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). others collapse most of the System. terns naturally arise in this System. Fi- Figures 2-4 show sand configura­ Note in Figure 2c the appearance of nally we show in Figure 4 the resuit of tions resulting from some simple ex- islands which hâve not yet tumbled. collapsing a column of 32,768 grains of periments with this model. Hère the Thèse hâve ail disappeared in Fig­ sand on a single site of an empty table. System is 190 by 178 sites, and the ure 2d, the final relaxed state. This is an Could that illustrate the origin of a threshold is chosen to be 3; so, a stable exact resuit, spécial to this model; once fractal universe from a "big bang?" configuration has Zx = 0,1, 2, or 3. We in the critical ensemble, it is impossible dénote thèse levels by black, red, blue, to trigger a set of avalanches which will Experiments or green, respectively. We use various leave an isolated island of unaltered Do the above dynamics really de- shades of yellow for unstable cells. The cells surrounded by disturbed ground. scribe sand? Of course not! The dis­ black boundary dénotes open edges. This further emphasizes that the ap- crète cellular automata dynamics are Thèse configurations were ail gener- parently random configuration in Fig­ not a realistic substitute for Newton's ated interactively on a Commodore ure 2a is, in fact, not. équation, gravity, friction, and other Amiga. Figure 2a shows a configura­ The model has a remarkable number spécifie materials properties of any- tion typical of the critical ensemble. At of exact results. For example, if an ava­ thing. Nevertheless, we hope that the first glance, the System appears quite random. There are, however, some

Figure 3. The "identity state" in the avalanche algebra. Figure 4. Final stable configuration from collapsing a column of height 32,768 at a single site.

MRS BULLETIN/JUNE1991 19 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

concept of criticality survives for the following reasons: First, the models studied are robust with respect to modifications. For in­ stance, one may remove a fraction of the bonds Connecting the individual sites, so that at some sites only three or two grains of sand are falling. This might represent the effect of "snow- screens" on the avalanches. After a transient period the system has been organized into a new critical state. The pile becomes steeper, but the large fluctuations remain. One may also let the critical height be a random vari­ able, or send a random non-integer amount of sand to the nearest neigh- bors. One may even study determinis- tic models with no randomness at ail in the évolution; instead of adding a grain of sand at a random position, one may hâve the heights Z be real numbers growing at a small uniform rate every- where. Starting from a random configu­ ration of heights, the usual dynamics apply and eventually the criticality con­ Figure 5. The IBM sand pile experiment. Sand was dropped from a rotating funnel to a dition is fulfilled. None of this affects circutar plate on a balance. The motor rotating the funnel, and the balance were controlled the criticality. and monitored by a PC. A power-law distribution of avalanches ivas obtained for small Second, one lesson from equilibrium plates once the System reached a stationary state. critical phenomena is that universal- ity may apply. The long-time, long- distance scaling behavior does not dépend on microscopic mechanisms, but only on symmetry and spatial di- served that the distribution of energy balances the increase of stress from the mensionality. This allows us to calcu- £ released during earthquakes follows tectonic plate motion. Simple models late the correct exponents by studying a power law of blocks connected with springs, slid- simple Ising models with the same sym­ ing on a rough surface can reproduce metry. We hope that sand models N(£) ~ £"b the Gutenberg Richter law. There may serve as "Ising" models for self- appears to be nothing spécial with organized criticality in real Systems. The magnitude, m, of earthquakes large earthquakes: they are simply A few experiments on real sand hâve on the Richter scale is roughly the loga- large versions of small earthquakes. If been reported.6,7 The expérimental rithm of the energy release. There is we had time to wait a couple of mil­ setup by Glenn Held's group at IBM is some ambiguity as to how to relate the lion years, there is no reason that we shown in Figure 5. The distribution of energy to any real observables for would not observe earthquakes of size avalanches of sand dropping off a sand earthquakes, but very recently Kagan8 10 or 11 following the same Gutenberg pile mounted on a balance was mea­ suggests from an analysis of the Har­ Richter law. sured. They measured a powerlaw dis­ vard earthquake catalogue that the The situation may not be very différ­ tribution similar to that found for the value b = 0.5-0.6 applies to earthquakes ent from what happens when a métal sand models. Also, the cutoff of ava­ everywhere. The Gutenberg Richter rod is slowly squeezed. In the begin- lanche size versus the size of the plate law appears to be valid over six to seven ning the déformations are elastic, but obeyed a finite-size scaling, another décades, more than for any measured at some point there will be irréversible indication of criticality. However, large equilibrium critical behavior. slips, caused, for instance, by disloca­ sand piles appear to behave differ- This observation can be interpreted tion motion. Eventually, the métal rod ently; the piles would oscillate be- as a conséquence of self-organized enters a plastic phase where further tween steep states and more shallow criticality.9 In active seismic régions, déformations occur at a constant pres­ states. This behavior was also observed tectonic plates grind into each other. sure. It would be interesting to monitor by Jaeger's group at the University of Most of the time the plates are at rest, the magnitude of slips in the station­ ary plastic phase. Bobrov and Lebyod- Chicago. It has been suggested that but now and then, as the pressure in- 15 this is due to the snowball effect caused creases, a rupture occurs somewhere. kin of the Institute of Solid State by ever accelerating grains of sand. This initial instability may propagate Physics, Chernogolovka, Moscow, hâve analogously to the avalanches in the reported some initial findings of such Earthquakes and the sand pile model. We assume that the "earthquakes" in aluminum and Elastic-to-Plastic Transition plates hâve been slowly pumped up to niobium rods, but it is not yet clear in Metals the critical stationary state where the whether or not they follow a power law distribution. In 1956 Gutenberg and Richter ob­ release of stress during earthquakes

