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Register a new userFinite System-Size Effects in Self-Organized Criticality Systems
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Markus J. Aschwanden
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We explore upper limits for the largest avalanches or catastrophes in nonlinear energy dissipation systems governed by self-organized criticality (SOC). We generalize the idealized straight power low size distribution and Pareto distribution functions in order to accomodate for incomplete sampling, limited instrumental sensitivity, finite system-size effects, Black-Swan and Dragon-King extreme events. Our findings are: (i) Solar flares show no finite system-size limits up to L < 200 Mm, but solar flare durations reveal an upper flare duration limit of < 6 hrs; (ii) Stellar flares observed with KEPLER exhibit inertial ranges of $E approx 10^{34}-10^{37}$ erg, finite system-size ranges at $E approx 10^{37}-10^{38}$ erg, and extreme events at $E =(1-5) times 10^{38}$ erg; (iii) The maximum flare energy of different spectral-type stars (M, K, G, F, A, Giants) reveal a positive correlation with the stellar radius, which indicates a finite system-size limit imposed by the stellar surface area. Fitting our finite system-size models to terrestrial data sets (Earth quakes, wildfires, city sizes, blackouts, terrorism, words, surnames, web-links) yields evidence (in half of the cases) for finite system-size limits and extreme events, which can be modeled with dual power law size distributions.
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Power law size distributions are the hallmarks of nonlinear energy dissipation processes governed by self-organized criticality. Here we analyze 75 data sets of stellar flare size distributions, mostly obtained from the {sl Extreme Ultra-Violet Explorer (EUVE)} and the {sl Kepler} mission. We aim to answer the following questions for size distributions of stellar flares: (i) What are the values and uncertainties of power law slopes? (ii) Do power law slopes vary with time ? (iii) Do power law slopes depend on the stellar spectral type? (iv) Are they compatible with solar flares? (v) Are they consistent with self-organized criticality (SOC) models? We find that the observed size distributions of stellar flare fluences (or energies) exhibit power law slopes of $alpha_E=2.09pm0.24$ for optical data sets observed with Kepler. The observed power law slopes do not show much time variability and do not depend on the stellar spectral type (M, K, G, F, A, Giants). In solar flares we find that background subtraction lowers the uncorrected value of $alpha_E=2.20pm0.22$ to $alpha_E=1.57pm0.19$. Furthermore, most of the stellar flares are temporally not resolved in low-cadence (30 min) Kepler data, which causes an additional bias. Taking these two biases into account, the stellar flare data sets are consistent with the theoretical prediction $N(x) propto x^{-alpha_x}$ of self-organized criticality models, i.e., $alpha_E=1.5$. Thus, accurate power law fits require automated detection of the inertial range and background subtraction, which can be modeled with the generalized Pareto distribution, finite-system size effects, and extreme event outliers.
This paper studies a stylized model of local interaction where agents choose from an ever increasing set of vertically ranked actions, e.g. technologies. The driving forces of the model are infrequent upward shifts (``updates), followed by a rapid process of local imitation (``diffusion). Our main focus is on the regularities displayed by the long-run distribution of diffusion waves and their implication on the performance of the system. By integrating analytical techniques and numerical simulations, we come to the following two main conclusions. (1) If dis-coordination costs are sufficiently high, the system behaves critically, in the sense customarily used in physics. (2) The performance of the system is optimal at the frontier of the critical region. Heuristically, this may be interpreted as an indication that (performance-sensitive) evolutionary forces induce the system to be placed ``at the edge of order and chaos
The original concept of self-organized criticality (Bak et al.~1987), applied to solar flare statistics (Lu and Hamilton 1991), assumed a slow-driven and stationary flaring rate, which warrants time scale separation (between flare durations and inter-flare waiting times), it reproduces power-law distributions for flare peak fluxes and durations, but predicts an exponential waiting time distribution. In contrast to these classical assumptions we observe: (i) multiple energy dissipation episodes during most flares, (ii) violation of the principle of time scale separation, (iii) a fast-driven and non-stationary flaring rate, (iv) a power law distribution for waiting times $Delta t$, with a slope of $alpha_{Delta t} approx 2.0$, as predicted from the universal reciprocality between mean flaring rates and mean waiting times; and (v) pulses with rise times and decay times of the dissipated magnetic free energy on time scales of $12pm6$ min, up to 13 times in long-duration ($lapprox 4$ hrs) flares. These results are inconsistent with coronal long-term energy storage (Rosner and Vaiana 1978), but require photospheric-chromospheric current injections into the corona.
Self-organized criticality (SOC) refers to the ability of complex systems to evolve towards a 2nd-order phase transition at which interactions between system components lead to scale-invariant events beneficial for system performance. For the last two decades, considerable experimental evidence accumulated that the mammalian cortex with its diversity in cell types and connections might exhibit SOC. Here we review experimental findings of isolated, layered cortex preparations to self-organize towards four dynamical motifs identified in the cortex in vivo: up-states, oscillations, neuronal avalanches, and coherence potentials. During up-states, the synchronization observed for nested theta/gamma-oscillations embeds scale-invariant neuronal avalanches that exhibit robust power law scaling in size with a slope of -3/2 and a critical branching parameter of 1. This dynamical coordination, tracked in the local field potential (nLFP) and pyramidal neuron activity using 2-photon imaging, emerges autonomously in superficial layers of organotypic cortex cultures and acute cortex slices, is homeostatically regulated, displays separation of time scales, and reveals unique size vs. quiet time dependencies. A threshold operation identifies coherence potentials; avalanches that in addition maintain the precise time course of propagated synchrony. Avalanches emerge under conditions of external driving. Control parameters are established by the balance of excitation and inhibition (E/I) and the neuromodulator dopamine. This rich dynamical repertoire is not observed in dissociated cortex cultures, which lack cortical layers and exhibit dynamics similar to a 1st-order phase transition. The precise interactions between up-states, nested oscillations, avalanches, and coherence potentials in superficial cortical layers provide compelling evidence for SOC in the brain.
The shape of clouds has proven to be essential for classifying them. Our analysis of images from fair weather cumulus clouds reveals that, besides by turbulence they are driven by self-organized criticality (SOC). Our observations yield exponents that support the fact the clouds, when projected to two dimensions (2D), exhibit conformal symmetry compatible with $c=-2$ conformal field theory (CFT), in contrast to 2D turbulence which has $c=0$ CFT. By using a combination of the Navier-Stokes equation, diffusion equations and a coupled map lattice (CML) we successfully simulated cloud formation, and obtained the same exponents.
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