Verfügbarkeit: Critical phenomena in natural sciences

Critical phenomena in natural sciences: chaos, fractals, selforganization, and disorder : concepts and tools

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Bibliographische Detailangaben
Beteilige Person:	Sornette, Didier 1957- (VerfasserIn)
Format:	Buch
Sprache:	Englisch
Veröffentlicht:	Berlin Springer 2004
Ausgabe:	Second edition
Schriftenreihe:	Springer series in synergetics Springer complexity
Schlagwörter:	Teoria de campos e ondas Critical phenomena (Physics) Fraktal Chaotisches System Naturwissenschaften Selbstorganisation Chaos Statistische Physik Kritisches Phänomen Wahrscheinlichkeit
Links:	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=010501660&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
Umfang:	XXII, 528 Seiten Illustrationen, Diagramme
ISBN:	9783540308829 3540407545

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MARC


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Datensatz im Suchindex

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adam_text	CONTENTS 1. USEFUL NOTIONS OF PROBABILITY THEORY ..................... 1 1.1 WHAT IS PROBABILITY? .................................. 1 1.1.1 FIRSTINTUITIVENOTIONS .......................... 1 1.1.2 OBJECTIVE VERSUS SUBJECTIVE PROBABILITY ........... 2 1.2 BAYESIANVIEWPOINT................................... 6 1.2.1 INTRODUCTION................................... 6 1.2.2 BAYES THEOREM................................ 7 1.2.3 BAYESIAN EXPLANATION FOR CHANGE OF BELIEF......... 9 1.2.4 BAYESIAN PROBABILITY AND THE DUTCH BOOK ......... 10 1.3 PROBABILITY DENSITY FUNCTION............................ 12 1.4 MEASURESOFCENTRALTENDENCY .......................... 13 1.5 MEASUREOFVARIATIONSFROMCENTRALTENDENCY............. 14 1.6 MOMENTSANDCHARACTERISTICFUNCTION.................... 15 1.7 CUMULANTS............................................ 16 1.8 MAXIMUM OF RANDOM VARIABLES AND EXTREME VALUE THEORY. 18 1.8.1 MAXIMUM VALUE AMONG N RANDOM VARIABLES ..... 19 1.8.2 STABLEEXTREMEVALUEDISTRIBUTIONS............... 23 1.8.3 FIRST HEURISTIC DERIVATION OFTHESTABLEGUMBELDISTRIBUTION................ 25 1.8.4 SECOND HEURISTIC DERIVATION OFTHESTABLEGUMBELDISTRIBUTION................ 26 1.8.5 PRACTICAL USE AND EXPRESSION OF THE COEFFICIENTS OFTHEGUMBELDISTRIBUTION...................... 28 1.8.6 THE GNEDENKO-PICKANDS-BALKEMA-DE HAAN THEO REM ANDTHEPDFOFPEAKS-OVER-THRESHOLD ............. 29 2. SUMS OF RANDOM VARIABLES, RANDOM WALKS AND THE CENTRAL LIMIT THEOREM ........................... 33 2.1 THERANDOMWALKPROBLEM ............................ 33 2.1.1 AVERAGEDRIFT.................................. 34 2.1.2 DIFFUSIONLAW.................................. 35 2.1.3 BROWNIAN MOTION AS SOLUTION OF A STOCHASTIC ODE . 35 2.1.4 FRACTALSTRUCTURE .............................. 37 XVI CONTENTS 2.1.5 SELF-AFFINITY ................................... 39 2.2 MASTER AND DIFFUSION (FOKKER-PLANCK) EQUATIONS .......... 41 2.2.1 SIMPLEFORMULATION............................. 41 2.2.2 GENERALFOKKER-PLANCKEQUATION................. 43 2.2.3 ITOVERSUSSTRATONOVICH ......................... 44 2.2.4 EXTRACTING MODEL EQUATIONS FROM EXPERIMENTAL DATA 47 2.3 THECENTRALLIMITTHEOREM ............................ 