Self-organized Critical State Research Articles

Signatures of self-organized dynamics in rapidly driven critical sandpiles.

We study two prototypical models of self-organized criticality, namely sandpile automata with deterministic (Bak-Tang-Wiesenfeld) and probabilistic (Manna model) dynamical rules, focusing on the nature of stress fluctuations induced by driving-adding grains during avalanche propagation, and dissipation through avalanches that hit the system boundary. Our analysis of stress evolution time series reveals robust cyclical trends modulated by collective fluctuations with dissipative avalanches. These modulated cycles attain higher harmonics, characterized by multifractal measures within a broad range of timescales. The features of the associated singularity spectra capture the differences in the dynamic rules behind the self-organized critical states at adiabatic driving and their pertinent response to the increased driving rate, which alters the process of stochasticity and causes a loss of avalanche scaling. In sequences of outflow current carried by dissipative avalanches, the first return distributions follow the q-Gaussian law in the adiabatic limit. They appear to follow different laws at an intermediate scale with an increased driving rate, describing different pathways to the gradual loss of cooperative behavior in these two models. The robust appearance of cyclical trends and their multifractal modulation thus represents another remarkable feature of self-organized dynamics beyond the scaling of avalanches. It can also help identify the prominence of self-organizational phenomenology in an empirical time series when underlying interactions and driving modes remain hidden.

Nanoindentation-induced serrated flow behavior and deformation mechanism in metallic glass: A combined experiments and molecular dynamics simulation

In this study, the deformation behaviors of bulk metallic glass (BMG) during the nanoindentation are presented via the experiments and molecular dynamics (MD). The relationship between shear transformation zone (STZ) formation and serrated flow dynamics are developed to explain the deformation characterization evolution of BMG. It is found that as the peak load increases, the hardness and elastic modulus significantly decrease. Creep behavior at room temperature was also revealed by analyzing the creep displacement curve and the stress index under different peak loads.The accumulation of free volume during the deformation process promotes more uniform creep deformation. Furthermore, the serrated flow behaviors were analyzed innovatively using the shear stress drops further statistically. At lower loads, the occurrence of serrated retention is due to energy exceeding the potential barrier required for flow unit activation, ultimately released in the form of kinetic energy. At higher loads, the transition from intermittent curve to smooth curve exhibits a typical self-organized critical (SOC) state dynamic characteristics, and ISE phenomenon is also observed. The results also suggested that the shear deformation of the spherical indenter is more pronounced, with the shear band forming at a 45° angle to the direction of the downward pressure. And based on the Cooperative shear model (CSM) theory, the STZ size was estimated and verify that the increment of STZ makes it easier to generate stronger plastic strain to dissipate STZ, ultimately leading to complete plastic deformation. Combining five fold symmetry and gradient atoms, the majority of atoms with five fold symmetry (LFFS>0.5) have stronger resistance to plastic deformation when the load rate is low.

Ergodic Concepts for a Self‐Organizing Trivalent Spin Network: A Path to (2+1)$(2+1)$‐D Black Hole Entropy

AbstractFrom a dynamical systems point of view, a trivalent spin network model in Loop Quantum Gravity is considered, which presents self‐organized criticality (SOC), arising from a spin propagation dynamics. A partition function is obtained for the domains of stability connecting gauge non‐invariant avalanches, leading to an entropy formula for the asymptotic SOC state. The microscopic origin of this SOC entropy is therefore given by the excitation‐relaxation spin dynamics in the avalanche cycle. The puncturing of trivalent spin networks (TSN) edges participating in the avalanche are counted in terms of an ensemble perimeter over the implicit avalanches. By identifying this perimeter with that of an isolated ‐D black hole horizon, it is conjectured that the SOC entropy reduces to the Bekenstein‐Hawking perimeter‐entropy law for the Bañados, Teitelboim, and Zanelli (BTZ) black hole, by an appropriate adjustment of a potential function based on the thermodynamical formalism of Sinai, Ruelle, and Bowen.

