DOCSLIB.ORG

Dynamics of Sand States Belonging to the Attractor Grows Exponentially with the Size of the Sys­ Tem

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Dynamics of Sand States Belonging to the Attractor Grows Exponentially with the Size of the Sys­ Tem collapse urider small perturbations. The attractor is high dimensional in the sensé that the motion is described by many variables. The number of Dynamics of Sand states belonging to the attractor grows exponentially with the size of the Sys­ tem. The complex dynamics cannot be Per Bak and Michael Creutz thought of as low-dimensional chaotic, or low dimensional anything. The most important property of the self-organized critical state is its resil- Introduction of ail sizes (limited only by the size of iency with respect to modifications of "Sand" represents a "state of matter" the system) following a power-law dis­ the system. Suppose that at some point, to which not much attention has been tribution. The criticality can also be one starts to use wet sand instead of paid. A sand pile can be formed into thought of as a critical chain reaction: dry. For a transient period the ava­ différent shapes; it can exist in many the instability caused by a single fall- lanches will be smaller, but eventually stable states, almost ail of which are ing grain of sand propagates by a the pile organizes itself into a steeper not the flat lowest energy state. Thus, branching process where the branch- state, where again there will be col- sand contains memory; one can write ing probability is precisely balanced by lapses of ail sizes. Also, if one builds letters in sand. If a heap of sand is per- the probability that the activity dies. snow screens locally to prevent slid- turbed, for instance by adding more Starting from an arbitrary config­ ing, the pile will again respond by sand, by tilting the pile, or by shaking uration of sand, the chain reaction is building up steeper states, and the it, the system goes from one metastable generally supercritical or suberitical, large avalanches will résume. The sand state to another. In some sensé, this but eventually the process of adding pile picture provides an intuitive pic- happens by a diffusion process, but this sand modifies the médium to the point ture of the consistency of a stable dy- process is very différent from the pro­ where it is critical. namical attractor, and the existence of cess which relaxes a glass of water to large fluctuations. Without this resil- equilibrium after shaking. The diffu­ iency one would not expect the concept sion process in sand can stop at any to apply to real Systems in nature. of many states, and the process is a threshold process, where nothing hap­ Simulations pens before the perturbation reaches a The convergence to the self-orga­ minimum magnitude. nized critical state can be demonstrated The threshold dynamics of sand is a by computer simulations on toy sand paradigm of many processes in nature. models. In the simplest model1 one de- Earthquakes occur only when the stress fines a "height" variable Z(i,j) on a somewhere on the crust of the Earth two-dimensional lattice. The pile is exceeds a critical value, and the earth- grown by adding sand, Z —» Z + 1 at quake takes the crust from one stable random position, one grain at a time. state to another. Economie Systems are When the height Z at some point ex­ driven by threshold processes: the in- ceeds a critical value Zcr, the pile re­ dividual agents change their behavior laxes by sending one grain of sand to only when certain factors reach a cer­ each of the four neighbors of that tain level. Biological species émerge point, i.e., Z —-* Z - 4, Znn —* Zn„ + 1. or die when spécifie conditions in the (Hère, Znn dénotes the nearest neigh­ ecology are fulfilled. Neurons in a net­ Figure 1. Evolution to the critical bors of Z.) At the edges or corners, only work fire when the input reaches a attractor in the space of metastable states. three or two neighbors, respectively, threshold level, etc. are affected, and sand is lost off the We shall be mostly concerned with a edge. When one or more of the neigh­ situation where the sand pile is built bors becomes critical the process is on a flat surface by randomly adding repeated at those sites, so a chain reac­ one grain at a time. In the beginning, Figure 1 schematically illustrâtes the tion, or avalanche, may take place. If at the particles stay more or less where configuration space of the sand pile some time several sites are supercriti­ they land, but after a while there will and other self-organized critical Sys­ cal, they ail relax simultaneously. In be small avalanches. Eventually the tems. Some states (the dots) represent some ways it may be more realistic to pile reaches a statistically stationary