48 2.3.1 CONVOLUTION ................................... 48 2.3.2 STATEMENT..................................... 50 2.3.3 CONDITIONS .................................... 50 2.3.4 COLLECTIVEPHENOMENON ......................... 51 2.3.5 RENORMALIZATIONGROUPDERIVATION ............... 52 2.3.6 RECURSION RELATION AND PERTURBATIVE ANALYSIS...... 55 3. LARGE DEVIATIONS ......................................... 59 3.1 CUMULANTEXPANSION................................... 59 3.2 LARGEDEVIATIONTHEOREM .............................. 60 3.2.1 QUANTIFICATION OF THE DEVIATION FROMTHECENTRALLIMITTHEOREM ................. 61 3.2.2 HEURISTIC DERIVATION OFTHELARGEDEVIATIONTHEOREM(3.9)............. 61 3.2.3 EXAMPLE:THEBINOMIALLAW ..................... 63 3.2.4 NON-IDENTICALLY DISTRIBUTED RANDOM VARIABLES ..... 64 3.3 LARGE DEVIATIONS WITH CONSTRAINTS ANDTHEBOLTZMANNFORMALISM .......................... 66 3.3.1 FREQUENCIES CONDITIONED BYLARGEDEVIATIONS ...... 66 3.3.2 PARTITIONFUNCTIONFORMALISM ................... 68 3.3.3 LARGEDEVIATIONSINTHEDICEGAME............... 70 3.3.4 MODEL CONSTRUCTION FROM LARGE DEVIATIONS ........ 73 3.3.5 LARGE DEVIATIONS IN THE GUTENBERG-RICHTER LAW ANDTHEGAMMALAW ........................... 76 3.4 EXTREMEDEVIATIONS.................................... 78 3.4.1 THE DEMOCRATIC RESULT ....................... 78 3.4.2 APPLICATION TO THE MULTIPLICATION OF RANDOM VARIABLES: AMECHANISMFORSTRETCHEDEXPONENTIALS .......... 80 3.4.3 APPLICATION TO TURBULENCE AND TO FRAGMENTATION ... 83 3.5 LARGE DEVIATIONS IN THE SUM OF VARIABLES WITHPOWERLAWDISTRIBUTIONS........................... 87 3.5.1 GENERAL CASE WITH EXPONENT µ 2 ............... 87 3.5.2 BORDERLINE CASE WITH EXPONENT µ =2............. 90 CONTENTS XVII 4. POWER LAW DISTRIBUTIONS ................................. 93 4.1 STABLE LAWS: GAUSSIAN AND L´EVYLAWS ................... 93 4.1.1 DEFINITION ..................................... 93 4.1.2 THE GAUSSIAN PROBABILITY DENSITY FUNCTION........ 93 4.1.3 THELOG-NORMALLAW........................... 94 4.1.4 THE L´EVYLAWS ................................ 96 4.1.5 TRUNCATED L´EVYLAWS...........................101 4.2 POWERLAWS........................................... 104 4.2.1 HOW DOES ONE TAME WILD DISTRIBUTIONS? ....... 105 4.2.2 MULTIFRACTALAPPROACH .......................... 110 4.3 ANOMALOUS DIFFUSION OF CONTAMINANTS INTHEEARTH SCRUSTANDTHEATMOSPHERE................. 112 4.3.1 GENERALINTUITIVEDERIVATION .....................113 4.3.2 MORE DETAILED MODEL OF TRACER DIFFUSION IN THE CRUST113 4.3.3 ANOMALOUSDIFFUSIONINAFLUID ..................115 4.4 INTUITIVE CALCULATION TOOLS FORPOWERLAWDISTRIBUTIONS ............................116 4.5 FOX FUNCTION, MITTAG-LEFFLER FUNCTION AND L´EVYDISTRIBUTIONS.................................118 5. FRACTALS AND MULTIFRACTALS ................................ 123 5.1 FRACTALS.............................................. 123 5.1.1 INTRODUCTION................................... 123 5.1.2 A FIRST CANONICAL EXAMPLE: THE TRIADIC CANTOR SET . 124 5.1.3 HOWLONGISTHECOASTOFBRITAIN?................ 