Self-organized Criticality Research Articles

Dark matter searches and energy accumulation and release in materials

Efforts to identify dark matter by detecting nuclear recoils produced by dark matter particles reveal low-energy backgrounds of unknown origin in different types of detectors. In many cases, energy accumulation and delayed burst-like releases of stored energy could provide an explanation. These dynamics follow Prigogine’s ideas on systems with energy flow and the general Self-Organized Criticality scenario. We compare these models with properties of excess backgrounds in cryogenic solid-state detectors, relaxation processes in glasses and crystals, our observations of delayed luminescence in NaI(Tl), and make predictions for more phenomena present in these systems and in superconducting photon detectors and qubits. Experiments to create accurate phenomenological models are needed.

Characterization of mesoscopic turbulent transport events with long-radial-range correlation in DIII-D H-mode plasmas

A dimensionless collisionality scan has been performed in H-mode plasmas on DIII-D tokamak, with detailed measurements of intermediate-to-high wavenumber turbulence using Doppler backscattering systems. It is found that the shorter wavelength turbulence develops into spatially asymmetric turbulent structures with a long-radial-range correlation (LRRC) in the mid-radius region of high collisionality discharges. Linear cgyro simulations indicate that the underlying turbulence is likely driven by the electron-temperature-gradient mode. The LRRC transport events are highly intermittent and show a power spectrum of Sñ(k⊥)∝k⊥−1 for density fluctuations, which is often associated with self-organized criticality. The magnitude and the radial scale of those turbulent structures increase significantly when the Er×B mean flow shearing rate decreases at higher collisionality. The enhanced LRRC transport events appear to be correlated with the degraded energy confinement time. The emergence of such LRRC transport events may serve as a candidate explanation for the degrading nature of H-mode core plasma confinement at high collisionality.

Superferromagnetic Sensors

The strong ferromagnetic nanoparticles are analyzed within the band structure-based shell model, accounting for discrete quantum levels of conducting electrons. As is demonstrated, such an approach allows for the description of the observed superparamagnetic features of these nanocrystals. Assemblies of such superparamagnets incorporated into nonmagnetic insulators, semiconductors, or metallic substrates are shown to display ferromagnetic coupling, resulting in a superferromagnetic ordering at sufficiently dense packing. Properties of such metamaterials are investigated by making use of the randomly jumping interacting moments model, accounting for quantum fluctuations induced by the discrete electronic levels and disorder. Employing the mean-field treatment for such superparamagnetic assemblies, we obtain the magnetic state equation, indicating conditions for an unstable behavior. Respectively, magnetic spinodal regions and critical points occur on the magnetic phase diagram of such ensembles. The respective magnetodynamics exhibit jerky behavior expressed as erratic stochastic jumps in magnetic induction curves. At critical points, magnetodynamics displays the features of self-organized criticality. Analyses of magnetic noise correlations are proposed as model-independent analytical tools employed in order to specify, quantify, and analyze the magnetic structure and origin of superferromagnetism. We discuss some results for a sensor-mode application of superferromagnetic reactivity associated with spatially local external fields, e.g., the detection of magnetic particles. The transport of electric charge carriers between superparamagnetic particles is considered tunneling and Landau-level state dynamics. The tunneling magnetoresistance is predicted to grow noticeably with decreasing nanomagnet size. The giant magnetoresistance is determined by the ratio of the respective times of flight and relaxation and can be significant at room temperature. Favorable designs for superferromagnetic systems with sensor implications are revealed.

