that the exponents, shown in Fig. 4, do not depend on the moment q. Here the exponents are determined from a fit in the range of 1-30 time steps.
Thus from both the experimentally determined approach to the critical state via the maximum slope of the pile as well as the temporal multi-scaling of the surface fluctu-ations, there is good agreement with the theoretical de-scriptions of SOC systems and our rice pile. On the one hand, the exponent of the Gap-equation which is deter-mined experimentally is in agreement [10] with the value determined from the exponents of the critical state [7]. On the other hand, the dependence of the growth exponent on the multi-scaling momentqis in good agreement with the theoretical prediction [11] as well. This indicates for in-stance that indeed, Paczuski’s criterion can be used to dis-tinguish generic critical behavior from SOC, in systems where the transient state is unavailable to the experiment and that the approach to the critical state in our rice pile is governed by extremal dynamics.
∗Division of Physics and Astronomy, Vrije Universiteit, De Boelelaan 1081, 1081HV Amsterdam, The Nether-lands.
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