When observing escape ramps around the world, three designs stand out: the sandpile bed, the gravity escape ramp, and the mechanical arrester ramp.
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The Bak–Tang–Wiesenfeld sandpile was mentioned on the Numb3rs episode "Rampage," as mathematician Charlie Eppes explains to his colleagues a solution to a criminal investigation. The computer game Hexplode is based around the Abelian sandpile model on a finite hexagonal grid where instead of random grain placement, grains are placed by players.
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The sandpile model can be expressed in 1D; however, instead of evolving to its critical state, the 1D sandpile model instead reaches a minimally stable state where every lattice site goes toward the critical slope. For two dimensions, the associated conformal field theory is suggested to be symplectic fermions with central charge c = −2.
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Then, new force chains form until the shear stress is less than the critical value, and so the sandpile maintains a constant angle of repose.
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The identity element of the sandpile group of a rectangular grid. Yellow pixels correspond to vertices carrying three particles, lilac to two particles, green to one, and black to zero. The Abelian sandpile model, also known as the Bak–Tang–Wiesenfeld model, was the first discovered example of a dynamical system displaying self-organized criticality. It was introduced by Per Bak, Chao Tang and Kurt Wiesenfeld in a 1987 paper.
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More than a hundred power-law distributions have been identified in physics (e.g. sandpile avalanches), biology (e.g. species extinction and body mass), and the social sciences (e.g. city sizes and income).
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The Mathematics of Chip-Firing is a textbook in mathematics on chip-firing games and abelian sandpile models. It was written by Caroline Klivans, and published in 2018 by the CRC Press.
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A subset of these states, the so-called critical states, form an abelian group under this addition operation. The abelian sandpile model applies this model to large grid graphs, with the black hole connected to the boundary vertices of the grid; in this formulation, with all eligible vertices selected simultaneously, it can also be interpreted as a cellular automaton. The identity element of the sandpile group often has an unusual fractal structure. The book covers these topics, and is divided into two parts.
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However, it has been argued that this model would actually generate 1/f2 noise rather than 1/f noise. This claim was based on untested scaling assumptions, and a more rigorous analysis showed that sandpile models generally produce 1/fa spectra, with a<2\. Other simulation models were proposed later that could produce true 1/f noise, and experimental sandpile models were observed to yield 1/f noise. In addition to the nonconservative theoretical model mentioned above, other theoretical models for SOC have been based upon information theory , mean field theory , the convergence of random variables , and cluster formation.
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New Brighton is almost exactly halfway between the equator and the north pole, with a latitude of 45 degrees. New Brighton is part of east–central Minnesota's glacial plain sandpile, which was flattened by glaciers during the most recent glacial advance. During the last glacial period, massive ice sheets at least thick ravaged the landscape of the town and sculpted its current terrain which can be easily seen in Long Lake Regional Park within the city. The Wisconsin glaciation left 12,000 years ago.
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This hypothesis also provides for an alternative paradigm to explain power law manifestations that have been attributed to self-organized criticality. There are various mathematical models to create pink noise. Although self-organised criticality has been able to reproduce pink noise in sandpile models, these do not have a Gaussian distribution or other expected statistical qualities. It can be generated on computer, for example, by filtering white noise, inverse Fourier transform, or by multirate variants on standard white noise generation.
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In his early career, he worked on problems in statistical physics, dynamical system and complex systems. In 1987, along with Per Bak and Kurt Wiesenfeld, he proposed the concept and developed the theory for self-organized criticality, which had and continues to have broad applications in complex systems with scale invariance. The model they used to illustrate the idea is referred to as the Bak-Tang-Wiesenfeld "sandpile" model. His current research interest is at the interface between physics and biology.
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Once the sandpile model reaches its critical state there is no correlation between the system's response to a perturbation and the details of a perturbation. Generally this means that dropping another grain of sand onto the pile may cause nothing to happen, or it may cause the entire pile to collapse in a massive slide. The model also displays 1/ƒ noise, a feature common to many complex systems in nature. This model only displays critical behaviour in two or more dimensions.
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While many single phenomena have been shown to exhibit scale-free properties over narrow ranges, a phenomenon offering by far a larger amount of data is solvent-accessible surface areas in globular proteins. These studies quantify the differential geometry of proteins, and resolve many evolutionary puzzles regarding the biological emergence of complexity. Despite the considerable interest and research output generated from the SOC hypothesis, there remains no general agreement with regards to its mechanisms in abstract mathematical form. Bak Tang and Wiesenfeld based their hypothesis on the behavior of their sandpile model.
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Thus, equivalently, the recurrent configurations are exactly those configurations which can be reached from the minimally stable configuration by only adding grains of sand and stabilizing. Not every non-negative stable configuration is recurrent. For example, in every sandpile model on a graph consisting of at least two connected non-sink vertices, every stable configuration where both vertices carry zero grains of sand is non-recurrent. To prove this, first note that the addition of grains of sand can only increase the total number of grains carried by the two vertices together.
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The first of these parts covers the basic theory outlined above, formulating chip-firing in terms of algebraic graph theory and the Laplacian matrix of the given graph. It describes an equivalence between states of the sandpile group and the spanning trees of the graph, and the group action on spanning trees, as well as similar connections to other combinatorial structures, and applications of these connections in algebraic combinatorics. And it studies chip-firing games on other classes of graphs than grids, including random graphs. The second part of the book has four chapters devoted to more advanced topics in chip-firing.