Some statistics about Tropical Sandpile Model

Tropical sandpile model (or linearized sandpile model) is the only known continuous geometric model exhibiting self-organised criticality. This model represents the scaling limit behavior of a small perturbation of the maximal stable sandpile state on a big subset of $\mathbb Z^2$. Given a set $P$ of points in a compact convex domain $\Omega\subset \mathbb R^2$ this linearized model produces a tropical polynomial $G_P{\bf 0}_\Omega$. Here we present some quantitative statistical characteristics of this model and some speculative explanations. Namely, we study the dependence between the number $n$ of randomly dropped points $P=\{p_1,\dots,p_n\}\subset[0,1]^2=\Omega$ and the degree of the tropical polynomial $G_{P}{\bf 0}_\Omega$. We also study the distributions of the coefficients of $G_{P}{\bf 0}_\Omega$ and the correlation between them. This paper's main (experimental) result is that the tropical curve $C(G_{P}{\bf 0}_\Omega)$ defined by $G_{P}{\bf 0}_\Omega$ is a small perturbation of the standard square grid lines. This explains a previously known fact that most of the edges of the tropical curve $C(G_{P}{\bf 0}_\Omega)$ are of directions $(1,0),(0,1),(1,1),(-1,1)$. The main theoretical result is that $C(G_{P}{\bf 0}_\Omega)\setminus (P\cap \partial\Omega)$, i.e. the tropical curve in $\Omega^\circ$ with marked points $P$ removed, is a tree.

Self-organized cascade collaborative optimization method for associated unit processes

The process manufacturing industry includes multiple continuous unit processes, which cooperate with each other to complete a specific production task. Owing to the coupling between processes, it is often difficult to optimize a unit process to achieve global optimization. Considering the influence of the energy-mass coupling between the associated unit processes and the self-organized motion in the system on the process optimization operation, this study proposes a self-organized cascade collaborative optimization method for associated unit processes. By gradually producing a description of the operating status and coupling relationship of the system, a global optimization model can be constructed. First, reaction efficiency is proposed to evaluate the operating status of the system, and an online estimation model is constructed. Second, self-organized criticality is introduced to describe the evolution of the operating status in the collaborative production unit, and the mechanism knowledge and production data characteristics are combined to construct a coupling relationship model describing coupling and collaboration in the cascade system. Finally, a global optimization method for the system is constructed according to the coupling relationship between the associated unit processes and the cascading characteristic of the cascade system and within the unit process. The experimental results show that the method can realize global optimization of the cascade system, which improves the economic benefits compared with the existing production conditions and ensures stability of the system.

A Game of Life Shifted toward a Critical Point

The Game of Life (GoL), which produces complex patterns of life, has been employed to describe biological systems through self-organized criticality and scale-free properties. This paper develops two novel GoL models. One model allows each cell to update the time for the state update following interactions with other local cells using parameter tuning. Thus, individual cells replace their behaviors from intermittent state updates with continuous ones. The system evolves unpredictably close to a critical point and occasionally close to extinction for the alive population if an adequate parameter is chosen. This event occurs with a power-law tailed time interval and presents synchronous behaviors, since individual cells modify their state-update intervals and create time continuity. The other model is the same except that the system evolves unpredictably without any parameter tuning. In the second model, actions of individual cells are tuned not by a fixed parameter but by the surrounding situation. We found that the GoL system in the second model behaved in a similar manner in the first model, which suggests that that model shifts toward a critical point autonomously.

Synchronization of Alternative Models in a Supermodel and the Learning of Critical Behavior