stable configurations of not-too-steep think of the Zs as the local slopes rather state where the amount of sand added piles. The states outside the surface than the local heights. on average balances what falls off the represent unstable states of steep piles. Starting from an empty pile, there edge. When a particle is added to one Starting anywhere, the dynamics will are no tumblings in the beginning. As configuration belonging to this state, a eventually carry the System to the at­ the pile grows (Z increases), there will collapse may occur, and the System tractor inside the stable volume. The be collapses, first small, then bigger ends up in another stable configura­ stable states outside the attractor can and bigger. Eventually, the avalanches tion belonging to the "attractor." We be reached only by carefully placing become large enough that the amount 1 shall argue that the attractor is critical the individual grains (like cards in a of sand added is balanced on average in the sensé that there are avalanches card house), but will catastrophically by the sand falling off the edges; so, MRS BULLETIN/JUNE1991 17 Downloaded from http:/www.cambridge.org/core. Brookhaven National Laboratory, on 21 Oct 2016 at 17:54:38, subject to the Cambridge Core terms of use, available at http:/www.cambridge.org/core/terms. http://dx.doi.org/10.1557/S0883769400056694 Dynamics of Sand (c) (d) Figure 2a-d. Evolving avalanche in a critical sand-pile. the growth has stopped. At this point where b = 1.1 in two dimensions, b » an analytic theory, such as the renor- there are avalanches of ail sizes, and 1.35 in three dimensions. Such power malization group theory for equili- the stationary state is critical. laws are fingerprints of a critical state, brium critical phenomena, so we could Monitoring how many avalanches and b is an example of a critical expo- estimate the exponents, and at the same there are of each size s in the station­ nent. The fractal dimension of, say, the time gain insights into the mechanisms ary state, one finds that the distribu­ set of tumbling sites can be seen as an- of self-organized criticality. We are not tion N(s) is a power law other critical exponent. yet at that point. However, in a remark- 2 N(s) ~ s~b, It would be highly désirable to hâve able séries of papers, Deepak Dhar has 18 MRS BULLETIN/JUNE1991 Downloaded from http:/www.cambridge.org/core. Brookhaven National Laboratory, on 21 Oct 2016 at 17:54:38, subject to the Cambridge Core terms of use, available at http:/www.cambridge.org/core/terms. http://dx.doi.org/10.1557/S0883769400056694 Dynamics of Sand calculated some properties of the sand subtle corrélations. For example, never lanche is started by adding a grain to a pile models, including the number of are two black cells adjacent, nor does site on the boundary, any other site in states belonging to the critical attrac- any site hâve four black neighbors. the System can tumble at most once. tor, and the rate of convergence to the To this configuration a small amount Complementarily, if a single grain of attractor. He introduced a complète al­ of sand was added to a site near the sand starts an avalanche at any site, at gebra, where the individual operator a-, center, triggering an avalanche. We most one tumbling can occur at any represents the effect of an avalanche trace this slide in the following figures given edge site. induced by adding a grain of sand to a by giving the cells which hâve tumbled Most of the exactly known proper­ site i. He exploited the fact that the a muddy red color. Figures 2b and ties of this System follow from the alge­ avalanche algebra is Abelian to calcu- 2c show intermediate active stages in bra of sand création operators studied late the number of operators, and thus the collapse. Yellow sites are still ac­ in Référence 2. Figure 3 shows a par­ the number of states of the attractor. tive. Figure 2d displays the final stable ticularly spécial state. This represents The number of states grows exponen- configuration. the identity operator under this alge­ tially with the number of sites of the This was a particularly large ava­ bra. The state is unique in that if ail its System, so there is an entropy associ- lanche, shown for illustration. In gêner­ ZiS are doubled, i.e., the state is added ated with the dynamics. Also, Dhar ai the ultimate size of the disturbance to itself, then it relaxes exactly back to and co-workers calculated the height- is unpredictable without actually run- itself. This state explicitly exhibits height corrélation function (represent- ning the simulation. Some avalanches structures involving a wide range of ing color corrélations in the critical may involve but a single tumbling, and length scales, showing how fractal pat- state illustrated in Figure 2).
