复杂性和临界状态

　　《复杂性和临界状态》是作者在从2000年开始给伦敦帝国理工学院研究生讲授统计力学的讲稿基础上形成的。

　　复杂性是21世纪的重点研究课题之一，而临界状态则是统计物理中已有相当深入研究的一个分支，《复杂性和临界状态》旨在采用统计力学的方法，以渗滤和伊辛模型为范例，讨论突破复杂性研究的途径。《复杂性和临界状态》共分3章，第一章讲述渗滤现象的研究方法，并就一维、二维渗滤的定义、点阵结构、块体的大小和数密度、关联函数、标度函数、临界指数和实空间重整化群的变换方法等方面都作了详尽的介绍。第二章的重点是讲述二维伊辛模型的相变理论，涉及相互作用自旋系统的自由能和配分函数、磁化强度和磁导率、能量和比热、响应函数、平均场理论、相变的朗道ˉ京茨堡理论、Widom标度假设、临界指数和Wilson重整化群论。第三章介绍自组织临界状态。本章从容易想象的所谓“沙堆”模型出发，讨论沙堆崩塌的物理处理方法，从中引入开放系统的平均场理论、二分叉理论和几率分布的矩分析、定态出现的条件等。本章还就地震和降雨的预测预报作了定性讨论。

　　《复杂性和临界状态》每章之后都有专门设计的练习，为了降低解题难度，每道题都细分为很多小题目，使解题思路十分明确。答案可从http://www.worldscibooks.com/physics/p365.htm/.查找。

　　为了使不同学科的读者克服数学上的困难，书后的8个附录把相关的数学知识和物理量都作了补充交代。

　　KimChristensen

　　伦敦帝国理工学院（ImperialCollegeLondon）理论物理教授。1990年于丹麦Arhus大学物理和天文研究所获得科学硕士。1993年于丹麦Arhus大学物理和天文研究所获得科学博士。

　　主要研究兴趣是外界因素引起非平衡系统复杂性变化的理论和数值研究，涉及统计力学、复杂性、标度不变性实验现象、自组织临界状态。

　　NicholasR.Moloney

　　伦敦帝国理工学院（ImperialCollegeLondon）Blackett实验室教授。

Contents
Preface
1.Percolation
1.1Introduction
1.1.1Definitionofsitepercolation
1.1.2Quantitiesofinterest
1.2Percolationind=1
1.2.1Clusternumberdensity
1.2.2Averageclustersize
1.2.3Transitiontopercolation
1.2.4Correlationfunction
1.2.5Criticaloccupationprobability
1.3PercolationontheBetheLattice
1.3.1DefinitionoftheBethelattice
1.3.2Criticaloccupationprobability
1.3.3Averageclustersize
1.3.4Transitiontopercolation
1.3.5Clusternumberdensity
1.3.6Correlationfunction
1.4Percolationind=2
1.4.1Transitiontopercolation
1.4.2Averageclustersize
1.4.3Clusternumberdensity-exact
1.4.4Clusternumberdensity-numerical
1.5ClusterNumberDensity-ScalingAnsatz
1.5.1Scalingfunctionanddatacollapse
1.5.2Scalingfunctionanddatacollapseind=1
1.5.3ScalingfunctionanddatacollapseontheBethelattice
1.5.4Scalingfunctionanddatacollapseind=2
1.6ScalingRelations
1.7GeometricPropertiesofClusters
1.7.1Self-similarityandfractaldimension
1.7.2Massofalargebutfiniteclusteratp=pc
1.7.3Correlationlength
1.7.4Massofthepercolatingclusterforp>pc
1.8Finite-SizeScaling
1.8.1Orderparameter
1.8.2Averageclustersizeandhighermoments
1.8.3Clusternumberdensity
1.9Non-UniversalCriticalOccupationProbabilities
1.10UniversalCriticalExponents
1.11Real-SpaceRenormalisation
1.11.1Self-similarityandthecorrelationlength
1.11.2Self-similarityandfixedpoints
1.11.3Coarsegrainingandrescaling
1.11.4Real-spacerenormalisationgroupprocedure
1.11.5Renormalisationind=1
1.11.6Renormalisationind=2onatriangularlattice
1.11.7Renormalisationind=2onasquarelattice
1.11.8Approximationviathetruncationofparameterspace
1.12Summary
Exercises