Abstract “Supermodeling” climate by allowing different models to assimilate data from one another in run time has been shown to give results superior to those of any one model and superior to any weighted average of model outputs. The only free parameters, connection strengths between corresponding variables in each pair of models, are determined using some form of machine learning. It is demonstrated that supermodeling succeeds because near critical states, interscale interactions are important but unresolved processes cannot be effectively represented diagnostically in any single parameterization scheme. In two examples, a pair of toy quasigeostrophic (QG) channel models of the midlatitudes and a pair of ECHAM5 models of the tropical Pacific atmosphere with a common ocean, supermodels dynamically combine parameterization schemes so as to capture criticality, associated critical structures, and the supporting scale interactions. The QG supermodeling scheme extends a previous configuration in which two such models synchronize with intermodel connections only between medium-scale components of the flow; here the connections are trained against a third “real” model. Intermittent blocking patterns characterize the critical behavior thus obtained, even where such patterns are missing in the constituent models. In the ECHAM-based climate supermodel, the corresponding critical structure is the single ITCZ pattern, a pattern that occurs in neither of the constituent models. For supermodels of both types, power spectra indicate enhanced interscale interactions in frequency or energy ranges of physical interest, in agreement with observed data, and supporting a generalized form of the self-organized criticality hypothesis. Significance Statement In a “supermodel” of Earth’s climate, alternative models (climate simulations), which differ in the way they represent processes on the smallest scales, are trained to exchange information as they run, adjusting to one another much as weather prediction models adjust to new observations. They form a consensus, capturing atmospheric behaviors that have eluded all the separate models. We demonstrate that simplified supermodels succeed, where no single approach can, by correctly representing critical phenomena involving sudden qualitative transitions, such as occur in El Niño events, that depend on interactions among atmospheric processes on many different scales in space and time. The correct reproduction of critical phenomena is vital both for predicting weather and for projecting the effects of climate change.

Repeating Fast Radio Bursts Reveal Memory from Minutes to an Hour

Fast radio bursts (FRBs) are brief, luminous pulses with unknown physical origin. The repetition pattern of FRBs contains essential information about their physical nature and emission mechanisms. Using the two largest samples of FRB 20121102 and FRB 20201124A, we report that the sources of the two FRBs reveal memory over a large range of timescales, from a few minutes to about an hour. The memory is detected from the coherent growths in burst-rate structures and the Hurst exponent. The waiting time distribution displays an approximate power-law tail, which is consistent with a Poisson model with a time-varying rate. From cellular automaton simulations, we find that these characteristics can be well understood within the physical framework of a self-organized criticality system driven in a correlation way, such as random walk functions. These properties indicate that the triggers of bursts are correlated, preferring the crustal failure mechanism of neutron stars.

Autonomic Nervous System Influences on Cardiovascular Self-Organized Criticality.

Cardiovascular self-organized criticality has recently been demonstrated. We studied a model of autonomic nervous system changes to better characterize heart rate variability self-organized criticality. The model included short and long-term autonomic changes associated with body position and physical training, respectively. Twelve professional soccer players took part in a 5-week training session divided into "Warm-up", "Intensive", and "Tapering" periods. A stand test was carried out at the beginning and end of each period. Heart rate variability was recorded beat by beat (Polar Team 2). Bradycardias, defined as successive heart rates with a decreasing value, were counted according to their length in number of heartbeat intervals. We checked whether bradycardias were distributed according to Zipf's law, a feature of self-organized criticality. Zipf's law draws a straight line when the rank of occurrence is plotted against the frequency of occurrence in a log-log graph. Bradycardias were distributed according to Zipf's law, regardless of body position or training. Bradycardias were much longer in the standing position than the supine position and Zipf's law was broken after a delay of four heartbeat intervals. Zipf's law could also be broken in some subjects with curved long bradycardia distributions by training. Zipf's law confirms the self-organized nature of heart rate variability and is strongly linked to autonomic standing adjustment. However, Zipf's law could be broken, the significance of which remains unclear.

Anomalous Self-Organization in Active Piles.

Inspired by recent observations on active self-organized critical (SOC) systems, we designed an active pile (or ant pile) model with two ingredients: beyond-threshold toppling and under-threshold active motions. By including the latter component, we were able to replace the typical power-law distribution for geometric observables with a stretched exponential fat-tailed distribution, where the exponent and decay rate are dependent on the activity's strength (ζ). This observation helped us to uncover a hidden connection between active SOC systems and α-stable Levy systems. We demonstrate that one can partially sweep α-stable Levy distributions by changing ζ. The system undergoes a crossover towards Bak-Tang-Weisenfeld (BTW) sandpiles with a power-law behavior (SOC fixed point) below a crossover point ζ<ζ*≈0.1.