Load more

Recommended publications
  • Seeking Empirical Evidence for Self-Organized Criticality in Open Source Software Evolution
    David R. Cheriton School of Computer Science, University of Waterloo, Ontario, Canada Seeking Empirical Evidence for Self-Organized Criticality in Open Source Software Evolution Jingwei Wu 1 and Richard Holt 2 Software Architecture Group (SWAG) David R. Cheriton School of Computer Science University of Waterloo Waterloo, ON, Canada Abstract We examine eleven open source software systems and present empirical evidence for the existence of fractal structures in software evolution. In our study, fractal structures are measured as power laws through the lifetime of a software system. We describe two specific power law related phenomena: the probability distribution of software changes decreases as a power function of change sizes; and the time series of software change exhibits long range correlations with power law behavior. The existence of such spatial (across the system) and temporal (over the system lifetime) power laws suggests that Self-Organized Criticality (SOC) occurs in the evolution of open source software systems. As a consequence, SOC may be useful as a conceptual framework for understanding software evolution dynamics (the cause and mechanism of change or growth). We give a qualitative explanation of software evolution based on SOC. We also discuss some potential implications of SOC to current software practices. 1 Introduction Laws of software evolution formulated by Lehman [Leh97] represent a major intellectual contribution to understanding software evolution dynamics (the underlying cause and mechanism of software change or growth). Lehman’s eight laws are empirically grounded on observing how closed source industrial software systems such as IBM OS/360 were developed and maintained within a single company using conventional management techniques [LRW+97].
    [Show full text]
  • How Nature Works: the Science of Self-Organized Criticality/ Per Bak
    how nature works c Springer Science+Business Media, LLC PERBAK how nature works © 1996 Springer Science+Business Media New York Originally published by Springer-Verlag New York Inc. in 1996 Softcover reprint of the hardcover 1st edition 1996 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher. All characters except for historical personages are ficticious. Library of Congress Cataloging-in-Publication Data Bak,P. (Per), 194?- How nature works: the science of self-organized criticality/ Per Bak. p. em. Includes bibliographical references and index. ISBN 978-0-387-98738-5 ISBN 978-1-4757-5426-1 (eBook) DOI 10.1007/978-14757-5426-1 r. Critical phenomena (Physics) 2. Complexity (Philosophy) 3· Physics-Philosophy. I. Title. QC173+C74B34 1996 00 31 .7-dc2o Printed on acid-free paper. Designed by N ikita Pristouris 9 8 7 6 5 4 3 2 SPIN 10523741 ISBN 978-o-387-98738-5 Who could ever calculate the path of a molecule? How do we know that the creations of worlds are not determined by falling grains of sand? -Victor Hugo, Les M;serables Contents Preface and Acknowledgments Xl Chapter 1 Complex~ty and Cr~t~cal~ty 1 The Laws of Physics Are Simple, but Nature Is Complex . .3 . Storytelling Versus Science . 7. What Can a Theory of Complexity Explain? . 9 Power Laws and Criticality . .27 . Systems in Balance Are Not Complex . .28 .