125 5.1.4 THEHAUSDORFFDIMENSION .......................127 5.1.5 EXAMPLESOFNATURALFRACTALS ....................127 5.2 MULTIFRACTALS.......................................... 141 5.2.1 DEFINITION ..................................... 141 5.2.2 CORRECTION METHOD FOR FINITE SIZE EFFECTS ANDIRREGULARGEOMETRIES........................ 143 5.2.3 ORIGIN OF MULTIFRACTALITY AND SOME EXACT RESULTS... 145 5.2.4 GENERALIZATION OF MULTIFRACTALITY: INFINITELY DIVISIBLE CASCADES ..................... 146 5.3 SCALEINVARIANCE....................................... 148 5.3.1 DEFINITION ..................................... 148 5.3.2 RELATION WITH DIMENSIONAL ANALYSIS............... 150 5.4 THEMULTIFRACTALRANDOMWALK .........................153 5.4.1 A FIRST STEP: THE FRACTIONAL BROWNIAN MOTION ..... 153 5.4.2 DEFINITION AND PROPERTIES OFTHEMULTIFRACTALRANDOMWALK.................154 5.5 COMPLEX FRACTAL DIMENSIONS ANDDISCRETESCALEINVARIANCE ........................... 156 5.5.1 DEFINITION OF DISCRETE SCALE INVARIANCE ............ 156 5.5.2 LOG-PERIODICITY AND COMPLEX EXPONENTS .......... 157 XVIII CONTENTS 5.5.3 IMPORTANCE AND USEFULNESS OFDISCRETESCALEINVARIANCE...................... 159 5.5.4 SCENARII LEADING TO DISCRETE SCALE INVARIANCE ...... 160 6. RANK-ORDERING STATISTICS AND HEAVY TAILS ................ 163 6.1 PROBABILITY DISTRIBUTIONS ............................... 163 6.2 DEFINITIONOFRANKORDERINGSTATISTICS.................... 164 6.3 NORMALANDLOG-NORMALDISTRIBUTIONS ................... 166 6.4 THEEXPONENTIALDISTRIBUTION...........................167 6.5 POWERLAWDISTRIBUTIONS ............................... 170 6.5.1 MAXIMUMLIKELIHOODESTIMATION................. 170 6.5.2 QUANTILESOFLARGEEVENTS ....................... 173 6.5.3 POWER LAWS WITH A GLOBAL CONSTRAINT: FRACTALPLATETECTONICS ........................174 6.6 THEGAMMALAW...................................... 179 6.7 THESTRETCHEDEXPONENTIALDISTRIBUTION.................. 180 6.8 MAXIMUM LIKELIHOOD AND OTHER ESTIMATORS OFSTRETCHEDEXPONENTIALDISTRIBUTIONS ................... 181 6.8.1 INTRODUCTION................................... 182 6.8.2 TWO-PARAMETER STRETCHED EXPONENTIAL DISTRIBUTION 185 6.8.3 THREE-PARAMETER WEIBULL DISTRIBUTION ............ 194 6.8.4 GENERALIZEDWEIBULLDISTRIBUTIONS ................196 7. STATISTICAL MECHANICS: PROBABILISTIC POINT OF VIEW AND THE CONCEPT OF TEMPERATURE ....................... 199 7.1 STATISTICAL DERIVATION OF THE CONCEPT OF TEMPERATURE....... 200 7.2 STATISTICALTHERMODYNAMICS ............................202 7.3 STATISTICAL MECHANICS AS PROBABILITY THEORY WITHCONSTRAINTS ...................................... 203 7.3.1 GENERALFORMULATION............................ 203 7.3.2 FIRSTLAWOFTHERMODYNAMICS ...................206 7.3.3 THERMODYNAMICPOTENTIALS ......................207 7.4 DOES THE CONCEPT OF TEMPERATURE APPLY TONON-THERMALSYSTEMS?...............................208 7.4.1 FORMULATIONOFTHEPROBLEM ..................... 208 7.4.2 AGENERALMODELINGSTRATEGY.................... 210 7.4.3 DISCRIMINATINGTESTS............................ 211 7.4.4 STATIONARY DISTRIBUTION WITH EXTERNAL NOISE ....... 213 7.4.5 EFFECTIVE TEMPERATURE GENERATED BYCHAOTICDYNAMICS........................... 214 7.4.6 PRINCIPLE OF LEAST ACTION FOR OUT-OF-EQUILIBRIUM SYSTEMS.................. 