Self-Organized Criticality and Cognitive Control Reasoned by Effort Minimization

We put forward a novel model for self-organized criticality in the dynamics of systems controlled by human actions. The model is based on two premises. First, without human control, the system in issue undergoes supercritical instability. Second, the subject’s actions are aimed at preventing the occurrence of critical fluctuations when the risk of control failure becomes essential rather than keeping the system in the stability region. The latter premise is reasoned as follows: (i) keeping the system rather far from the instability boundary is not justified from the standpoint of effort minimization, and (ii) keeping it in the immediate proximity to the instability onset also requires considerable effort because of the bounded capacity of human cognition. The concept of dynamical traps is used in the mathematical description of this type of subject’s behavior. Numerical simulation demonstrates that the proposed model does predict the emergence of fluctuations with the power-law distribution. In conclusion, we discuss that the self-organized criticality of social systems is possible due to the basic features of the human mind.

Space–Time Structures in the Auroral Oval: Approaches to Modeling

The interaction of the magnetospheric–ionospheric (MI) system surrounding the Earth with the environment (solar wind) occurs in the form of a series of transient processes at different scales. The largest of them, magnetic storms, are obviously triggered by disturbances in the solar wind (direct driving). The role of the internal dynamics of the MI system, which is caused to a large extent by the nonlinearity and temporal delays of the loading–unloading processes of energy and particle from the solar wind into the magnetosphere, becomes more significant at smaller scales (substorms, pseudobreakups, injections, and activations). A typical dynamic state of the MI system is characterized as self-organized criticality or turbulence, which are characterized by statistical scale invariance (scaling) in the fluctuation distributions of many characteristics. The dynamics of the MI system is projected into the region of the auroral oval, the very existence of which is due to this dynamics. The space–time structure of auroral disturbances largely reflects the structure of processes in the MI plasma. The description of this structure is important both for studying the fundamental study of plasma processes and for many topical applied problems related to the propagation of radio waves in the ionosphere and vital activity at high latitudes. The paper discusses approaches and developments for constructing a model of the space–time structure of the auroral oval, based on fractal and multifractal characteristics.

On the Possibility of Reproducing Utsu's Law for Earthquakes with a Spring-Block SOC Model.

The Olami, Feder and Christensen (OFC) spring-block model has proven to be a powerful tool for analyzing and comparing synthetic and real earthquakes. This work proposes the possible reproduction of Utsu's law for earthquakes in the OFC model. Based on our previous works, several simulations characterizing real seismic regions were performed. We located the maximum earthquake in these regions and applied Utsu's formulae to identify a possible aftershock area and made comparisons between synthetic and real earthquakes. The research compares several equations to calculate the aftershock area and proposes a new one with the available data. Subsequently, the team performed new simulations and chose a mainshock to analyze the behavior of the surrounding events, so as to identify whether they could be catalogued as aftershocks and relate them to the aftershock area previously determined using the formula proposed. Additionally, the spatial location of those events was considered in order to classify them as aftershocks. Finally, we plot the epicenters of the mainshock, and the possible aftershocks comprised in the calculated area resembling the original work of Utsu. Having analyzed the results, it is likely to say that Utsu's law is reproducible using a spring-block model with a self-organized criticality (SOC) model.

Fog Intermittency and Critical Behavior

The intermittency of fog occurrence (the switching between fog and no-fog) is a key stochastic feature that plays a role in its duration and the amount of moisture available. Here, fog intermittency is studied by using the visibility time series collected during the month of July 2022 on Sable Island, Canada. In addition to the visibility, time series of air relative humidity and turbulent kinetic energy, putative variables akin to the formation and breakup conditions of fog, respectively, are also analyzed in the same framework to establish links between fog intermittency and the underlying atmospheric variables. Intermittency in the time series is quantified with their binary telegraph approximations to isolate clustering behavior from amplitude variations. It is shown that relative humidity and turbulent kinetic energy bound many stochastic features of visibility, including its spectral exponent, clustering exponent, and the growth of its block entropy slope. Although not diagnostic, the visibility time series displays features consistent with Pomeau–Manneville Type-III intermittency in its quiescent phase duration PDF scaling (−3/2), power spectrum scaling (−1/2), and signal amplitude PDF scaling (−2). The binary fog time series exhibits properties of self-organized criticality in the relation between its power spectrum scaling and quiescent phase duration distribution.