    [Show full text]
  • Can We Model Darwin?: Reducing Darwin to a Set of Equations May Never Be P...Le Principles of Evolution - 12 March 1994 - Print Article - New Scientist
    Can we model Darwin?: Reducing Darwin to a set of equations may never be p...le principles of evolution - 12 March 1994 - Print Article - New Scientist HOME |NEWS |EXPLORE BY SUBJECT |LAST WORD |SUBSCRIBE |SEARCH |ARCHIVE |RSS |JOBS Click to Print Can we model Darwin?: Reducing Darwin to a set of equations may never be possible. But a promising computer model shows that mass extinctions could have happened naturally as a consequence of the simple principles of evolution ● 12 March 1994 ● From New Scientist Print Edition. Subscribe and get 4 free issues. ● PER BAK, HENRIK FLYVBJERG and KIM SNEPPEN Physics has been immensely successful in finding mathematical laws that help it describe the Universe. All matter obeys Einstein's equations of general relativity. The way elementary particles interact are described mathematically by the Standard Model. And Schrodinger's equation describes atoms, their nuclei, chemical compounds, crystals and many other states of matter. Even the life and death of stars have well-established formulas. But when it comes to the most complex system, mathematics has so far failed. Life on Earth involves a myriad species interacting with each other in ways that constantly change as they evolve, differentiate and become extinct. There are no 'Darwin's equations' to describe the evolution of life on Earth into complex, interacting ecologies. Scientists accept that Darwin and his followers have convincingly described the principles governing the evolution of life. Darwin's theory of natural selection identifies in a qualitative way the cause of evolutionary change: natural selection operating through a struggle among individual organisms for reproductive success.
    [Show full text]
  • 25 Years of Self-Organized Criticality: Concepts and Controversies
    Space Sci Rev DOI 10.1007/s11214-015-0155-x 25 Years of Self-organized Criticality: Concepts and Controversies Nicholas W. Watkins1,2,3,4 · Gunnar Pruessner5 · Sandra C. Chapman4,6 · Norma B. Crosby7 · Henrik J. Jensen5 Received: 24 January 2015 / Accepted: 4 May 2015 © The Author(s) 2015. This article is published with open access at Springerlink.com Abstract Introduced by the late Per Bak and his colleagues, self-organized criticality (SOC) has been one of the most stimulating concepts to come out of statistical mechanics and con- densed matter theory in the last few decades, and has played a significant role in the de- velopment of complexity science. SOC, and more generally fractals and power laws, have attracted much comment, ranging from the very positive to the polemical. The other pa- pers (Aschwanden et al. in Space Sci. Rev., 2014, this issue; McAteer et al. in Space Sci. Rev., 2015, this issue; Sharma et al. in Space Sci. Rev. 2015, in preparation) in this special issue showcase the considerable body of observations in solar, magnetospheric and fusion plasma inspired by the SOC idea, and expose the fertile role the new paradigm has played in approaches to modeling and understanding multiscale plasma instabilities. This very broad impact, and the necessary process of adapting a scientific hypothesis to the conditions of a given physical system, has meant that SOC as studied in these fields has sometimes dif- B N.W. Watkins [email protected] G. Pruessner [email protected] S.C. Chapman [email protected] N.B.
    [Show full text]
  • SOC Computer Simulations
    Chapter 1 SOC computer simulations Gunnar Pruessner Abstract The following chapter provides an overview of the techniques used to understand Self-Organised Criticality (SOC) by performing computer simulations. Those are of particular significance in SOC, given its very paradigm, the BTW (Bak-Tang-Wiesenfeld) sandpile, was introduced on the basis of a process that is conveniently implemented as a computer program. The chapter is divided into three sections: In the first section a number of key concepts are introduced, followed by four brief presentations of SOC models which are most commonly investigated or which have played an important part in the development of the field as a whole. The second section is concerned with the basics of scaling with particular emphasis of its rˆole in numerical models of SOC, introducing a number of basic tools for data analysis such as binning, moment analysis and error estimation. The third section is devoted to numerical methods and algorithms as applied to SOC models, address- ing typical computational questions with the particular application of SOC in mind. The present chapter is rather technical, but hands-on at the same time, providing practical advice and even code snippets (in C) wherever possible. 1.1 Introduction The concept of Self-Organised Criticality (SOC)1 was introduced by Bak et al. (1987) on the basis of a computer model, the famous BTW Sandpile. The notion of “computer model” and “simulation” used here is subtle and can be misleading. Often the models are not meant to mimic a particular (natural) phenomenon, but are intended to capture merely what is considered to be the essential interaction ob- Gunnar Pruessner arXiv:1301.2918v1 [cond-mat.stat-mech] 14 Jan 2013 Imperial College London Department of Mathematics e-mail: [email protected] 1 A more extensive review on the present subject area can be found in (Pruessner, 2012c).
