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

539 Pages·2006·3.982 MB·English

by  Didier Sornette

Springer Complexity
SpringerComplexityisapublicationprogram,cuttingacrossalltraditionaldis-
ciplinesofsciencesaswellasengineering,economics,medicine,psychologyand
computersciences,whichisaimedatresearchers,studentsandpractitionerswor-
kinginthefieldofcomplexsystems.ComplexSystemsaresystemsthatcomprise
manyinteractingpartswiththeabilitytogenerateanewqualityofmacroscopic
collectivebehaviorthroughself-organization,e.g.,thespontaneousformationof
temporal,spatialorfunctionalstructures.Thisrecognition,thatthecollectivebe-
haviorofthewholesystemcannotbesimplyinferredfromtheunderstandingof
thebehavioroftheindividualcomponents,hasledtovariousnewconceptsand
sophisticatedtoolsofcomplexity.Themainconceptsandtools–withsometimes
overlapping contents and methodologies – are the theories of self-organization,
complexsystems,synergetics,dynamicalsystems,turbulence,catastrophes,insta-
bilities,nonlinearity,stochasticprocesses,chaos,neuralnetworks,cellularauto-
mata,adaptivesystems,andgeneticalgorithms.
The topics treated within Springer Complexity are as diverse as lasers or
fluids in physics, machine cutting phenomena of workpieces or electric circuits
withfeedbackinengineering,growthofcrystalsorpatternformationinchemistry,
morphogenesisinbiology,brainfunctioninneurology,behaviorofstockexchange
rates in economics, or the formation of public opinion in sociology. All these
seeminglyquitedifferentkindsofstructureformationhaveanumberofimportant
featuresandunderlyingstructuresincommon.Thesedeepstructuralsimilarities
canbeexploitedtotransferanalyticalmethodsandunderstandingfromonefield
to another. The Springer Complexity program therefore seeks to foster cross-
fertilization between the disciplines and a dialogue between theoreticians and
experimentalistsforadeeperunderstandingofthegeneralstructureandbehavior
ofcomplexsystems.
The program consists of individual books, books series such as “Springer
Series in Synergetics”, “Institute of Nonlinear Science”, “Physics of Neural Net-
works”,and“UnderstandingComplexSystems”,aswellasvariousjournals.
Springer Series in Synergetics
SeriesEditor
HermannHaken
Institutfu¨rTheoretischePhysik
undSynergetik
derUniversit¨atStuttgart
70550Stuttgart,Germany
and
CenterforComplexSystems
FloridaAtlanticUniversity
BocaRaton,FL33431,USA
MembersoftheEditorialBoard
ÅkeAndersson,Stockholm,Sweden
GerhardErtl,Berlin,Germany
BernoldFiedler,Berlin,Germany
YoshikiKuramoto,Sapporo,Japan
Ju¨rgenKurths,Potsdam,Germany
LuigiLugiato,Milan,Italy
Ju¨rgenParisi,Oldenburg,Germany
PeterSchuster,Wien,Austria
FrankSchweitzer,Zu¨rich,Switzerland
DidierSornette,Zu¨rich,Switzerland,andNice,France
ManuelG.Velarde,Madrid,Spain
SSSyn–AnInterdisciplinarySeriesonComplexSystems
ThesuccessoftheSpringerSeriesinSynergeticshasbeenmadepossiblebythe
contributionsofoutstandingauthorswhopresentedtheirquiteoftenpioneering
resultstothesciencecommunitywellbeyondthebordersofaspecialdiscipline.
Indeed,interdisciplinarityisoneofthemainfeaturesofthisseries.Butinterdis-
ciplinarity is not enough: The main goal is the search for common features of
self-organizingsystemsinagreatvarietyofseeminglyquitedifferentsystems,or,
stillmorepreciselyspeaking,thesearchforgeneralprinciplesunderlyingthespon-
taneousformationofspatial,temporalorfunctionalstructures.Thetopicstreated
maybeasdiverseaslasersandfluidsinphysics,patternformationinchemistry,
morphogenesis in biology, brain functions in neurology or self-organization in
a city. As is witnessed by several volumes, great attention is being paid to the
pivotalinterplaybetweendeterministicandstochasticprocesses,aswellastothe
dialoguebetweentheoreticiansandexperimentalists.Allthishascontributedtoa
remarkablecross-fertilizationbetweendisciplinesandtoadeeperunderstanding
of complex systems. The timeliness and potential of such an approach are also
mirrored – among other indicators – by numerous interdisciplinary workshops
andconferencesallovertheworld.
