Mapping Self-Organized Criticality Onto Criticality Didier Sornette, Anders Johansen, Ivan Dornic
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Mapping Self-Organized Criticality onto Criticality Didier Sornette, Anders Johansen, Ivan Dornic To cite this version: Didier Sornette, Anders Johansen, Ivan Dornic. Mapping Self-Organized Criticality onto Criticality. Journal de Physique I, EDP Sciences, 1995, 5 (3), pp.325-335. 10.1051/jp1:1995129. jpa-00247058 HAL Id: jpa-00247058 https://hal.archives-ouvertes.fr/jpa-00247058 Submitted on 1 Jan 1995 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. (1995) Phys. J. I IYance 325-335 5 1995, 325 MARCH PAGE Classification ysics PA Abstracts 64.60H 05.70L 05.40 CriticaEty Mapping Self.Organized CriticaEty enta Sornette, Didier Anders Johansen Dornic and Ivan (*), Physique Laboratoire de Sciences, de la Matière Condensée Université des 70, B-P. Parc Valrose, 2, Nice Cedex France 06108 (Received 1994, received November accepted in final November 14 1994, November form 30 25 1994) Résumé. stratégie générale responsable identifier Nous mécanisme le proposons une pour phénomènes duto-organisés, critiques qu'ils simplement traduction, des basée l'idée la sont sur paramètres choisis, dynamique dans de critique d'un instable standard. La point espace un ajusté auto-organisée résulte paramètre positive criticalité du valeur contrôle du d'ordre à une zéro, paramètre correspondant qui tendant automatiquement de contrôle le que vers ce assure critique cale valeur critique sous"jacente. Ce résultat de la de transition exactement se sur sa forçage explique particulier joué le le rôle infiniment lent caractère à est tous par qui commun un auto-organisés. systèmes critiques appliquons sable, les Nous idées modèles de de tas ces aux décrochage forêts, modèles de tremblements de feux de transitions de de terre, et aux aux aux qui fractale, été proposés d'exemples caractéristiques modèles croissance de ont autant comme duto-organisée. criticalité de la (SOC), general conceptual self-organized criticality Abstract. framework for We present a nothing recognition that expression, but the "unfolded" based the it is suitable param- on m a precisely, underlying dynarnical SOC of unstable is shown critical More to point. eter space, an value, thus vanishingly small, positive of the order result from the but tuning parameter to a corresponding under- exactly critical control value for the that lies the its parameter at ensunng driving ail lying This clarifies the rote of the slow transition. and rate to nature very common earthquakes, sandpiles, exhibiting apply SOC. This mechanism of shown models systems to to is growth tires, proposed exarnples SOC. depinning, fractal and forest been of which have as Introduction 1. insightful early witnessed clear [1-6], Following of decade has studies number the past a a by acknowledgement phenomena described law statistics. that natural be must power many of these developed origin understand the Correspondingly, activity has order in to intense an criticality' of'self-organized particular This has led the ubiquitous law tails. in concept to power CNRS URA190. (* © Physique Les Editions de 1995 PHYSIQUE JOURNAL I N°3 DE 326 spatially extended evolve dynamically (SOC) driven according which certain systems to 8], [7, dynamical characteristic globally with time stationary critical spontaneously towards state no a length scales- or equilibrium phase underlying that, SOC unlike transitions statis- in is fundamental idea The fine-tuning of control reached without the need critical parameter, physics, the is tical state a SOC, ideas Bak dynamics. of illustrate the basic of the To the critical is 1-e- attractor state an pile of inspired by avalanches of the creation cellular in used and co-workers automaton a a slope until local by grain added lattice becomes models, grain "sand" is Sand. In these a on a pile stationary initiated. the critical reaches avalanche this In unstable, state, is and way, a an pile of will fall grains slope, which sand off the via critical additional by in characterized a according lifetime and scale, lattice distributed grain from size in avalanches of ail sizes to to law. power a physical general effort, which conditions the under large theoretical In of system spite a a established: largely Some facts however been still unknown. have exhibits SOC are obeys diffusion large qualify equation evolution if Some SOC their scale systems a as . scale) (possibly global satisfies characteristic which non-linear but with time con- no a loi. law servation [9, obey but of which diffusion-like do Mass exhibit There systems exists not response a a . il1-14]. nevertheless exhibit SOC the global In law and these conservation to cases, seem deep if for SOC still clear exist there rela- underlying