2.IsingModel
2.1Introduction
2.1.1DefinitionoftheIsingmodel
2.1.2Reviewofequilibriumstatisticalmechanics
2.1.3Thermodynamiclimit
2.2SystemofNon-InteractingSpins
2.2.1Partitionfunctionandfreeenergy
2.2.2Magnetisationandsusceptibility
2.2.3Energyandspecificheat
2.3QuantitiesofInterest
2.3.1Magnetisation
2.3.2Responsefunctions
2.3.3Correlationlengthandspin-spincorrelationfunction
2.3.4Criticaltemperatureandexternalfield
2.3.5Symmetrybreaking
2.4IsingModelind=1
2.4.1Partitionfunction
2.4.2Freeenergy
2.4.3Magnetisationandsusceptibility
2.4.4Energyandspecificheat
2.4.5Correlationfunction
2.4.6Criticaltemperature
2.5Mean-FieldTheoryoftheIsingModel
2.5.1Partitionfunctionandfreeenergy
2.5.2Magnetisationandsusceptibility
2.5.3Energyandspecificheat
2.6LandauTheoryoftheIsingModel
2.6.1Freeenergy
2.6.2Magnetisationandsusceptibility
2.6.3Specificheat
2.7LandauTheoryofContinuousPhaseTransitions
2.8IsingModelind=2
2.8.1Partitionfunction
2.8.2Magnetisationandsusceptibility
2.8.3Energyandspecificheat
2.8.4Criticaltemperature
2.9WidomScalingAnsatz
2.9.1Scalingansatzforthefreeenergy
2.9.2Scalingansatzforthespecificheat
2.9.3Scalingansatzforthemagnetisation
2.9.4Scalingansatzforthesusceptibility
2.9.5Scalingansatzforthespin-spincorrelationfunction
2.10ScalingRelations
2.11WidomScalingFormandCriticalExponentsind=1
2.12Non-UniversalCriticalTemperatures
2.13UniversalCriticalExponents
2.14GinzburgCriterion
2.15Real-SpaceRenormalisation
2.15.1Kadanoffsblockspintransformation
2.15.2Kadanoffsblockspinandthefreeenergy
2.15.3Kadanoffsblockspinandthecorrelationfunction
2.15.4Renormalisationind=1
2.15.5Renormalisationind=2onasquarelattice
2.16WilsonsRenormalisationGroupTheory
2.16.1Couplingspaceandrenormalisationgroupflow
2.16.2Self-similarityandfixedpoints
2.16.3Basinofattractionoffixedpoints
2.16.4RGflowincouplingandconfigurationalspace
2.16.5UniversalityandRGflownearfixedpoint
2.16.6Widomscalingform
2.17Summary
Exercises

3.Self-OrganisedCriticality
3.1Introduction
3.1.1Sandpilemetaphor
3.2BTWModelind=1
3.2.1AlgorithmoftheBTWmodelind=1
3.2.2Transientandrecurrentconfigurations
3.2.3Avalanchetimeseries
3.2.4Avalanche-sizeprobability
3.3Mean-FieldTheoryoftheBTWModel
3.3.1RandomneighbourBTWmodel
3.3.2AlgorithmoftherandomneighbourBTWmodel
3.3.3Steadystateandtheaverageavalanchesize
3.4BranchingProcess
3.4.1Branchingratio
3.4.2Avalanche-sizeprobability-exact
3.4.3Avalanche-sizeprobability-scalingform
3.5Avalanche-SizeProbability-ScalingAnsatz
3.6ScalingRelations
3.7MomentAnalysisofAvalanche-SizeProbability
3.8BTWModelind=2
3.8.1AlgorithmoftheBTWmodelind=2
3.8.2Steadystateandtheaverageavalanchesize
3.8.3Avalanchetimeseries
3.8.4Avalanche-sizeprobability
3.9RicepileExperimentandtheOsloModel
3.9.1Ricepileexperiment
3.9.2Ricepileavalanchetimeseries
3.9.3Ricepileavalanche-sizeprobabilitydensity
3.9.4Ricepilemodelling
3.9.5AlgorithmoftheOslomodel
3.9.6Transientandrecurrentconfigurations
3.9.7Avalanchetimeseries
3.9.8Avalanche-sizeprobability
3.10EarthquakesandtheOFCModel
3.10.1Earthquakemechanism
3.10.2Earthquaketimeseries
3.10.3Earthquake-sizefrequency
3.10.4Earthquakemodelling
3.10.5AlgorithmoftheOFCmodel
3.10.6Steadystateandtheaverageavalanchesize
3.10.7Avalanchetimeseries
3.10.8Avalanche-sizeprobability
3.11Rainfall
3.11.1Rainfallmechanism
3.11.2Rainfalltimeseries
3.11.3Rainfall-sizenumberdensity
3.12Summary
Exercises

AppendixATaylorExpansion
AppendixBHyperbolicFunctions
AppendixCHomogeneousandScalingFunctions
AppendixDFractals
AppendixEDataBinning
AppendixFBoltzmannDistribution
AppendixGFreeEnergy
AppendixHMetropolisAlgorithm
Bibliography
ListofSymbols