Dynamical complexity in microscale disk-wind systems

Context. Powerful winds at accretion-disk scales have been observed in the past 20 years in many active galactic nuclei (AGN). These are the so-called ultrafast outflows (UFOs). Outflows are intimately related to mass accretion through the conservation of angular momentum, and they are therefore a key ingredient of most accretion disk models around black holes (BHs). At the same time, nuclear winds and outflows can provide the feedback that regulates the joint BH and galaxy growth. Aims. We reconsidered UFO observations in the framework of disk-wind scenarios, both magnetohydrodynamic disk winds and radiatively driven winds. Methods. We studied the statistical properties of observed UFOs from the literature and derived the distribution functions of the ratio ω̄ of the mass-outflow and -inflow rates and the ratio λw of the mass-outflow and the Eddington accretion rates. We studied the links between ω̄ and λw and the Eddington ratio λ = Lbol/LEdd. We derived the typical wind-activity history in our sources by assuming that it can be statistically described by population functions. Results. We find that the distribution functions of ω̄ and λw can be described as power laws above some thresholds, suggesting that there may be many wind subevents for each major wind event in each AGN activity cycle, which is a fractal behavior. We then introduced a simple cellular automaton to investigate how the dynamical properties of an idealized disk-wind system change following the introduction of simple feedback rules. We find that without feedback, the system is overcritical. Conversely, when feedback is present, regardless of whether it is magnetic or radiation driven, the system can be driven toward a self-organized critical state. Conclusions. Our results corroborate the hypothesis that AGN feedback is a necessary key ingredient in disk-wind systems, and following this, in shaping the coevolution of galaxies and supermassive BHs.

The role of intervention mechanisms on a self-organized system: dynamics of a sandpile with site reinforcement

We revisit the sandpile model and examine the effect of introducing site-dependent thresholds that increase over time based on the generated avalanche size. This is inspired by the simplest means of introducing stability into a self-organized system: the locations of collapse are repaired and reinforced. Statistically, for the case of finite driving times, we observe that the site-dependent reinforcements decrease the occurrence of very large avalanches, leading to an effective global stabilization. Interestingly, however, long simulation runs indicate that the system will persist in a state of self-organized criticality (SOC), recovering the power-law distributions with a different exponent as the original sandpile. These results suggest that tipping the heavy-tailed power-laws into more equitable and normal statistics may require unrealistic scales of intervention for real-world systems, and that, in the long run, SOC mechanisms still emerge. This may help explain the robustness of power-law statistics for many complex systems.

Do Successful Researchers Reach the Self-Organized Critical Point?

The index of success of the researchers is now mostly measured using the Hirsch index (h). Our recent precise demonstration, that statistically h∼Nc∼Np, where Np and Nc denote, respectively, the total number of publications and total citations for the researcher, suggests that average number of citations per paper (Nc/Np), and hence h, are statistical numbers (Dunbar numbers) depending on the community or network to which the researcher belongs. We show here, extending our earlier observations, that the indications of success are not reflected by the total citations Nc, rather by the inequalities among citations from publications to publications. Specifically, we show that for highly successful authors, the yearly variations in the Gini index (g, giving the average inequality of citations for the publications) and the Kolkata index (k, giving the fraction of total citations received by the top (1−k) fraction of publications; k=0.80 corresponds to Pareto’s 80/20 law) approach each other to g=k≃0.82, signaling a precursor for the arrival of (or departure from) the self-organized critical (SOC) state of his/her publication statistics. Analyzing the citation statistics (from Google Scholar) of thirty successful scientists throughout their recorded publication history, we find that the g and k for the highly successful among them (mostly Nobel laureates, highest rank Stanford cite-scorers, and a few others) reach and hover just above (and then) below that g=k≃0.82 mark, while for others they remain below that mark. We also find that all the lower (than the SOC mark 0.82) values of k and g fit a linear relationship, k=1/2+cg, with c=0.39, as suggested by an approximate Landau-type expansion of the Lorenz function, and this also indicates k=g≃0.82 for the (extrapolated) SOC precursor mark.

Gyrokinetic analysis of turbulent transport by electromagnetic turbulence in finite β plasmas with weak magnetic shear on HL-2A