mechanism is to trot seems even a synchronization coupled oscillators problem of of of threshold relaxation tionship the with [8,15-18]. of the domain occurring in parameter space some which, of usual from the generally, perspective feedback mechanism More operate must a . the descnbes the of the order control phenomena, action critical parameter onto parame- dynamics for critical mechanism This then and the suggests to ter state. attracts [19] a a transforming phase We critical SOC call usual "unstable transitions" unstable into [19]. self-organized. phenomena which those critical not are understanding fragmented theoretical of In the SOC rather real with is present summary, no unifying goal general theoretical Our here perspective. framework present attempt to to a is SOC, for nothing recognition expression, based the that the "unfolded" it but is in on a underlying suitable correspondence of unstable critical This genuine parameter point. space, an provides, fundamental what believe mechanism for SOC and the information is, we more on obtained relevant be framework. the within this critical exponents con Self-Organized Criticality The Nature of 2. approach of GENERAL MECHANISM. The be few summarized 2.1. in essence our con a sen- Ising ferromagnet phase Consider "standard" unstable such transition, critical the tences. as a percolation. analogously, assigned Here, bond down, spin, each with site to or, a up or is an Furthermore, exchange coupling neighbour defines J. be sites constant to two nearest con- one e~~~/~B~' probability field, nected with if both spin have For 1 externat this p a up. zero = bond-density defines order critical below which the the T~ temperature parameter p~ mû, or a T)P probability cluster, infinite of magnetization the and behaves is (T~ mû an or zero as oc (T diverging length by characterized above. further T~)~" correlation The is ( transition a oc (T T~)~~ susceptibility quantifying approached, and spatial fluctua- hence the is T~ x oc as Suppose the of be natural the order that for under tions it parameter. turns out to system now MAPPING SELF.ORGANIZED N°3 CRITICALITY ONTO CRITICALITY 327 controlling "operator" that, T, consideration instead of the controls the order parameter mû fixing limiting furthermore of arbitrary positive takes but and the it small value. The to case a o+ (Specifically, Tj. equivalent condition the T is above to mû scenano comes more - - only strength" cluster, infinite natural the of "zero where in bond needs be context to a one ferromagnet.) broken, words, of thon in that other the In is the of critical value system at a therefore fluctuations the unstable point critical exhibit and ail scales its in must at response. nothing underlying of This is but the hallmark the unstable critical As point. will be shown following applies explicitly examples, the naturally this out-of-equilibrium in most to scenario driven systems. precisely, exhibiting More shall that SOC genuine critical systems present tran- argue we a by forced often form generalized suitable control the when of force sition in parameter, a a (torque depinning systems). earthquake models, sandpile models, for for force for Then, stress SOC controls drives the the order of the critical via system parameter appears as soon as one or (it that, these the order also the of the transition is conjugate in systems, parameter tums out equations). control of mechanical Hamilton-Jacobi the The order in parameter parameter sense general velocity form of flux. control of the hence takes the The in order parameter, a or a o+ out-of-equilibrium flux, driving natural The condition that the M is in systems. at is - by dTiuing special played the of slow role illuminates the constraint Tate to veTy a common o+ exhibiting being SOC, the control ail the order condition systems parameter to exact at as of positioning which the critical value the control parameter. at exact ensures general develop by idea, detailed shall this We illustrate discussion of rive and now a ex- pinned-depinned Charge-Density-Waves amples: sandpiles, earthquake models, lines the or models, forest tires. growth fractal and processes SANDPILE THE 2.2. sandpile Model cellular consider trie of Unstable Transition. Let 2.2.1. automaton us an mspired namely rotating and from that of model it in geometry [20], experiments put [7] a a cylinder. cylinder cylinder Trie horizontal and trie trie Trie axis is rotation is axis. same as partially initially flat filled horizontal interface below trie presenting with "sand" axis. is an by cylinder fixed frame which Suppose spring trie that of trie held torsion to a on one axis is a cylinder position surface If trie takes trie such that trie 0, controlled T. T torque exert a can = non-vanishing angle If horizontal, 9 of trie sand trie rotation o.