Accelerated Tribo-Films Formation in Complex Adaptive Surface-Engineered Systems under the Extreme Tribological Conditions of Ultra-High-Performance Machining

This study investigates accelerated physical–chemical processes in a complex adaptive surface-engineered system represented by a nano-multilayer TiAlCrSiYN/TiAlCrN PVD coating under the extreme tribological conditions of ultra-high-performance dry machining of hardened H 13 tool steel. These processes are similar to the different catalyzing phenomena. Experimental results of tool life vs. wear rate, SEM/TEM data of the worn surfaces, XPS and EDS data of tribo-films formed on the friction surfaces, and chip surface morphology are presented in this study. The corresponding relationships between self-organization, self-organized criticality, and various catalyzing phenomena were evaluated on the basis of the accrued data. A method of enhancing these processes through the variation of machining conditions is also outlined, which resulted in the improvement of coated tool life by 35%.

Seismic risk assessment of a large metropolitan area by means of simulated earthquakes

The determination of seismic risk in urban settlements has received increasing attention in the scientific community during the last decades since it allows to identify the most vulnerable portions of urban areas and therefore to plan appropriate strategies for seismic risk reduction. In order to accurately evaluate the seismic risk of urban settlements it should be necessary to estimate in detail the seismic vulnerability of all the existing buildings in the considered area. This task could be very cumbersome due to both the great number of information needed to accurately characterize each building and the huge related computational effort. Several simplified methods for the assessment of the seismic vulnerability of existing buildings have been therefore presented in the literature. In order to estimate the occurrence of damage in buildings due to possible seismic phenomena, the published studies usually refer to response spectra evaluated according to seismic events expected in the territory with assumed probabilities. In the present paper seismic events are instead simulated using a modified Olami–Feder–Christensen (OFC) model, within the framework of self-organized criticality. The proposed methodology takes into account some geological parameters in the evaluation of the seismic intensities perceived by each single building, extending the approach presented in a previous study of some of the authors. Here, a large territory in the Sicilian oriental coast, the metropolitan area of Catania, which includes several urbanized zones with different features, has been considered as a new case study. Applications of the procedure are presented first with reference to seismic sequences of variable intensity, whose occurrence is rather frequent in seismic territories, showing that the damage can be progressively accumulated in the buildings and may lead to their collapse even when the intensities of each single event are moderate. Moreover, statistically significant simulations of single major seismic events, equivalent to a given sequence in terms of produced damages on buildings, are also performed. The latter match well with a novel a-priori risk index, introduced with the aim of characterizing the seismic risk of each single municipality in the considered metropolitan area. The proposed procedure can be applied to any large urbanized territory and, allowing to identify the most vulnerable areas, can represent a useful tool to prioritize the allocation of funds. This could be a novelty for risk policies in many countries in which public subsidies are currently assigned on a case-by-case basis, taking into account only hazard and vulnerability. The use of an a-priori risk index in the allocation process will allow to take into due account the relevant role of exposure.