Turbulent transport and zonal flow (ZF) dynamics in the mixed kinetic ballooning mode (KBM) and ion temperature gradient (ITG) mode dominated turbulence states are analyzed by gyrokinetic simulations based on the experimental observations of KBMs in the HL-2A Internal Transport Barrier (ITB) plasmas with weak magnetic shear configuration. Results have shown that KBMs and ITGs are the main candidates for turbulent transport and the inclusion of fast ions (FIs) can destabilize both instabilities, which is resulted by the decrease of ballooning parameter αMHD hence reduced Shafranov shift stabilization due to the negative FI density gradient, making them more unstable according to the ŝ-αMHD diagram. In addition, the ITGs are suppressed by both dilution and finite-β stabilization effects caused by FIs. Nonlinear simulations excluding the effects of FIs indicate that the transport is minimum at βe=βeexp as the turbulence predominated by ITGs is strongly suppressed by the nonlinear electromagnetic (EM) stabilization and enhancement of ZFs. However, the transport is further increased with β e although the ZFs become stronger due to the transition from ITG to KBM dominated turbulence state. The presence of FIs can modify the relation between ZF shearing rate and β e. The transport level is insensitive to β e when βe⩽βeexp but increases significantly because of the KBM destabilization. Meanwhile, the ZF amplitude reaches maximum at βe=βeexp whereas it suffers an erosion at higher values, implying that the plasma might be a self-organized critical state owning to the interactions among ZFs, KBMs and FI effects hence setting the plasma β around β exp. The results have also provided a possible explanation that the destabilization of EM turbulence is responsible for the ion heat transport stiffness under weak magnetic shear configurations in the off-axis neutral beam injection heated ITB plasmas on HL-2A.

Relationship between structural heterogeneity and serrated flow in nanoscale phase separated Zr-Cu-Fe-Al bulk metallic glasses

In this work, the relationship between structural heterogeneity and serrated flow in nanoscale phase separated Zr-Cu-Fe-Al bulk metallic glasses with different plasticity was investigated. The results showed that liquid-liquid phase separation would occur during rapid cooling of Zr59(Cu1-xFex)33Al8 (x = 0.4 and 0.5) alloys, resulting in the microstructure formation of amorphous FeZr-rich nanoparticles embedded in the amorphous CuZr-rich matrix. The compression test and statistical analysis indicated that serration dynamics transforms from a chaotic state with a characteristic length scale of stress drop to a self-organized critical (SOC) state with the increase of the atomic ratio of Cu to Fe. This transformation is ascribed to the increase of the number of nucleation sites for shear transformation zones with the increase of structural heterogeneity, which is conducive to the formation of multiple shear bands. This work will be helpful to understand the relationship between the microstructure and deformation behavior.

Ductile Ni-based bulk metallic glasses at room temperature

In this work, a series of new Ni76-xCoxNb4P16B4 (x = 5, 10, 15, in at.%) bulk metallic glasses (BMGs) with the maximum diameter in the range of 1.2–1.6 mm were prepared by combining fluxing and J-quenching technology. The effects of Co content on glass forming ability (GFA), thermal stability, mechanical properties, serrated-flow behaviors and corrosion performance of the present Ni-based BMGs were systematically investigated. The ranges of Tg and ΔTx are 664–668 K and 21–31 K, respectively. Uniaxial compressive tests revealed that the present Ni-based BMGs have the compressive plastic strain (εp), yield strength (δy), and fracture strength (δf) that range from 16% to 27%, 2.1–2.6 GPa, and 2.2–3.6 GPa, respectively. The unprecedented mechanical properties originate from more icosahedral-like clusters (ILCs) and crystal-like clusters (CLCs) in the present Ni-based BMGs. It is found that the serrated-flow behaviors of the present Ni-based BMGs transition self-organized critical state (SOCs) into chaotic state (Cs) as plasticity decreases. The stick-slip dynamics were used to calculate the average shear-band velocity (ASBV), which shows an increasing trend as the Co content rises from 5 at.% to 15 at.%. Electrochemical measurements in 3.5 wt.% NaCl solution showed that the Ni71Co5Nb4P16B4 BMG has a self-corrosion current density of 3.30 × 10−8 A/cm2 and significantly stronger corrosion resistance than austenite 316L and 304L stainless steel. The present Ni-based BMGs have excellent comprehensive performance, which has great potential for industrial applications.

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.

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.

Response of cross-correlations between high PM2.5 and O3 with increasing time scales to the COVID-19: different trends in BTH and PRD.