218 7.4.7 SUPERSTATISTICS ................................. 219 CONTENTS XIX 8. LONG-RANGE CORRELATIONS ................................. 223 8.1 CRITERIONFORTHERELEVANCEOFCORRELATIONS................ 223 8.2 STATISTICALINTERPRETATION ............................... 226 8.3 AN APPLICATION: SUPER-DIFFUSION IN A LAYERED FLUID WITHRANDOMVELOCITIES ................................ 228 8.4 ADVANCEDRESULTSONCORRELATIONS .......................229 8.4.1 CORRELATIONANDDEPENDENCE.....................229 8.4.2 STATISTICAL TIME REVERSAL SYMMETRY .............. 231 8.4.3 FRACTIONAL DERIVATION AND LONG-TIME CORRELATIONS . 236 9. PHASE TRANSITIONS: CRITICAL PHENOMENA AND FIRST-ORDER TRANSITIONS ............................... 241 9.1 DEFINITION ............................................241 9.2 SPINMODELSATTHEIRCRITICALPOINTS..................... 242 9.2.1 DEFINITIONOFTHESPINMODEL..................... 242 9.2.2 CRITICALBEHAVIOR...............................245 9.2.3 LONG-RANGE CORRELATIONS OF SPIN MODELS ATTHEIRCRITICALPOINTS.......................... 246 9.3 FIRST-ORDERVERSUSCRITICALTRANSITIONS...................248 9.3.1 DEFINITIONANDBASICPROPERTIES ..................248 9.3.2 DYNAMICAL LANDAU-GINZBURG FORMULATION......... 250 9.3.3 THE SCALING HYPOTHESIS: DYNAMICAL LENGTH SCALES FORORDERING...................................253 10. TRANSITIONS, BIFURCATIONS AND PRECURSORS .................. 255 10.1 SUPERCRITICAL BIFURCATION ............................. 256 10.2 CRITICALPRECURSORYFLUCTUATIONS.........................258 10.3 SUBCRITICAL BIFURCATION ............................... 262 10.4 SCALINGANDPRECURSORSNEARSPINODALS ................... 264 1 0 .5 SE LE C TIO NO FA NATTR A C TO RINTHEABS E NC E OFAPOTENTIAL ......................................... 265 11. THE RENORMALIZATION GROUP .............................. 267 11.1 GENERALFRAMEWORK ...................................267 11.2 AN EXPLICIT EXAMPLE: SPINS ON A HIERARCHICAL NETWORK ..... 269 11.2.1 RENORMALIZATION GROUP CALCULATION............... 269 11.2.2 FIXED POINTS, STABLE PHASES AND CRITICAL POINTS .... 273 11.2.3 SINGULARITIES AND CRITICAL EXPONENTS .............. 275 11.2.4 COMPLEX EXPONENTS ANDLOG-PERIODICCORRECTIONSTOSCALING...........276 11.2.5 WEIERSTRASS-TYPE FUNCTIONS FROM DISCRETE RENORMALIZATION GROUP EQUATIONS ... 279 11.3 CRITICALITY AND THE RENORMALIZATION GROUP ONEUCLIDEANSYSTEMS..................................283 XX CONTENTS 11.4 A NOVEL APPLICATION TO THE CONSTRUCTION OFFUNCTIONALAPPROXIMANTS ............................287 11.4.1 GENERALCONCEPTS ..............................287 11.4.2 SELF-SIMILARAPPROXIMANTS....................... 288 11.5 TOWARDSAHIERARCHICALVIEWOFTHEWORLD ................ 291 12. THE PERCOLATION MODEL ................................... 293 12.1 PERCOLATIONASAMODELOFCRACKING ......................293 12.2 EFFECTIVE MEDIUM THEORY AND PERCOLATION ................ 296 12.3 RENORMALIZATION GROUP APPROACH TO PERCOLATION ANDGENERALIZATIONS....................................298 12.3.1 CELL-TO-SITETRANSFORMATION...................... 299 12.3.2 A WORD OF CAUTION ON REAL SPACE RENORMALIZATION GROUP TECHNIQUES .. 