Didier Sornette
Critical Phenomena
in Natural Sciences
Chaos, Fractals,
Selforganization and Disorder:
Concepts and Tools
SecondEdition
With102Figures
123
ProfessorDidierSornette
ChairofEntrepreneurialRisks
DepartmentofManagement,
TechnologyandEconomics(D-MTEC)
ETHZentrum
8092Zürich,Switzerland
and
LaboratoiredePhysiquedelaMatièreCondens´ee
CNRSUMR6622
UniversitédeNice-SophiaAntipolis
FacultédesSciences,B.P.71
06108NiceCedex2,France
LibraryofCongressControlNumber:2006920906
2ndPrintingoftheHardcoverEditionwithISBN3-540-40754-5
ISSN0172-7389
ISBN-103-540-30882-2 2ndEdition SpringerBerlinHeidelbergNewYork
ISBN-13978-3-540-30882-9 2ndEdition SpringerBerlinHeidelbergNewYork
Thisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeorpartofthematerialis
concerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broadcasting,
reproductiononmicrofilmorinanyotherway,andstorageindatabanks.Duplicationofthispublication
orpartsthereofispermittedonlyundertheprovisionsoftheGermanCopyrightLawofSeptember9,1965,
initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer.Violationsareliable
toprosecutionundertheGermanCopyrightLaw.
SpringerisapartofSpringerScience+BusinessMedia
springeronline.com
©Springer-VerlagBerlinHeidelberg2006
PrintedinGermany
Theuseofgeneraldescriptivenames,registerednames,trademarks,etc.inthispublicationdoesnotimply,
evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevantprotectivelaws
andregulationsandthereforefreeforgeneraluse.
Typesetting:bytheauthor
Production:LE-TEXJelonek,Schmidt&VöcklerGbR,Leipzig
Coverdesign:ErichKirchner,Heidelberg
Printedonacid-freepaper 54/3100/YL 54321
To Anne, Jaufray and Paul
Preface
Since its first edition, the ideas discussed in this book have expanded signif-
icantly as a result of very active research in the general domain of complex
systems. I have also seen with pleasure different communities in the geo-,
medical and social sciences becoming more aware of the usefulness of the
concepts and techniques presented here.
In this second edition, I have first corrected, made more precise and ex-
panded a large number of points. I have also added a significant amount of
novel material which I describe briefly below.
Chapter 1 has been expanded by inclusion of stricter and more in-depth
discussions of the differences between objective and subjective (Bayesian)
probabilities with, in particular, the addition of the Dutch book argument.
ApresentationoftheGnedenko–Pickands–Balkema–deHaantheoremforthe
distribution of peaks-over-threshold has been added, which derives the gen-
eralized Pareto distribution of the asymptotic distribution for independent
random variables from the extreme value distributions. I have also added
a formal treatment of the expectation of the sum over the maximum of
random variables, for fat-tailed and non-fat-tailed probability distribution
functions (pdf’s).
In Chap. 2, I have added a section on the extraction of model equations
from experimental data.
InChap.4,theexplicitrepresentationofstableL´evydistributionsisgiven
intermsofFoxfunctionsandtheusefulpropertiesofthegeneralizedMittag–
Leffler exponentials and of Fox functions are described. Chapter 4 also con-
tainsadditionalinformationontheexpectationofthesumoverthemaximum
of random variables for fat-tailed pdf’s.
Chapter 5 contains a new section on the multifractal random walk
(MRW), a recently introduced stochastic process that generalized the frac-
tional Brownian motion by having an exact multifractal structure in the
continuous limit.
Chapter 6 contains a new section on conditional power law distributions
withapplicationto“fractalplatetectonics”andasignificantdevelopmenton
Wilkstatisticsofembeddedhypothesistestingtocomparetherelativemerits
of power law versus stretched exponential distributions. A novel embedding
of the family of power law pdf’s within the family of stretched exponentials
VIII Preface
ispresentedandthegeneralformulasforthe covarianceoftheestimatorsare
given.
New figures have been added to Chap. 7 to clarify and enhance the dis-
cussion on the relevance of the concept of temperature to out-of-equilibrium
systems. Chapter 7 also contains a new section on the Beck–Cohen super-
statistics which provides a dynamical origin of non-extensive Tsallis-type
statistics.
Chapter 8 contains a new presentation of fractional diffusion equations,
their relationship with L´evy laws and the associated anomalous diffusion.
Chapter 10 contains applications of the critical precursors and critical
dynamics to explain for instance the way our internal hearing organ, the
cochlea, works.
