Self-Organised Criticality

518 Pages·2012·4.605 MB·English

by  Pruessner G.

Self-OrganisedCriticality
Theory,ModelsandCharacterisation
Givingadetailedoverviewofthesubject,thisbooktakesintheresultsandmethodsthat
havearisensincetheterm‘self-organisedcriticality’wascoinedtwentyyearsago.
Providinganoverviewofnumericalandanalyticalmethods,fromtheirtheoreticalfoun-
dation to actual application and implementation, the book is an easy access point to im-
portantresultsandsophisticatedmethods.StartingwiththefamousBak–Tang–Wiesenfeld
sandpile,tenkeymodelsarecarefullydefined,togetherwiththeirresultsandapplications.
Comprehensive tables of numerical results are collected in one volume for the first time,
makingtheinformationreadilyaccessibletoreaders.
Writtenforgraduatestudentsandpractisingresearchersinarangeofdisciplines,from
physicsandmathematicstobiology,sociology,finance,medicineandengineering,thebook
givesapractical,hands-onapproachthroughout.Methodsandresultsareappliedinways
thatwillrelatetothereader’sownresearch.
GunnarPruessnerisaLecturerinMathematicalPhysicsintheDepartmentofMathematics
atImperialCollegeLondon.Hisresearchrangesfromcomplexity,throughfieldtheoretic
methods and applications, to synchronisation and the application of statistical mechanics
inthemedicalsciences.

Self-Organised Criticality
Theory, Models and Characterisation
GUNNAR PRUESSNER
ImperialCollegeLondon
CAMBRIDGE UNIVERSITY PRESS
Cambridge,NewYork,Melbourne,Madrid,CapeTown,
Singapore,Sa˜oPaulo,Delhi,MexicoCity
CambridgeUniversityPress
TheEdinburghBuilding,CambridgeCB28RU,UK
PublishedintheUnitedStatesofAmericabyCambridgeUniversityPress,NewYork
www.cambridge.org
Informationonthistitle:www.cambridge.org/9780521853354
(cid:1)C G.Pruessner2012
Thispublicationisincopyright.Subjecttostatutoryexception
andtotheprovisionsofrelevantcollectivelicensingagreements,
noreproductionofanypartmaytakeplacewithoutthewritten
permissionofCambridgeUniversityPress.
Firstpublished2012
PrintedintheUnitedKingdomattheUniversityPress,Cambridge
AcataloguerecordforthispublicationisavailablefromtheBritishLibrary
ISBN 978-0-521-85335-4Hardback
CambridgeUniversityPresshasnoresponsibilityforthepersistenceor
accuracyofURLsforexternalorthird-partyinternetwebsitesreferredto
inthispublication,anddoesnotguaranteethatanycontentonsuch
websitesis,orwillremain,accurateorappropriate.
InmemoryofourfriendHolgerBruhn.
Toseetheworldinagrainofsand,
andtoseeheaveninawildflower,
holdinfinityinthepalmofyourhands,
andeternityinanhour.
WilliamBlake,AuguriesofInnocence,ca1803

Contents
Listoftables pagex
ForewordbyHenrikJ.Jensen xi
Preface xiii
Listofsymbols xvii
PartI Introduction 1
1 Introduction 3
1.1 Reviews 5
1.2 Basicingredients 7
1.3 Basicobservablesandobservations 11
1.4 Universality 18
2 Scaling 25
2.1 Distributionfunctions 26
2.2 Moments 37
2.3 Algebraiccorrelations 46
2.4 Multiscaling 50
3 Experimentsandobservations 52
3.1 Granularmedia 53
3.2 Superconductors 58
3.3 Barkhauseneffect 62
3.4 Earthquakes 66
3.5 Evolution 68
3.6 Neuralnetworks 69
3.7 Othersystems 71
3.8 Averybriefconclusion 76
PartII Modelsandnumerics 79
4 Deterministicsandpiles 85
4.1 TheBAK–TANG–WIESENFELDModel 85
4.2 Dhar’sAbelianSandpileModel 91
4.3 TheZHANGModel 104
vii
viii Contents
5 Dissipativemodels 111
5.1 TheBAK–CHEN–TANGForestFireModel 112
5.2 TheDROSSEL–SCHWABLForestFireModel 115
5.3 TheOLAMI–FEDER–CHRISTENSENModel 125
5.4 TheBAK–SNEPPENModel 141
6 Stochasticsandpiles 162
6.1 TheMANNAModel 162
6.2 Dhar’sAbelianMANNAModel 166
6.3 TheOSLOModel 183
7 Numericalmethodsanddataanalysis 210
7.1 Independenceandcorrelations 211
7.2 Subsampling 215
7.3 Momentanalysis 220
7.4 PDFestimation 229
PartIII Theory 241
8 Analyticalresults 243
8.1 Randomwalkersandmean-fieldapproaches 243
8.2 Renormalisation 265
8.3 Abeliansymmetryandtheoperatorapproach 269
8.4 Directedmodels 285
8.5 Surfaces,interfaces,depinningandgrowthphenomena 298
9 Mechanisms 317
9.1 SOCandordinarycriticality 318
9.2 Dissipativetransportinopensystems 321
9.3 TheASmechanism 329
9.4 Othermechanisms 342
10 Summaryanddiscussion 346
10.1 Convenientfallaciesandinconvenienttruths 348
10.2 Outlook 355
Appendix TheOLAMI–FEDER–CHRISTENSENModelinC 357
A.1 Stacks 362
A.2 Moments 363
A.3 Boxes 365
A.4 Main 369
A.5 Outputandtermination 383
A.6 Othertechniques 384