Evolving cycles and self-organised criticality in social dynamics

In many complex systems, self-organised criticality (SOC) provides a mechanism for the diversity of spatiotemporal scales that optimises the system’s response to omnipresent driving forces. Signatures of SOC are increasingly more evidenced in collective social behaviours. However, the spontaneous occurrence of critical states and their role in maintaining the system’s functional properties still need to be better understood; the reason can be related to the complexity of human interactions and the ubiquitous presence of cycles in social dynamics. In this work, we shed new light on these issues based on a critical survey and the extensive data analysis of online social dynamics. Firstly, we highlight prominent features of human activity patterns, conditioned by circadian cycles and content-related interactions, that can affect the course of the dynamics from the elemental to the global scale. We then analyse the prototypal time series of emotion-driven communications in the online social network MySpace to demonstrate the coexistence of SOC states with the modulated cyclical trends. Precisely, we determine avalanches of emotional comments exhibiting multifractal scaling, scale-invariant inter-avalanching behaviours and temporal correlations coexist with the cyclical trends of broad singularity spectra. We demonstrate that similar multi-harmonic cycles occur in entirely different datasets, particularly the negative emotion-driven Diggs and the infection-rate data from recent epidemics. Our results reveal the dynamical regime where the modulated cycles coexist with self-organised critical states; in contrast, in the cycles-dominated regime, exemplified by the infection time series, the nature of collective dynamics remains hidden behind the cycle modulation.

Critical Drift in a Neuro-Inspired Adaptive Network.

It has been postulated that the brain operates in a self-organized critical state that brings multiple benefits, such as optimal sensitivity to input. Thus far, self-organized criticality has typically been depicted as a one-dimensional process, where one parameter is tuned to a critical value. However, the number of adjustable parameters in the brain is vast, and hence critical states can be expected to occupy a high-dimensional manifold inside a high-dimensional parameter space. Here, we show that adaptation rules inspired by homeostatic plasticity drive a neuro-inspired network to drift on a critical manifold, where the system is poised between inactivity and persistent activity. During the drift, global network parameters continue to change while the system remains at criticality.

Space–Time Structures in the Auroral Oval: Approaches to Modeling

The interaction of the magnetospheric–ionospheric (MI) system surrounding the Earth with the environment (solar wind) occurs in the form of a series of transient processes at different scales. The largest of them, magnetic storms, are obviously triggered by disturbances in the solar wind (direct driving). The role of the internal dynamics of the MI system, which is caused to a large extent by the nonlinearity and temporal delays of the loading–unloading processes of energy and particle from the solar wind into the magnetosphere, becomes more significant at smaller scales (substorms, pseudobreakups, injections, and activations). A typical dynamic state of the MI system is characterized as self-organized criticality or turbulence, which are characterized by statistical scale invariance (scaling) in the fluctuation distributions of many characteristics. The dynamics of the MI system is projected into the region of the auroral oval, the very existence of which is due to this dynamics. The space–time structure of auroral disturbances largely reflects the structure of processes in the MI plasma. The description of this structure is important both for studying the fundamental study of plasma processes and for many topical applied problems related to the propagation of radio waves in the ionosphere and vital activity at high latitudes. The paper discusses approaches and developments for constructing a model of the space–time structure of the auroral oval, based on fractal and multifractal characteristics.

Sandpile Universality in Social Inequality: Gini and Kolkata Measures.

Social inequalities are ubiquitous and evolve towards a universal limit. Herein, we extensively review the values of inequality measures, namely the Gini (g) index and the Kolkata (k) index, two standard measures of inequality used in the analysis of various social sectors through data analysis. The Kolkata index, denoted as k, indicates the proportion of the 'wealth' owned by (1-k) fraction of the 'people'. Our findings suggest that both the Gini index and the Kolkata index tend to converge to similar values (around g=k≈0.87, starting from the point of perfect equality, where g=0 and k=0.5) as competition increases in different social institutions, such as markets, movies, elections, universities, prize winning, battle fields, sports (Olympics), etc., under conditions of unrestricted competition (no social welfare or support mechanism). In this review, we present the concept of a generalized form of Pareto's 80/20 law (k=0.80), where the coincidence of inequality indices is observed. The observation of this coincidence is consistent with the precursor values of the g and k indices for the self-organized critical (SOC) state in self-tuned physical systems such as sand piles. These results provide quantitative support for the view that interacting socioeconomic systems can be understood within the framework of SOC, which has been hypothesized for many years. These findings suggest that the SOC model can be extended to capture the dynamics of complex socioeconomic systems and help us better understand their behavior.