The air pollution in China currently is characterized by high fine particulate matter (PM2.5) and ozone (O3) concentrations. Compared with single high pollution events, such double high pollution (DHP) events (both PM2.5 and O3 are above the National Ambient Air Quality Standards (NAAQS)) pose a greater threat to public health and environment. In 2020, the outbreak of COVID-19 provided a special time window to further understand the cross-correlation between PM2.5 and O3. Based on this background, a novel detrended cross-correlation analysis (DCCA) based on maximum time series of variable time scales (VM-DCCA) method is established in this paper to compare the cross-correlation between high PM2.5 and O3 in Beijing-Tianjin-Heibei (BTH) and Pearl River Delta (PRD). At first, the results show that PM2.5 decreased while O3 increased in most cities due to the effect of COVID-19, and the increase in O3 is more significant in PRD than in BTH. Secondly, through DCCA, the results show that the PM2.5-O3 DCCA exponents α decrease by an average of 4.40% and 2.35% in BTH and PRD respectively during COVID-19 period compared with non-COVID-19 period. Further, through VM-DCCA, the results show that the PM2.5-O3 VM-DCCA exponents [Formula: see text] in PRD weaken rapidly with the increase of time scales, with decline range of about 23.53% and 22.90% during the non-COVID-19 period and COVID-19 period respectively at 28-h time scale. BTH is completely different. Without significant tendency, its [Formula: see text] is always higher than that in PRD at different time scales. Finally, we explain the above results with the self-organized criticality (SOC) theory. The impact of meteorological conditions and atmospheric oxidation capacity (AOC) variation during the COVID-19 period on SOC state are further discussed. The results show that the characteristics of cross-correlation between high PM2.5 and O3 are the manifestation of the SOC theory of atmospheric system. Relevant conclusions are important for the establishment of regionally targeted PM2.5-O3 DHP coordinated control strategies.

Metaphorical language change is Self-Organized Criticality

Abstract One way to resolve the actuation problem of metaphorical language change is to provide a statistical profile of metaphorical constructions and generative rules with antecedent conditions. Based on arguments from the view of language as complex systems and the dynamic view of metaphor, this paper argues that metaphorical language change qualifies as a Self-Organized Criticality state and the linguistic expressions of a metaphor can be profiled as a fractal with spatio-temporal correlations. Synchronously, these metaphorical expressions self-organize into a self-similar, scale-invariant fractal that follows a power-law distribution; temporally, long range interdependence constrains the self-organization process by the way of transformation rules that are intrinsic of a language system. This argument is verified in the paper with statistical analyses of twelve randomly selected Chinese verb metaphors in a large-scale diachronic corpus.

Shear transformation zones and serrated flow dynamics of metallic glasses revealed by nanoindentation

The relationship between shear transformation zones (STZs) and serrated flow dynamics of Co-, Fe-, Zr-, La- and Al-based metallic glasses (MGs) with different plasticity were analyzed by nanoindentation. The serrated flow dynamic behaviors were analyzed using statistical analysis of shear stress drops, Δτ. The STZ size was estimated via the nanoindentation creep method based on the cooperative shear model. Examination of two distinct types of serrated flow dynamics were performed. For MGs (e.g., Al- or Zr-based) where STZs could be easily activated or at low loading rates, shear stress exhibited a decreasing trend and followed a power-law distribution, which was an indicator of a self-organized critical (SOC) state in the dynamics. In contrast, a Gaussian law distribution was found for STZs in hard MGs (e.g., La-, Co-, and Fe-based) or at high loading rate. In the case of Gaussian distribution case, serrated flow dynamics were in a chaotic state, which indicated that STZ aggregation reduced the stresses required for plastic deformation.

Avalanches of magnetic flux rope in the state of self-organized criticality

ABSTRACT The self-organized criticality (SOC) is a universal theory to explain the ubiquitous power-law size distributions of astrophysical instabilities such as solar eruptions. One way to understand the dynamical processes of an SOC system is through cellular automaton (CA) simulations. Here, we develop a three-dimensional solar CA model that assumes a twisted magnetic flux rope (MFR), in which the avalanche takes place when a local magnetic vector potential exceeds a Gaussian distributed instability criterion, triggered by a global and space-dependent energy driving mechanism. To avoid non-physical released energies, an energy screening mechanism is applied to calculate the avalanche energies of each time-step. Our results show that the statistics of the CA simulated flaring events are comparable to the frequency distributions of observed solar flares originating from an individual active region. Due to the fact of the universality of MFRs, the CA model can be applied to many other astrophysical SOC systems.

Application of the Theory of Self-organized Criticality to the Analysis of the Liberal Agenda in the Press of 1815-1825.