20 MRS BULLETIN/JUNE 1991 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

Other Applications Références 44 (3) (1991) p. 9. In materials science we usually deal 1. P. Bak, C. Tang, and K. Wiesenfeld, Phys. 15. W. S. Bobrov and M. Lebyodkin, private with large interactive dynamical Sys­ Rev. Letf. 59 (1987) p. 381; Phys. Rev. A 38 communication. D tems, so we might frequently expect to (1988) p. 364; P. Bak and K. Chen, Sci. Am. 264 (1) (1991) p. 48. Per Bak is a senior scientist at Brookhaven encounter the kind of behavior de- 10 2. D. Dhar and R. Ramaswamy, Phys. Rev. National Laboratory, where he is studying scribed hère. Babcock and Westervelt Lett. 63 (1989) p. 1659; D. Dhar, Phys. Rev. self-organized criticality in physics, biology, hâve reported self-organized critical Lett. 64 (1990) p. 1613; D. Dhar and S. N. and économies. In 1974 he received his PhD avalanches in magnetic domain pat- Majumdar, /. Phys. A 23 (1990) p. 4333. in physics from the Technical University of terns of magnetic films. Che and Suhl" 3. B. Mandelbrot, The Fractal Geometry of Na­ Denmark. Working for Nordita Denmark, suggest that noise associated with ture (W.H. Freeman, San Francisco, 1982). the IBM T.). Watson Research Center and magnetic tape is a self-organized criti­ 4. P. Bak and K. Chen, Physica D 38 (1989) the University of Copenhagen, he pursued p. 5. cal phenomenon. his interests in statistical physics, dynami­ Some of the more interesting ap­ 5. M. Creutz, Computers in Physics (1991), in press. cal Systems (chaos), low-dimensional con- plications may be outside materials 6. G.A. Held, D.H. Solina, D.T. Keane, W.J. ductors and quasicrystals. He is a fellow of science. Perhaps économie Systems op- Haag, P.M. Horn, and G. Grinstein, Phys. the AAAS, member of the Danish Academy erate at the critical point and ob- Rev Lett. 65 (1990) p. 1120. of Science, and member of the editorial served fluctuations are an unavoidable 7. H.M. Jaeger, C. Liu, and S.R. Nagel, Phys. boards of Journal of Statistical Physics conséquence of criticality. Kauffman Rev. Lett. 69 (1989) p. 40. and Physica A. and Johnson13 hâve suggested that mass 8. Y.Y. Kagan, "Seismic Moment Distribu­ extinction in biology can be described tion," preprint. Michael Creutz, senior physicist at Brook­ as the conséquence of co-evolutionary 9. P. Bak and C. Tang, /. Ceophys. Res. B 94 haven National Laboratory, is a theoretical avalanches in ecological Systems. Thus, (1989) p. 15,635; K. Ito and M. Matsuzaki, /. physicist specializing in computational Ceophys. Res. B 95 (1989) p. 6853; A. Sornette no external cataclysmic mechanism and D. Sornette, Europhys. Lett. 9 (1989) methods for lattice gauge theory and statis­ may be needed in order to explain the p. 192; K. Chen, P. Bak, and S.A. Obukhov, tical Systems. He has been at Brookhaven extinction of the dinosaurs, or the col- Phys. Reu A 43 (1991) p. 625. since 1972 and served as leader of the High lapses on Wall Street in 1929 and 1987. 10. K.L. Babcock and R.M. Westervelt, Phys. Energy Theory Group from 1984 to 1987. It has even been hinted that the scaling Rev. Lett. 64 (1989) p. 2168. He received his undergraduate degree from behavior of vortices in turbulent Sys­ 11. X. Che and H. Suhl, Phys. Reu. Lett. 64 Caltech in 1966, and his PhD from Stanford tems is a self-organized critical phe­ (1989) p. 1670. University in 1970. He has served on the nomenon.14 But ail this is yet at a very 12. P. Bak, K. Chen, J. Scheinkman, and M. High Energy Physics Advisory Panel to the preliminary state. Woodford (1991), preprint. Department of Energy, the Program Advi­ 13. S. A. Kauffman and S. Johnson, "Co- evolution to the Edge of Chaos: Coupled sory Committèe for the Division of Advanced Acknowledgments Fitness Landscapes, Poised States, and Co- Scientific Computing of the National Sci­ Supported by the U.S. Department of evolutionary Avalanches" (1991), preprint. ence Foundation, and the Editorial Boardfor Energy under contract DE-AC02-76- 14. P. Bak, K. Chen, and C. Tang, Phys. Lett. Physical Review D. Creutz is a fellow of CH00016. 147 (1990) p. 297; L.P. Kadanoff, Phys. Today the Americal Physical Society.

