On the critical nature of plastic flow: one and two dimensional models

Steady state plastic flows have been compared to developed turbulence because the two phenomena share the inherent complexity of particle trajectories, the scale free spatial patterns and the power law statistics of fluctuations. The origin of the apparently chaotic and at the same time highly correlated microscopic response in plasticity remains hidden behind conventional engineering models which are based on smooth fitting functions. To regain access to fluctuations, we study in this paper a minimal mesoscopic model whose goal is to elucidate the origin of scale free behavior in plasticity. We limit our description to fcc type crystals and leave out both temperature and rate effects. We provide simple illustrations of the fact that complexity in rate independent athermal plastic flows is due to marginal stability of the underlying elastic system. Our conclusions are based on a reduction of an over-damped visco-elasticity problem for a system with a rugged elastic energy landscape to an integer valued automaton. We start with an overdamped one dimensional model and show that it reproduces the main macroscopic phenomenology of rate independent plastic behavior but falls short of generating self similar structure of fluctuations. We then provide evidence that a two dimensional model is already adequate for describing power law statistics of avalanches and fractal character of dislocation patterning. In addition to capturing experimentally measured critical exponents, the proposed minimal model shows finite size scaling collapse and generates realistic shape functions in the scaling laws.Comment: 72 pages, 40 Figures, International Journal of Engineering Science for the special issue in honor of Victor Berdichevsky, 201

25 Years of Self-Organized Criticality: Numerical Detection Methods

The detection and characterization of self-organized criticality (SOC), in both real and simulated data, has undergone many significant revisions over the past 25 years. The explosive advances in the many numerical methods available for detecting, discriminating, and ultimately testing, SOC have played a critical role in developing our understanding of how systems experience and exhibit SOC. In this article, methods of detecting SOC are reviewed; from correlations to complexity to critical quantities. A description of the basic autocorrelation method leads into a detailed analysis of application-oriented methods developed in the last 25 years. In the second half of this manuscript space-based, time-based and spatial-temporal methods are reviewed and the prevalence of power laws in nature is described, with an emphasis on event detection and characterization. The search for numerical methods to clearly and unambiguously detect SOC in data often leads us outside the comfort zone of our own disciplines - the answers to these questions are often obtained by studying the advances made in other fields of study. In addition, numerical detection methods often provide the optimum link between simulations and experiments in scientific research. We seek to explore this boundary where the rubber meets the road, to review this expanding field of research of numerical detection of SOC systems over the past 25 years, and to iterate forwards so as to provide some foresight and guidance into developing breakthroughs in this subject over the next quarter of a century.Comment: Space Science Review series on SO

25 Years of Self-Organized Criticality: Solar and Astrophysics

Shortly after the seminal paper {\sl "Self-Organized Criticality: An explanation of 1/f noise"} by Bak, Tang, and Wiesenfeld (1987), the idea has been applied to solar physics, in {\sl "Avalanches and the Distribution of Solar Flares"} by Lu and Hamilton (1991). In the following years, an inspiring cross-fertilization from complexity theory to solar and astrophysics took place, where the SOC concept was initially applied to solar flares, stellar flares, and magnetospheric substorms, and later extended to the radiation belt, the heliosphere, lunar craters, the asteroid belt, the Saturn ring, pulsar glitches, soft X-ray repeaters, blazars, black-hole objects, cosmic rays, and boson clouds. The application of SOC concepts has been performed by numerical cellular automaton simulations, by analytical calculations of statistical (powerlaw-like) distributions based on physical scaling laws, and by observational tests of theoretically predicted size distributions and waiting time distributions. Attempts have been undertaken to import physical models into the numerical SOC toy models, such as the discretization of magneto-hydrodynamics (MHD) processes. The novel applications stimulated also vigorous debates about the discrimination between SOC models, SOC-like, and non-SOC processes, such as phase transitions, turbulence, random-walk diffusion, percolation, branching processes, network theory, chaos theory, fractality, multi-scale, and other complexity phenomena. We review SOC studies from the last 25 years and highlight new trends, open questions, and future challenges, as discussed during two recent ISSI workshops on this theme.Comment: 139 pages, 28 figures, Review based on ISSI workshops "Self-Organized Criticality and Turbulence" (2012, 2013, Bern, Switzerland