The subject of the research in this paper is the liberal agenda in the Russian press of the pre-Decabrist period. The object of the study is the newspapers published during this period. The novelty of the work lies in the fact that the proposed study searches for pink noise in the data that were obtained from the press of the first quarter of the XIX century. The paper shows that the public consciousness of this period was in a state of self-organized criticality. Previously, the state of self-organized criticality could be found only in systems that arose at the end of the XIX century or later. The difficulty of the problem considered in this paper is that there are almost no mass sources for such an early historical period, and very few of the available ones lend themselves to formalization. The novelty of the conducted research lies in the application of the scientific tool of the theory of self-organized criticality to data having origins in the first quarter of the XIX century. The main conclusion made by the authors of the article is that the public consciousness in the pre-Decabrist period was in a state of self-organized criticality. For the analysis, statistics of publications in newspapers and magazines were collected, which served as a reflection of the liberal agenda relevant to the period of the genesis of the Decembrists. The paper shows that the sequence of publications on liberal information issues in the Russian press in the period 1815-1825 contains pink noise. Fourier analysis was used to determine it in the dynamic series.

Near universal values of social inequality indices in self-organized critical models

We have studied few social inequality measures associated with the sub-critical dynamical features (measured in terms of the avalanche size distributions) of four self-organized critical models while the corresponding systems approach their respective stationary critical states. It has been observed that these inequality measures (specifically the Gini and Kolkata indices) exhibit nearly universal values though the models studied here are widely different, namely the Bak–Tang–Wiesenfeld sandpile, the Manna sandpile and the quenched Edwards–Wilkinson interface, and the fiber bundle interface. These observations suggest that the self-organized critical systems have broad similarity in terms of these inequality measures. A comparison with similar earlier observations in the data of socio-economic systems with unrestricted competitions suggest the emergent inequality as a result of the possible proximity to the self-organized critical states.

Mechanical and tribological behaviors of metallic glass/graphene film with a laminated structure

Metallic glass (MG) film often suffer from high friction and wear, which considerably limit their lifespan in sliding contact service. In this study, graphene (GR) is chosen as an interfacial material to develop a MG/GR multilayer with enhanced mechanical and tribological properties. We report on the fabrication of this multilayer by repeated deposition of Cu50Zr50 MG alternating with transfer of GR. Upon deformation, it is found that the incorporation of GR can lead to extensive shear bands and shear band/interface interactions, which are confirmed by the transition of deformation mode from a chaotic to a self-organized critical state. In comparison with the monolithic MG film, the coefficient of friction and wear rate of the multilayer are decreased by ∼ 35% and 65% respectively, which is attributed to the enhanced mechanical properties and lubricating effect induced by the GR layer. The current work not only broadens our understanding of the positive role of GR in improving the frictional behavior of MG, but opens a new path for designing of novel MG films with a combination of exceptional mechanical and tribological properties.

Early Warning Signals for Critical Transitions in Sandpile Cellular Automata

The sandpile cellular automata, despite the simplicity of their basic rules, are adequate mathematical models of real-world systems, primarily open nonlinear systems capable to self-organize into the critical state. Such systems surround us everywhere. Starting from processes at microscopic distances in the human brain and ending with large-scale water flows in the oceans. The detection of critical transitions precursors in sandpile cellular automata will allow progress significantly in the search for effective early warning signals for critical transitions in complex real systems. The presented paper is devoted to the detection and investigation of such signals based on multifractal analysis of the time series of falls of the cellular automaton cells. We examined cellular automata in square lattice and random graphs using standard and facilitated rules. It has been established that log wavelet leaders cumulant are effective early warning measures of the critical transitions. Common features and differences in the behavior of the log cumulants when cellular automata transit into the self-organized critical state and the self-organized bistability state are also established.

Self-organization of salt marsh patches on mudflats: Field evidence using the UAV technique

The salt marsh system on the tidal flat is one of the most productively ecological wetlands with high biological productivity and blue carbon sequestration levels. The evolution of salt marshes-bare flats boundaries and the shapes of salt marsh margins are of scientific and engineering prospects due to the capability on indicating the morphodynamic patterns of tidal flats. Previous studies are almost qualitative, so far, quantitative analysis on this issue is scarce. To improve our understandings on the spatio-temporal characteristics of salt marsh margins and the mechanism underpinning, we extracted datasets on Spartina alterniflora using orthoimages derived from the quarterly-surveyed UAV photographs and satellite imagery. Results show that the patch-size distribution corresponded to the power law, indicating the marsh patches are in the state of self-organized criticality and the system is scale-independent. It was observed that the external dynamic conditions (i.e., the sediment erosion and accretion on the tidal flat) imposed negative impacts on the self-organization process. In particular, when the short-term morphological changes were negligible, R2 of the power-law relationship notably increased to nearly 1, indicating the salt marsh patches tended to adhere to a high-order self-organization structure in this process.