Thermodynamic formalism: the mathematical structures of equilibrium statistical mechanics

195 Pages·2004·0.792 MB·English

by  David Ruelle

ThermodynamicFormalism
TheMathematicalStructuresofEquilibriumStatisticalMechanics
SecondEdition
ReissuedintheCambridgeMathematicalLibrarythisclassicbookoutlinesthetheory
ofthermodynamicformalismwhichwasdevelopedtodescribethepropertiesof
certainphysicalsystemsconsistingofalargenumberofsubunits.Itisaimedat
mathematiciansinterestedinergodictheory,topologicaldynamics,constructive
quantumfieldtheory,andthestudyofcertaindifferentiabledynamicalsystems,
notablyAnosovdiffeomorphismsandflows.Itisalsoofinteresttotheoretical
physicistsconcernedwiththecomputationalbasisofequilibriumstatisticalmechanics.
Thelevelofthepresentationisgenerallyadvanced,theobjectivebeingtoprovidean
efficientresearchtoolandatextforuseingraduateteaching.Backgroundmaterialon
physicshasbeencollectedinappendicestohelpthereader.Extramaterialisgivenin
theformofupdatesofproblemsthatwereopenattheoriginaltimeofwritingandasa
newprefacespeciallywrittenforthiseditionbytheauthor.
David RuelleisaProfessorEmeritusattheInstitutdesHautesEtudes
Scientifiques,Bures-sur-Yvette,Paris.
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Thermodynamic Formalism
The Mathematical Structures of Equilibrium
Statistical Mechanics
SecondEdition
DAVID RUELLE
InstitutdesHautesEtudesScientifiques
CAMBRIDGEUNIVERSITYPRESS
Cambridge,NewYork,Melbourne,Madrid,CapeTown,Singapore,SãoPaulo
Cambridge University Press
TheEdinburghBuilding,CambridgeCB22RU,UK
PublishedintheUnitedStatesofAmericabyCambridgeUniversityPress,NewYork
www.cambridge.org
Information on this title: www.cambridg e.org /9780521546492
©FirsteditionAddison-WesleyPublishingCompany,Inc.1978
Second edition Cambridge University Press 2004
Thispublicationisincopyright.Subjecttostatutoryexceptionandtotheprovisionof
relevantcollectivelicensingagreements,noreproductionofanypartmaytakeplace
withoutthewrittenpermissionofCambridgeUniversityPress.
Firstpublishedinprintformat 2004
ISBN-13 978-0-511-26581-5 eBook (NetLibrary)
ISBN-10 0-511-26581-6 eBook (NetLibrary)
ISBN-13 978-0-521-54649-2 paperback
ISBN-10 0-521-54649-4 paperback
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forexternalorthird-partyinternetwebsitesreferredtointhispublication,anddoesnot
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orthelast,either.
Youknowus.
J.Vinograd

Contents
Forewordtothefirstedition pagexv
Prefacetothefirstedition xvii
Prefacetothesecondedition xix
Introduction 1
0.1 Generalities 1
0.2 Descriptionofthethermodynamicformalism 3
0.3 Summaryofcontents 9
1 TheoryofGibbsstates 11
1.1 Configurationspace 11
1.2 Interactions 12
1.3 Gibbsensemblesandthermodynamiclimit 13
1.4 Proposition 14
1.5 Gibbsstates 14
1.6 ThermodynamiclimitofGibbsensembles 15
1.7 Boundaryterms 16
1.8 Theorem 18
1.9 Theorem 18
1.10 Algebraatinfinity 19
1.11 Theorem(characterizationofpureGibbsstates) 20
1.12 TheoperatorsM 20
(cid:3)
1.13 Theorem(characterizationofuniqueGibbsstates) 21
1.14 Remark 22
Notes 23
Exercises 23
vii
viii Contents
2 Gibbsstates:complements 24
2.1 Morphismsoflatticesystems 24
2.2 Example 25
2.3 Theinteraction F∗(cid:4) 25
2.4 Lemma 26
2.5 Proposition 26
2.6 Remarks 27
2.7 Systemsofconditionalprobabilities 28
2.8 PropertiesofGibbsstates 29
2.9 Remark 30
Notes 30
Exercises 31
3 Translationinvariance.Theoryofequilibriumstates 33
3.1 Translationinvariance 33
3.2 Thefunction A(cid:4) 34
3.3 Partitionfunctions 35
3.4 Theorem 36
3.5 Invariantstates 39
3.6 Proposition 39
3.7 Theorem 40
3.8 Entropy 42
3.9 InfinitelimitinthesenseofvanHove 43
3.10 Theorem 43
3.11 Lemma 45
3.12 Theorem 45
3.13 Corollary 47
3.14 Corollary 48
3.15 Physicalinterpretation 48
3.16 Theorem 49
3.17 Corollary 49
3.18 Approximationofinvariantstatesbyequilibriumstates 50
3.19 Lemma 50
3.20 Theorem 52
3.21 Coexistenceofphases 53
Notes 54
Exercises 54
Contents ix
4 ConnectionbetweenGibbsstatesandEquilibriumstates 57
4.1 Generalities 57
4.2 Theorem 58
4.3 Physicalinterpretation 59
4.4 Proposition 59
4.5 Remark 60
4.6 Strictconvexityofthepressure 61
4.7 Proposition 61
ν ν
4.8 Z -latticesystemsandZ -morphisms 62
4.9 Proposition 62
4.10 Corollary 63
4.11 Remark 63
4.12 Proposition 64
ν
4.13 Restrictionof Z toasubgroupG 64
4.14 Proposition 65
4.15 Undecidabilityandnon-periodicity 65
Notes 66
Exercises 66
5 One-dimensionalsystems 69
5.1 Lemma 70
5.2 Theorem 70
5.3 Theorem 71
5.4 Lemma 72
5.5 Proofoftheorems5.2and5.3 73
5.6 Corollariestotheorems5.2and5.3 75
5.7 Theorem 76
5.8 MixingZ-latticesystems 78
5.9 Lemma 78
5.10 Theorem 79
5.11 ThetransfermatrixandtheoperatorL 80
5.12 Thefunctionψ> 81
5.13 Proposition 81
5.14 TheoperatorS 82
5.15 Lemma 82
5.16 Proposition 82
5.17 Remark 83
5.18 Exponentiallydecreasinginteractions 83