Attention Workshop on Thin Films MRS Members in Japan for the 21st Century MRS members in Japan may now order MRS books from July28-August2,1991 AIMS Corporation in fapan at MRS member prices. MRS Northwestern University members in Japan may still order the books directly from Key scientists, technologists, educators, and industrie and govem- MRS in the United States or take advantage of the speed ment leaders will gather to ask where thin filmtechnolog y will be 10 to 20 years foomnow , what fondamentalscientifi c issues will bring about and economy of purchasing books through AIMS. future developments, what future manufacturing requirements will Contact: be, and if current éducation in this area is adéquate. AIMS Corporation or Materials Research Society Topics: Si and group IV materials, compound semiconductors, 2-8-17 Shinbashi Minato-ku 9800 McKnisht Road polymer/biological/organic insulators, superconductors/ceramics, Tokyo 105, Japan Pittsbursh, PA 15237 U.S.A. magnetic materials. Phone (81 ) (3) 501-1059 Phone (412) 367-3003 Invited intemationaOy distinguished speakers will give talks each Fax (81) (3) 501-1388 Fax (412) 367-4373 moming, followed by panel discussions in the afternoon and contrib- uted poster présentations in the evening. For information contact: Judy Tumer, Northwestern University, Ma­ terials Research Center, 2145 Sheridan Road, Evanston, IL 60208- 3116; phone (708) 491-3606; fax (708) 491-4181. Sponsored by the NSF and Northwestern University; co-sponsored by the Interna­ tional Union of Materials Research Societies and MRS.

MRS BULLETIN/JUNE1991 21

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 ROYAL SOCIETY OF CHEMISTRY

Announcing a NEW international journal... Journal of Materials JS^nalof Chemistry An interdisciplinary journal dealing with the synthesis, structures, properties and applications of materials, particularly those associated with advanced technology

Scientific Editor: Professor A. R. West, University of Aberdeen, UK ISSN 0959-9428 Published Bimonthly (6 issues per annum) 1991 Subscription: EC £175.00 Overseas £195.00 US$395.00

This new international journal, which commenced publication in 1991, represents an exciting and radical new publishing initiative on the part of the RSC. It breaks away from the traditional divisions of inorganic, organic and physical chemistry, by recognizing the interdisciplinary nature of materials research. The Journal of Materials Chemistry contains original research reports (both full-length papers and short communications), occasional review articles, book reviews, détails of forthcoming conférences and a cumulative author index, together with colour photographs and diagrams where appropriate. It has an International Advisory Editorial Board consisting of some of the world's leading authorities on materials chemistry. The areas to be covered in the Journal of Materials Chemistry include:

MATERIALS PROPERTIES AND Magnetic properties: APPLICATIONS ferro, ferri- and antiferro- Inorganics: magnetism, spin glass behaviour, ceramics; layered materials; Electrical properties: organic magnetism, magnetic microporous solids and zeolites; semi-, metallic and super- bubbles and information storage. silicates and synthetic minerais; conductivity; ionic conductivity; biogenic minerais. mixed ionic/electronic conductivity; Chemical properties: ferro-, pyrô-, and piezo-electricity; ion exchange, molecular séparation, Organics: electroceramics; dielectrics. catalytic action, sensor action, organometallic precursors for thin topochemical control of reactions. films/ceramics; novel molecular Optical properties: solids and synthetic polymers with luminescence, phosphorescence, Structural properties: materials applications; polymer laser action; non-linear optical structural ceramics, refractories; composites; biopolymers; effects; photoconductivity; photo- hard materials; protective coatings; biocompatible and biodégradable and electro-chromism, resists, composites, adhesives; prosthetic polymers; liquid crystals (both glasses, amorphous semiconductors, applications. iyotropic and thermotropic); optical modulation and switching. Langmuir-Blodgett films. Thermodynamic properties and phase behaviour

For further information and a sample issue of the Journal of Materials Chemistry, simply complète and return the slip below:

• Please send me further information and a sample issue of the Please retum to: ROYAL Journal of Materials Chemistry Sales and Promotion Department, Royal SOCIETY OF Society of Chemistry, Thomas Graham CHEMISTRY House, Science Parte, Milton Road, Name Cambridge CB4 4WF, United Kingdom. To order please phone Position (0462) 672555 quoting your crédit card rv^nnicrrrinn détails - we now accept Access/Visa/ Information ^ryulllMJMUII Mastercard/EurocarMnstarrnrri/Fumrnrrid Services Address. Or write to the address above enclosing a chèque mode payable to the Royal Society of Chemistry. We can also issue pro-forma Downloaded from http:/www.cambridge.org/core. Brookhaven National Laboratory, on 21 Oct 2016 at 17:54:38, subject to the Cambridgeinvoice Cores termsif required of use,. available at http:/www.cambridge.org/core/terms. http://dx.doi.org/10.1557/S0883769400056694Circle No. 30 on Reader Service Card.