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The original interest behind the model stemmed from the fact that in simulations on lattices, it is attracted to its critical state, at which point the correlation length of the system and the correlation time of the system go to infinity, without any fine tuning of a system parameter. This contrasts with earlier examples of critical phenomena, such as the phase transitions between solid and liquid, or liquid and gas, where the critical point can only be reached by precise tuning (e.g., of temperature). Hence, in the sandpile model we can say that the criticality is self-organized.
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A strongly related model is the so called divisible sandpile model, introduced by Levine and Peres in 2008, in which, instead of a discrete number of particles in each site x, there is a real number s(x) representing the amount of mass on the site. In case such mass is negative, one can understand it as a hole. The topple occurs whenever a site has mass larger than 1; it topples the excess evenly between its neighbors resulting in the situation that, if a site is full at time t, it will be full for all later times.
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This led to a renewal of physiology in the 1980s through the application of chaos theory, for example, in the study of pathological cardiac cycles. In 1987, Per Bak, Chao Tang and Kurt Wiesenfeld published a paper in Physical Review Letters However, the conclusions of this article have been subject to dispute. . See especially: describing for the first time self- organized criticality (SOC), considered one of the mechanisms by which complexity arises in nature. Alongside largely lab-based approaches such as the Bak–Tang–Wiesenfeld sandpile, many other investigations have focused on large-scale natural or social systems that are known (or suspected) to display scale-invariant behavior.
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In 1987, he and two postdoctoral researchers, Chao Tang and Kurt Wiesenfeld, published an article in Physical Review Letters setting a new concept they called self-organized criticality. The first discovered example of a dynamical system displaying such self-organized criticality, the Bak-Tang-Wiesenfeld sandpile model, was named after them. Faced with many skeptics, Bak pursued the implications of his theory at a number of institutions, including the Brookhaven National Laboratory, the Santa Fe Institute, the Niels Bohr Institute in Copenhagen, and Imperial College London, where he became a professor in 2000. In 1996, he took his ideas to a broader audience with his ambitiously entitled book, How Nature Works.
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Ouchy M2 station, Showing the angle iron guide bars, the I-beam roll ways and the bumper posts sandpile, in the Montreal Metro near the Beaugrand Station, showing the unusual inverted U cross-section of the guide bars, precast concrete roll ways and conventional track The rubber-tyred metro systems that incorporate track have angle irons as guide bars, or guiding bars, outside of the two roll ways. The Busan Subway Line 4, that lacks a rail track, has I-beams installed as guide bars. The flanges are vertical. The Sapporo Municipal Subway, that lacks a rail track as well, has no guide bars.
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Chain of transmission of stress forces in a granular mediaCoulomb regarded internal forces between granular particles as a friction process, and proposed the friction law, that the force of friction of solid particles is proportional to the normal pressure between them and the static friction coefficient is greater than the kinetic friction coefficient. He studied the collapse of piles of sand and found empirically two critical angles: the maximal stable angle \theta_m and the minimum angle of repose \theta_r.When the sandpile slope reaches the maximum stable angle, the sand particles on the surface of the pile begin to fall. The process stops when the surface inclination angle is equal to the angle of repose.
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Kurt Wiesenfeld received his Bachelor of Science in Physics from the Massachusetts Institute of Technology in 1979, after which he moved to University of California, Berkeley and received his doctorate in 1985. From 1984 to 1985 he was a Lecturer and Research Scientist at the University of California at Santa Cruz. In 1987, as a post-doctoral research scientist in the Solid State Theory Group of Brookhaven National Laboratory, he and another fellow post-doctoral scientist, Chao Tang, along with their mentor, Per Bak, presented new ideas in group organization with a concept they coined self-organized criticality in their paper in Physical Review Letters. The first discovered example of a dynamical system displaying such self- organized criticality was named after them as the Bak–Tang–Wiesenfeld "sandpile" model.
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The identity element of an abelian sandpile model A chip-firing game, in its most basic form, is a process on an undirected graph, with each vertex of the graph containing some number of chips. At each step, a vertex with more chips than incident edges is selected, and one of its chips is sent to each of its neighbors. If a single vertex is designated as a "black hole", meaning that chips sent to it vanish, then the result of the process is the same no matter what order the other vertices are selected. The stable states of this process are the ones in which no vertex has enough chips to be selected; two stable states can be added by combining their chips and then stabilizing the result.
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Crucially, however, the paper emphasized that the complexity observed emerged in a robust manner that did not depend on finely tuned details of the system: variable parameters in the model could be changed widely without affecting the emergence of critical behavior: hence, self- organized criticality. Thus, the key result of BTW's paper was its discovery of a mechanism by which the emergence of complexity from simple local interactions could be spontaneous--and therefore plausible as a source of natural complexity--rather than something that was only possible in artificial situations in which control parameters are tuned to precise critical values. The publication of this research sparked considerable interest from both theoreticians and experimentalists, producing some of the most cited papers in the scientific literature. Due to BTW's metaphorical visualization of their model as a "sandpile" on which new sand grains were being slowly sprinkled to cause "avalanches", much of the initial experimental work tended to focus on examining real avalanches in granular matter, the most famous and extensive such study probably being the Oslo ricepile experiment.
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