Deterministic Abelian Sandpile Models and Patterns

In this thesis we want to study the ASM in connection with its capability to produce interesting patterns. it is a surprising example of model that shows the emergence of patterns but maintains the property of being analytically tractable. Then it is qualitatively different from other typical growth models --like Eden model, the diffusion limit aggregation, or the surface deposition -- indeed while in these models the growth of the patterns is confined on the surfaces and the inner structures, once formed, are frozen and do not evolve anymore, in the ASM the patterns formed grow in size but at the same time the internal structures aquire structure, as it has been noted in several papers. There have been several earlier studies of the spatial patterns in sandpile models. The first of them was by Liu et.al. The asymptotic shape of the boundaries of the patterns produced in centrally seeded sandpile model on different periodic backgrounds was discussed in a work of Dhar of 1999. Borgne et.al. obtained bounds on the rate of growth of these boundaries, and later these bounds were improved by Fey et.al. and Levine et.al. An analysis of different periodic structures found in the patterns were first carried out by Ostojic who also first noted the exact quadratic nature of the toppling function within a patch. Wilson et.al. have developed a very efficient algorithm to generate patterns for a large numbers of particles added, which allows them to generate pictures of patterns with N up to 2^26. There are other models, which are related to the Abelian Sandpile Model,e.g., the Internal Diffusion-Limited Aggregation (IDLA), Eulerian walkers (also called the rotor-router model), and the infinitely-divisible sandpile, which also show similar structure. For the IDLA, Gravner and Quastel showed that the asymptotic shape of the growth pattern is related to the classical Stefan problem in hydrodynamics, and determined the exact radius of the pattern with a single point source. Levine and Peres have studied patterns with multiple sources in these models, and proved the existence of a limit shape. Limiting shapes for the non-Abelian sandpile has recently been studied by Fey et.al. The results of our investigation toward a comprehension of the patterns emerging in the ASM are reported along the thesis. In chapter 3 we will introduce some new algebraic operators, $a^\dagger_i$ and $\Pi_i$ in addition to $a_i$, over the space of the sandpile configurations, that will be in the following basic ingredients in the creation of patterns in the sandpile. We derive some Temperley-Lieb like relations they satisfy. At the end of the chapter we show how do they are closely related to multitopplings and which consequences has that relation on the action of $\Pi_i$ on recurrent configurations. In chapter 4 we search for a closed formula to characterize the Identity configuration of the ASM. At this scope we study the ASM on the square lattice, in different geometries, and in a variant with directed edges, the F-lattice or pseudo-Manhattan lattice. Cylinders, through their extra symmetry, allow an easy characterization of the identity which is a homogeneous function. In the directed version, the pseudo-Manhattan lattice, we see a remarkable exact self-similar structure at different sizes, which results in the possibility to give a closed formula for the identity, this work has been published. In chapter 5 we reach the cardinal point of our study, here we present the theory of strings and patches. The regions of a configuration periodic in space, called patches, are the ingredients of pattern formation. In a last paper of Dhar, a condition on the shape of patch interfaces has been established, and proven at a coarse-grained level. We discuss how this result is strengthened by avoiding the coarsening, and describe the emerging fine-level structures, including linear interfaces and rigid domain walls with a residual one-dimensional translational invariance. These structures, that we shall call strings, are macroscopically extended in their periodic direction, while showing thickness in a full range of scales between the microscopic lattice spacing and the macroscopic volume size. We first explore the relations among these objects and then we present full classification of them, which leads to the construction and explanation of a Sierpinski triangular structure, which displays patterns of all the possible patches

Subsampling effects in neuronal avalanche distributions recorded in vivo

Background Many systems in nature are characterized by complex behaviour where large cascades of events, or avalanches, unpredictably alternate with periods of little activity. Snow avalanches are an example. Often the size distribution f(s) of a system's avalanches follows a power law, and the branching parameter sigma, the average number of events triggered by a single preceding event, is unity. A power law for f(s), and sigma=1, are hallmark features of self-organized critical (SOC) systems, and both have been found for neuronal activity in vitro. Therefore, and since SOC systems and neuronal activity both show large variability, long-term stability and memory capabilities, SOC has been proposed to govern neuronal dynamics in vivo. Testing this hypothesis is difficult because neuronal activity is spatially or temporally subsampled, while theories of SOC systems assume full sampling. To close this gap, we investigated how subsampling affects f(s) and sigma by imposing subsampling on three different SOC models. We then compared f(s) and sigma of the subsampled models with those of multielectrode local field potential (LFP) activity recorded in three macaque monkeys performing a short term memory task. Results Neither the LFP nor the subsampled SOC models showed a power law for f(s). Both, f(s) and sigma, depended sensitively on the subsampling geometry and the dynamics of the model. Only one of the SOC models, the Abelian Sandpile Model, exhibited f(s) and sigma similar to those calculated from LFP activity. Conclusions Since subsampling can prevent the observation of the characteristic power law and sigma in SOC systems, misclassifications of critical systems as sub- or supercritical are possible. Nevertheless, the system specific scaling of f(s) and sigma under subsampling conditions may prove useful to select physiologically motivated models of brain function. Models that better reproduce f(s) and sigma calculated from the physiological recordings may be selected over alternatives