301 12.3.3 THE PERCOLATION MODEL ONTHEHIERARCHICALDIAMONDLATTICE.............. 303 12.4 DIRECTEDPERCOLATION...................................304 12.4.1 DEFINITIONS .................................... 304 12.4.2 UNIVERSALITYCLASS..............................306 12.4.3 FIELD THEORY: STOCHASTIC PARTIAL DIFFERENTIAL EQUA TION WITHMULTIPLICATIVENOISE........................ 308 12.4.4 SELF-ORGANIZED FORMULATION OF DIRECTED PERCOLATION ANDSCALINGLAWS...............................309 13. RUPTURE MODELS .......................................... 313 13.1 THEBRANCHINGMODEL ................................. 314 13.1.1 MEAN FIELD VERSION OR BRANCHING ONTHEBETHELATTICE............................314 13.1.2 A BRANCHING-AGGREGATION MODEL AUTOMATICALLY FUNCTIONING AT ITS CRITICAL POINT .... 316 13.1.3 GENERALIZATION OF CRITICAL BRANCHING MODELS ....... 317 13.2 FIBER BUNDLE MODELS AND THE EFFECTS OFSTRESSREDISTRIBUTION ................................ 318 13.2.1 ONE-DIMENSIONAL SYSTEM OFFIBERSASSOCIATEDINSERIES....................318 13.2.2 DEMOCRATIC FIBER BUNDLE MODEL (DANIELS, 1945).... 320 13.3 HIERARCHICALMODEL ....................................323 13.3.1 THE SIMPLEST HIERARCHICAL MODEL OF RUPTURE ....... 323 13.3.2 QUASI-STATIC HIERARCHICAL FIBER RUPTURE MODEL..... 326 13.3.3 HIERARCHICAL FIBER RUPTURE MODEL WITHTIME-DEPENDENCE.......................... 328 13.4 QUASI-STATICMODELSINEUCLIDEANSPACES ................. 330 13.5 A DYNAMICAL MODEL OF RUPTURE WITHOUT ELASTO-DYNAMICS: THE THERMALFUSEMODEL .............................335 CONTENTS XXI 13.6 TIME-TO-FAILUREANDRUPTURECRITICALITY.................. 339 13.6.1 CRITICALTIME-TO-FAILUREANALYSIS................. 339 13.6.2 TIME-TO-FAILURE BEHAVIOR INTHEDIETERICHFRICTIONLAW .................... 343 14. MECHANISMS FOR POWER LAWS .............................. 345 14.1 TEMPORAL COPERNICAN PRINCIPLE AND µ =1 UNIVERSAL DISTRIBUTION OF RESIDUAL LIFETIMES ..... 346 14.2 CHANGEOFVARIABLE .................................... 348 14.2.1 POWER LAW CHANGE OF VARIABLE CLOSE TO THE ORIGIN . 348 14.2.2 COMBINATIONOFEXPONENTIALS ....................354 14.3 MAXIMIZATION OF THE GENERALIZED TSALLIS ENTROPY .......... 356 14.4 SUPERPOSITIONOFDISTRIBUTIONS........................... 359 14.4.1 POWERLAWDISTRIBUTIONOFWIDTHS................ 359 14.4.2 SUM OF STRETCHED EXPONENTIALS (CHAP. 3).......... 362 14.4.3 DOUBLE PARETO DISTRIBUTION BY SUPERPOSITION OFLOG-NORMALPDF S ............................ 362 14.5 RANDOM WALKS: DISTRIBUTION OF RETURN TIMES TO THE ORIGIN. 363 14.5.1 DERIVATION.....................................364 14.5.2 APPLICATIONS................................... 365 14.6 SWEEPING OF A CONTROL PARAMETER TOWARDS AN INSTABILITY.... 367 14.7 GROWTHWITHPREFERENTIALATTACHMENT....................370 14.8 MULTIPLICATIVENOISEWITHCONSTRAINTS.................... 373 14.8.1 DEFINITIONOFTHEPROCESS ........................ 373 14.8.2 THE KESTEN MULTIPLICATIVE STOCHASTIC PROCESS ...... 374 14.8.3 RANDOMWALKANALOGY ......................... 375 14.8.4 EXACT DERIVATION, GENERALIZATION AND APPLICATIONS.. 378 14.9 THE COHERENT-NOISE MECHANISM.......................381 14.10 AVALANCHESIN HYSTERETIC LOOPSAND FIRST-ORDERTRANSITIONS WITHRANDOMNESS...................................... 