    [Show full text]
  • Self-Organized Criticality and Pattern Emergence Through the Lens of Tropical Geometry
    Self-Organized Criticality and Pattern Emergence through the lens of Tropical Geometry N. Kalinin,1 A. Guzman-S´ aenz,´ 2 Y. Prieto, 3 M. Shkolnikov, 4 V. Kalinina,5 E. Lupercio 6∗ 1Higher School of Economics, The Laboratory of Game Theory and Decision Making Saint-Petersburg, Russia 2Computational Genomics, IBM T.J. Watson Research Center, Yorktown Heights, NY, USA 3Institut de Mathematiques´ de Toulouse, Universite´ de Toulouse III Paul Sabatier Toulouse, France 4Institute of Science and Technology Austria Klosterneuburg, Austria 5Institute of Cytology, Russian Academy of Science Saint-Petersburg, Russia 6CINVESTAV, Department of Mathematics Campus Zacatenco, CDMX, Mexico ∗To whom correspondence should be addressed; e-mail: [email protected] Tropical Geometry, an established field in pure mathematics, is a place where String Theory, Mirror Symmetry, Computational Algebra, Auction Theory, arXiv:1806.09153v1 [nlin.AO] 24 Jun 2018 etc, meet and influence each other. In this paper, we report on our discov- ery of a tropical model with self-organized criticality (SOC) behavior. Our model is continuous, in contrast to all known models of SOC, and is a cer- tain scaling limit of the sandpile model, the first and archetypical model of SOC. We describe how our model is related to pattern formation and propor- tional growth phenomena, and discuss the dichotomy between continuous and 1 discrete models in several contexts. Our aim in this context is to present an idealized tropical toy-model (cf. Turing reaction-diffusion model), requiring further investigation. Self-Organized Criticality The statistics concerning earthquakes in a particular extended region of the Earth during a given period of time obey a power law known as the Gutenberg-Richter law (1): the logarithm of the energy of an earthquake is a linear function of the logarithm of the frequency of earthquakes of such energy.
    [Show full text]
  • Complexity, Contingency, and Criticality (Macroevolution/Macroeconomics/Punctuated Equilibrium)
    Proc. Natl. Acad. Sci. USA Vol. 92, pp. 6689-6696, July 1995 Colloquium Paper This paper was presented at a colloquium entitled "Physics: The Opening to Complexity," organized by Philip W. Anderson, held June 26 and 27, 1994, at the National Academy of Sciences, in Irvine, CA. Complexity, contingency, and criticality (macroevolution/macroeconomics/punctuated equilibrium) PER BAK AND MAYA PACZUSKI Department of Physics, Brookhaven National Laboratory, Upton, NY 11973 ABSTRACT Complexity originates from the tendency of difficult; history is complicated but perhaps easier. Historical large dynamical systems to organize themselves into a critical events depend on freak accidents, so that if the tape of history state, with avalanches or "punctuations" of all sizes. In the were replayed many times, the outcome would differ vastly critical state, events which would otherwise be uncoupled each time. The mysterious occurrence of incidents leading to become correlated. The apparent, historical corntingency in dramatic outcomes has fascinated historians as well as fiction many sciences, including geology, biology, and economics, writers. Would the Second World War have occurred if Hitler finds a natural interpretation as a self-organized critical had not been born? Historians explain events in a narrative phenomenon. These ideas are discussed in the context of language where event A leads to event B because event C simple mathematical models of sandpiles and biological evo- previously had led to events D and E. But suppose that event lution. Insights are gained not only from numerical simula- C did not happen; then, the course of history would have tions but also from rigorous mathematical analysis.