Tropical Geometry: new directions

The workshop "Tropical Geometry: New Directions" was devoted to a wide discussion and exchange of ideas between the leading experts representing various points of view on the subject, notably, to new phenomena that have opened themselves in the course of the last 4 years. This includes, in particular, refined enumerative geometry (using positive integer q-numbers instead of positive integer numbers), unexpected appearance of tropical curves in scaling limits of Abelian sandpile models, as well as a significant progress in more traditional areas of tropical research, such as tropical moduli spaces, tropical homology and tropical correspondence theorems

Sandpile-simulation-based graph data model for MVD generative design of shield tunnel lining using information entropy

BIM standard development is central to the performance and behavior of BIM model application across transmission, visualization, and information management perspectives. Tremendous effort has been made to ease the implementation of IFC data model in practice. Yet, the complexity of IFC data model hurdles the implementation of the import and export functionality by software vendors. To overcome this, buildingSMART introduced the concept of Model View Definitions to define which parts of an IFC data model need to be implemented for a specific data exchange scenario. With such, the certification of compatibility for software products with the IFC standard is formed. The Model View Definition is use case orientated to determine whether the specific information should be included in an IFC partial model. With the creation of ad-hoc, project-specific Exchange Requirements increasing, associated MVD development requires much more work to incorporate standard development. To resolve this issue, this paper attempts to exploit the potential of information entropy which has proven itself extremely crucial in many other industries in terms of information management, and then integrates it with sandpile simulation to propose a Top-down hierarchy to structure as well as interpret IFC partial model via Model View Definition. The proposed information entropy shifted MVD development approach would manage to unify the MVD development process that enables the reduction on confusion for various end users, specific organization, or project needs. Moreover, to better translate the BIM standard topology into sandpile simulations, a new notion system is proposed. Sandpile simulations are further implemented to prove their applicability, during the simulation, self-organized criticality is identified, and the existence of chaos is observed

Functional Role of Critical Dynamics in Flexible Visual Information Processing

Recent experimental and theoretical work has established the hypothesis that cortical neurons operate close to a critical state which signifies a phase transition from chaotic to ordered dynamics. Critical dynamics are suggested to optimize several aspects of neuronal information processing. However, although signatures of critical dynamics have been demonstrated in recordings of spontaneously active cortical neurons, little is known about how these dynamics are affected by task-dependent changes in neuronal activity when the cortex is engaged in stimulus processing. In fact, some in vivo investigations of the awake and active cortex report either an absence of signatures of criticality or relatively weak ones. In addition, the functional role of criticality in optimizing computation is often reported in abstract theoretical studies, adopting minimalistic models with homogeneous topology and slowly-driven networks. Consequently, there is a lack of concrete links between information theoretical benefits of the critical state and neuronal networks performing a behaviourally relevant task. In this thesis we explore such concrete links by focusing on the visual system, which needs to meet major computational challenges on a daily basis. Among others, the visual system is responsible for the rapid integration of relevant information from a large number of single channels, and in a flexible manner depending on the behavioral and environmental contexts. We postulate that critical neuronal dynamics in the form of cascades of activity spanning large populations of neurons may support such quick and complex computations. Specifically, we consider two notable examples of well-known phenomena in visual information processing: First the enhancement of object discriminability under selective attention, and second, a feature integration and figure-ground segregation scenario. In the first example, we model the top-down modulation of the activity of visuocortical neurons in order to selectively improve the processing of an attended region in a visual scene. In the second example, we model how neuronal activity may be modulated in a bottom-up fashion by the properties of the visual stimulus itself, which makes it possible to perceive different shapes and objects. We find in both scenarios that the task performance may be improved by employing critical networks. In addition, we suggest that the specific task- or stimulus-dependent modulations of information processing may be optimally supported by the tuning of relevant local neuronal networks towards or away from the critical point. Thus, the relevance of this dissertation is summarized by the following points: We formally extend the existing models of criticality to inhomogeneous systems subject to a strong external drive. We present concrete functional benefits for networks operating near the critical point in well-known experimental paradigms. Importantly, we find emergent critical dynamics only in the parts of the network which are processing the behaviourally relevant information. We suggest that the implied locality of critical dynamics in space and time may help explain why some studies report no signatures of criticality in the active cortex