Chapter 11 has been significantly expanded to include a section of func-
tional reconstructionof approximantsbased on renormalizationgroupideas,
which have been shown to be an improvement over the Pad´e approximants.
Chapter 11 also contains a new section on the Weierstrass and Weierstrass-
type functions and concludes with recalling Anderson’s message “more is
different.”
New figures have been added to Chap. 13 to clarify and enhance the
discussion of quasi-dynamical rupture models.
Chapter 14, already a favorite in the first edition, has been significantly
enhanced by including severalother mechanisms for the generationof power
law distributions. The discussion of the Kesten process in terms of multi-
plicativenoisehasbeenexpanded.Anewsectionpresentstheclassofgrowth
models withpreferentialattachment,whichhasawiderangeofapplications.
A new section discusses the superposition of log-normal pdf’s. Another sec-
tion presents the coherent-noise models and their limits for the application
to earthquakes.
Chapter15expandsonthemechanismofself-organizedcriticalityinterms
ofthefeedbackoftheorderparameterontothecontrolparameter.Anewsec-
tionalsopresentsthelinearfractionalstablemotionsfor extremaldynamics.
Chapter 16 contains a new section reviewing the fundamental Kol-
mogorov’s theorem on fragmentation models which played a fundamental
roleinattractingtheattentionontheimportanceoflog-normaldistributions
for general multiplicative processes.
I would have liked to enrich this second edition much more and remain
frustrated by the limits of its achievements. Nevertheless, I hope that the
readers,andespeciallythe“students”intheextraordinaryrichfieldsofcom-
plex dynamical systems, will find this new edition valuable.
In addition to the many collaborators and colleagues mentioned in the
preface of the first edition and who contributed to my understanding, this
second edition owes a lot to V.F. Pisarenko who provided numerous com-
ments and suggestions on the first edition, as well as detailed explanations
on subtle points in the field of mathematical statistics. The errors remain
Preface IX
mine.T.P.O’Brienhasbeenalsoverystimulatinginhispenetrativequestions
and remarks. In addition to my colleagues saluted in the first edition, I ac-
knowledge inspiring exchanges with Y. Ageon, S. Gluzman, A. Helmstetter,
K. Ide, Y.Y. Kagan, T. Lux, Y. Malevergne, M.E.J. Newman, A. Saichev,
H. Takayasu, H.J. Viljoen, V.I. Yukalov, and W.-X. Zhou.
UCLA and Nice,
October 2003 Didier Sornette
Preface to the First Edition:
Variability and Fluctuations
Life is fundamentally risky, reflecting the pervasive out-of-equilibrium na-
ture of the surrounding world. Risk is synonymous with uncertainty about
the future, leading not only to potential losses and perils, but also to gains.
This uncertainty results from the numerous dynamical factors entering our
life, giving it spice and color as well as its dangerous flavor. Life consists
of a succession of choices that have to be made with often limited knowl-
edge and in a complex and changing environment. These choices result in
a sequence of often unpredictable outcomes, whose accumulation defines the
specific trajectory characterizing each individual, somewhat similar to the
apparentrandomtrajectoryofaleafcarriedbyaturbulentwind.Thenotion
of risk is probably one of the most general concepts pervading all the facets
of our life [285, 794].
Risk is a companion to most of our daily activities, professional or pri-
vate. Crossing a street or driving a car involves risk that is quantified by the
statistics of traffic accidents and police reports and which impacts on our
insurancepremium.Stayingathomeisalsorisky:falling,burning,electrocu-
tion, plane crash, earthquakes, hurricanes, etc. Risk is present in the choice
of a career, in the selection of a college and university program as well as in
the effect ofsocialinteractionsonthe developmentofchildren.Any choiceis
intrinsically risky, since the existence of a choice implies several alternatives
that are all thought to be possible outcomes, albeit with possibly different
likelihood. In industry, companies have to face a multitude of risks: R&D,
choice of a niche, capital, production, sales, competition, etc., encompassing
all types of risks that, ideally, have to be optimized at each instant. The
apparent random nature of price variations in both organized and emerging
stock markets leads to risky investment choices, with impact on the global
economy and our welfare (retirement funds).
The Earth provides its share of risks, partly overcome with the devel-
opment of technology, but hurricanes, earthquakes, tsunamis, volcanic erup-
tions and meteorites bring episodic destruction each year, constituting as
many Damocles’ swords over our heads. Neither is biological risk negligible,
with endemic epidemics and the emergence of novel diseases. Human soci-
ety, with its technical development and population growth, introduces new
risks: unemployment, strike, dysfunction of cities, rupture of sensitive tech-