    [Show full text]
  • TOPICS on ENVIRONMENTAL and PHYSICAL GEODESY Compiled By
    TOPICS ON ENVIRONMENTAL AND PHYSICAL GEODESY Compiled by Jose M. Redondo Dept. Fisica Aplicada UPC, Barcelona Tech. November 2014 Contents 1 Vector calculus identities 1 1.1 Operator notations ........................................... 1 1.1.1 Gradient ............................................ 1 1.1.2 Divergence .......................................... 1 1.1.3 Curl .............................................. 1 1.1.4 Laplacian ........................................... 1 1.1.5 Special notations ....................................... 1 1.2 Properties ............................................... 2 1.2.1 Distributive properties .................................... 2 1.2.2 Product rule for the gradient ................................. 2 1.2.3 Product of a scalar and a vector ................................ 2 1.2.4 Quotient rule ......................................... 2 1.2.5 Chain rule ........................................... 2 1.2.6 Vector dot product ...................................... 2 1.2.7 Vector cross product ..................................... 2 1.3 Second derivatives ........................................... 2 1.3.1 Curl of the gradient ...................................... 2 1.3.2 Divergence of the curl ..................................... 2 1.3.3 Divergence of the gradient .................................. 2 1.3.4 Curl of the curl ........................................ 3 1.4 Summary of important identities ................................... 3 1.4.1 Addition and multiplication .................................
    [Show full text]
  • 25 Years of Self-Organized Criticality: Concepts and Controversies 3 Troversy
    25 Years of Self-Organized Criticality: Concepts and Controversies Nicholas W. Watkins1,2,3,4, Gunnar Pruessner5, Sandra C. Chapman6,7, Norma B. Crosby8, and Henrik J. Jensen9 Abstract Introduced by the late Per Bak and his colleagues, self-organized critical- ity (SOC) has been one of the most stimulating concepts to come out of statistical mechanics and condensed matter theory in the last few decades, and has played a significant role in the development of complexity science. SOC, and more generally fractals and power laws, have attracted much comment, ranging from the very pos- itive to the polemical. The other papers (Aschwanden et al., 2014; McAteer et al., 2014; Sharma et al., 2015) in this special issue showcase the considerable body of observations in solar, magnetospheric and fusion plasma inspired by the SOC idea, and expose the fertile role the new paradigm has played in approaches to modeling and understanding multiscale plasma instabilities. This very broad impact, and the necessary process of adapting a scientific hypothesis to the conditions of a given physical system, has meant that SOC as studied in these fields has sometimes dif- fered significantly from the definition originally given by its creators. In Bak’s own field of theoretical physics there are significant observational and theoretical open questions, even 25 years on (Pruessner, 2012). One aim of the present review is to address the dichotomy between the great reception SOC has received in some areas, and its shortcomings, as they became manifest in the controversies it triggered. Our article tries to clear up what we think are misunderstandings of SOC in fields more remote from its origins in statistical mechanics, condensed matter and dynamical systems by revisiting Bak, Tang and Wiesenfeld’s original papers.
    [Show full text]
  • Downloaded to Major Neurons and Is Thereby Capable of Communicating Its State to the Outside World’ (2006, 396F.)