386 14.11 HIGHLYOPTIMIZEDTOLERANT (HOT)SYSTEMS.............389 14.11.1 MECHANISM FOR THE POWER LAW DISTRIBUTION OF FIRE SIZES ......................................... 390 14.11.2 CONSTRAINEDOPTIMIZATIONWITHLIMITEDDEVIATIONS (COLD).......................................393 14.11.3 HOTVERSUSPERCOLATION.........................393 15. SELF-ORGANIZED CRITICALITY ................................. 395 15.1 WHATISSELF-ORGANIZEDCRITICALITY?......................395 15.1.1 INTRODUCTION................................... 395 15.1.2 DEFINITION ..................................... 397 15.2 SANDPILEMODELS.......................................398 15.2.1 GENERALITIES.................................... 398 15.2.2 THEABELIANSANDPILE........................... 398 15.3 THRESHOLDDYNAMICS................................... 402 XXII CONTENTS 15.3.1 GENERALIZATION .................................402 15.3.2 ILLUSTRATION OF SELF-ORGANIZED CRITICALITY WITHINTHEEARTH SCRUST........................ 404 15.4 SCENARIOSFORSELF-ORGANIZEDCRITICALITY...................406 15.4.1 GENERALITIES.................................... 406 15.4.2 NONLINEAR FEEDBACK OF THE ORDER PARAMETER ONTOTHE CONTROLPARAMETER .................... 407 15.4.3 GENERICSCALEINVARIANCE ........................ 409 15.4.4 MAPPINGONTOACRITICALPOINT ................... 414 15.4.5 MAPPINGTOCONTACTPROCESSES ...................422 15.4.6 CRITICALDESYNCHRONIZATION....................... 424 15.4.7 EXTREMALDYNAMICS.............................427 15.4.8 DYNAMICAL SYSTEM THEORY OF SELF-ORGANIZED CRITI CALITY .........................................435 15.5 TESTS OF SELF-ORGANIZED CRITICALITY IN COMPLEX SYSTEMS: THEEXAMPLEOFTHEEARTH SCRUST........................ 438 16. INTRODUCTION TO THE PHYSICS OF RANDOM SYSTEMS .......... 441 16.1 GENERALITIES...........................................441 16.2 THERANDOMENERGYMODEL............................. 445 16.3 NON-SELF-AVERAGINGPROPERTIES ..........................449 16.3.1 DEFINITIONS .................................... 449 16.3.2 FRAGMENTATIONMODELS ..........................451 17. RANDOMNESS AND LONG-RANGE LAPLACIAN INTERACTIONS ...... 457 17.1 L´EVY DISTRIBUTIONS FROM RANDOM DISTRIBUTIONS OF SOURCES WITHLONG-RANGEINTERACTIONS........................... 457 17.1.1 HOLTSMARK S GRAVITATIONAL FORCE DISTRIBUTION ...... 457 17.1.2 GENERALIZATION TO OTHER FIELDS (ELECTRIC,ELASTIC,HYDRODYNAMICS)................ 461 17.2 LONG-RANGE FIELD FLUCTUATIONS DUE TO IRREGULAR ARRAYS OF SOURCES AT BOUNDARIES ............................... 463 17.2.1 PROBLEMANDMAINRESULTS ......................463 17.2.2 CALCULATIONMETHODS............................ 464 17.2.3 APPLICATIONS................................... 471 REFERENCES .................................................... 477 INDEX ......................................................... 525
any_adam_object	1
author	Sornette, Didier 1957-
author_GND	(DE-588)171887921
author_facet	Sornette, Didier 1957-
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dewey-hundreds	500 - Natural sciences and mathematics
dewey-ones	530 - Physics
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dewey-tens	530 - Physics