    1 Review Article for Progress in Biophysics and Molecular Biology, Volume 131, December, 2017, Special Issue: Integral Biomathics: “The Necessary Conjunction of the Western and Eastern Thought Traditions for Exploring the Nature of Mind and Life" (Guest Editors: Plamen L. Simeonov, Arran Gare, Koichiro Matsuno and Abir U. Igamberdiev, pp.61-91. https://doi.org/10.1016/j.pbiomolbio.2017.08.010 Chreods, Homeorhesis and Biofields: FINDING THE RIGHT PATH FOR SCIENCE THROUGH DAOISM1 Arran Gare Abstract: C.H. Waddington’s concepts of ‘chreods’ (canalized paths of development) and ‘homeorhesis’ (the tendency to return to a path), each associated with ‘morphogenetic fields’, were conceived by him as a contribution to complexity theory. Subsequent developments in complexity theory have largely ignored Waddington’s work and efforts to advance it. Waddington explained the development of the concept of chreod as the influence on his work of Alfred North Whitehead’s process philosophy, notably, the concept of concrescence as a self-causing process. Processes were recognized as having their own dynamics, rather than being explicable through their components or external agents. Whitehead recognized the tendency to think only in terms of such ‘substances’ as a bias of European thought, claiming in his own philosophy ‘to approximate more to some strains of Indian, or Chinese, thought, than to western Asiatic, or European, thought.’ Significantly, the theoretical biologist who comes closest to advancing Waddington’s research program, also marginalized,
    [Show full text]
  • Evidence of Self-Organization in Time Series of Capital Markets
    Evidence of self-organization in time series of capital markets Leopoldo Sánchez-Cantú1, Carlos Soto-Campos, Andriy Kryvko *ESIME, Instituto Politécnico Nacional, Ciudad de México, México **Universidad Autónoma del Estado de Hidalgo, Pachuca, México HIGHLIGHTS • We study the dynamics of prices in multiple equity market indices. • The decrease in stock prices (draw-downs) were identified as units of study. • A range of price differentials larger than a critical level conforms to a power law. • This level is interpreted as a phase transition into a self-organized system. ABSTRACT A methodology is developed to identify, as units of study, each decrease in the value of a stock from a given maximum price level. A critical level in the amount of price declines is found to separate a segment operating under a random walk from a segment operating under a power law. This level is interpreted as a point of phase transition into a self-organized system. Evidence of self-organization was found in all the stock market indices studied but in none of the control synthetic random series. Findings partially explain the fractal structure characteristic of financial time series and suggest that price fluctuations adopt two different operating regimes. We propose to identify downward movements larger than the critical level apparently subject to the power law, as self-organized states, and price decreases smaller than the critical level, as a random walk with the Markov property. Keywords: power law; heavy-tailed distribution; self-organization; Self Organized
    [Show full text]
  • Luminous Matter May Arise from a Turbulent Plasma State of the Early
    View metadata, citationLuminous and similar matter papers at core.ac.uk may arise from a turbulent plasma state of the early universe.brought to you by CORE provided by CERN Document Server Per Bak and Maya Paczuski Niels Bohr Institute, Blegdamsvejz 17, Copenhagen,y Denmark. Departmentz of Mathematics, Imperial College, London SW7 2BZ, UK. y (April 4, 2002) The almost perfect uniformity of the cosmic microwave background (CMB) radiation, discovered by Penzias and Wilson in 1965 [1], appears to present clearcut evidence that the universe was uniform and in equilibrium at the decoupling transition when a plasma of protons and electrons condensed into 5 a gas of Hydrogen. COBE indicates that only very small ripples of order 10− existed at decoupling. Gravity then caused hydrogen to cluster and possibly reheat parts of the universe to form the luminous matter that we observe today. We suggest an alternative scenario, where a spatially intermittent structure of extremely hot matter already existed in an otherwise uniform plasma state at the decoupling transition. The plasma was not in equilibrium but in a very high Reynolds number turbulent state. The sparse bursts would not affect the uniformity of the CMB radiation. Luminous matter originates from localized hot bursts already present in the plasma state prior to decoupling. No reheating, and no exotic matter is needed to get luminous matter. The decoupling transition occurred when the temperature of the universe reached Hydrogen’s ionization energy, Tc =4; 200K degrees approximately 300,000 years after the Big Bang. Since radiation interacts with the electrons in the plasma, but not with the Hydrogen, the radiation thereafter simply cooled uniformly, reducing the temperature to the much smaller value T =2:736K that we observe today.
    [Show full text]