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id	DE-604.BV017428788
illustrated	Illustrated
indexdate	2024-12-20T11:19:09Z
institution	BVB
isbn	9783540308829 3540407545
language	English
oai_aleph_id	oai:aleph.bib-bvb.de:BVB01-010501660
oclc_num	52819765
open_access_boolean
owner	DE-703 DE-20 DE-19 DE-BY-UBM DE-83 DE-11 DE-384
owner_facet	DE-703 DE-20 DE-19 DE-BY-UBM DE-83 DE-11 DE-384
physical	XXII, 528 Seiten Illustrationen, Diagramme
publishDate	2004
publishDateSearch	2004
publishDateSort	2004
publisher	Springer
record_format	marc
series2	Springer series in synergetics Springer complexity
spellingShingle	Sornette, Didier 1957- Critical phenomena in natural sciences chaos, fractals, selforganization, and disorder : concepts and tools Teoria de campos e ondas larpcal Critical phenomena (Physics) Fraktal (DE-588)4123220-3 gnd Chaotisches System (DE-588)4316104-2 gnd Naturwissenschaften (DE-588)4041421-8 gnd Selbstorganisation (DE-588)4126830-1 gnd Chaos (DE-588)4191419-3 gnd Statistische Physik (DE-588)4057000-9 gnd Kritisches Phänomen (DE-588)4165788-3 gnd Wahrscheinlichkeit (DE-588)4137007-7 gnd
subject_GND	(DE-588)4123220-3 (DE-588)4316104-2 (DE-588)4041421-8 (DE-588)4126830-1 (DE-588)4191419-3 (DE-588)4057000-9 (DE-588)4165788-3 (DE-588)4137007-7
title	Critical phenomena in natural sciences chaos, fractals, selforganization, and disorder : concepts and tools
title_auth	Critical phenomena in natural sciences chaos, fractals, selforganization, and disorder : concepts and tools
title_exact_search	Critical phenomena in natural sciences chaos, fractals, selforganization, and disorder : concepts and tools
title_full	Critical phenomena in natural sciences chaos, fractals, selforganization, and disorder : concepts and tools Didier Sornette
title_fullStr	Critical phenomena in natural sciences chaos, fractals, selforganization, and disorder : concepts and tools Didier Sornette
title_full_unstemmed	Critical phenomena in natural sciences chaos, fractals, selforganization, and disorder : concepts and tools Didier Sornette
title_short	Critical phenomena in natural sciences
title_sort	critical phenomena in natural sciences chaos fractals selforganization and disorder concepts and tools
title_sub	chaos, fractals, selforganization, and disorder : concepts and tools
topic	Teoria de campos e ondas larpcal Critical phenomena (Physics) Fraktal (DE-588)4123220-3 gnd Chaotisches System (DE-588)4316104-2 gnd Naturwissenschaften (DE-588)4041421-8 gnd Selbstorganisation (DE-588)4126830-1 gnd Chaos (DE-588)4191419-3 gnd Statistische Physik (DE-588)4057000-9 gnd Kritisches Phänomen (DE-588)4165788-3 gnd Wahrscheinlichkeit (DE-588)4137007-7 gnd
topic_facet	Teoria de campos e ondas Critical phenomena (Physics) Fraktal Chaotisches System Naturwissenschaften Selbstorganisation Chaos Statistische Physik Kritisches Phänomen Wahrscheinlichkeit
url	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=010501660&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
work_keys_str_mv	AT sornettedidier criticalphenomenainnaturalscienceschaosfractalsselforganizationanddisorderconceptsandtools

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