研究生前沿教材书系:复杂性和临界状态 电子书下载 PDF下载

研究生前沿教材书系:复杂性和临界状态
内容简介
《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W ilson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。 ·查看全部>>
目录
Contents
Preface
1. Percolation
1.1 Introduction
1.1.1 Definition of site percolation
1.1.2 Quantities of interest
1.2 Percolation in d=1
1.2.1 Cluster number density
1.2.2 Average cluster size
1.2.3 Transition to percolation
1.2.4 Correlation function
1.2.5 Critical occupation probability
1.3 Percolation on the Bethe Lattice
1.3.1 Definition of the Bethe lattice
1.3.2 Critical occupation probability
1.3.3 Average cluster size
1.3.4 Transition to percolation
1.3.5 Cluster number density
1.3.6 Correlation function
1.4 Percolation in d=2
1.4.1 Transition to percolation
1.4.2 Average cluster size
1.4.3 Cluster number density- exact
1.4.4 Cluster number density - numerical
1.5 Cluster Number Density- Scaling Ansatz
1.5.1 Scaling function and data collapse
1.5.2 Scaling function and data collapse in d = 1
1.5.3 Scaling function and data collapse on the Bethe lattice
1.5.4 Scaling function and data collapse in d = 2
1.6 Scaling Relations
1.7 Geometric Properties of Clusters
1.7.1 Self-similarity and fractal dimension
1.7.2 Mass of a large but finite cluster at p = pc
1.7.3 Correlation length
1.7.4 Mass of the percolating cluster for p > pc
1.8 Finite-Size Scaling
1.8.1 Order parameter
1.8.2 Average cluster size and higher moments
1.8.3 Cluster number density
1.9 Non-Universal Critical Occupation Probabilities
1.10 Universal Critical Exponents
1.11 Real-Space Renormalisation
1.11.1 Self-similarity and the correlation length
1.11.2 Self-similarity and fixed points
1.11.3 Coarse graining and rescaling
1.11.4 Real-space renormalisation group procedure
1.11.5 Renormalisation in d = 1
1.11.6 Renormalisation in d = 2 on a triangular lattice
1.11.7 Renormalisation in d = 2 on a square lattice
1.11.8 Approximation via the truncation of parameter space
1.12 Summary
Exercises

2. Ising Model
2.1 Introduction
2.1.1 Definition of the Ising model
2.1.2 Review of equilibrium statistical mechanics
2.1.3 Thermodynamic limit
2.2 System of Non-Interacting Spins
2.2.1 Partition function and free energy
2.2.2 Magnetisation and susceptibility
2.2.3 Energy and specific heat
2.3 Quantities of Interest
2.3.1 Magnetisation
2.3.2 Response functions
2.3.3 Correlation length and spin-spin correlation function
2.3.4 Critical temperature and external field
2.3.5 Symmetry breaking
2.4 Ising Model in d = 1
2.4.1 Partition function
2.4.2 Free energy
2.4.3 Magnetisation and susceptibility
2.4.4 Energy and specific heat
2.4.5 Correlation function
2.4.6 Critical temperature
2.5 Mean-Field Theory of the Ising Model
2.5.1 Partition function and free energy
2.5.2 Magnetisation and susceptibility
2.5.3 Energy and specific heat
2.6 Landau Theory of the Ising Model
2.6.1 Free energy
2.6.2 Magnetisation and susceptibility
2.6.3 Specific heat
2.7 Landau Theory of Continuous Phase Transitions
2.8 Ising Model in d = 2
2.8.1 Partition function
2.8.2 Magnetisation and susceptibility
2.8.3 Energy and specific heat
2.8.4 Critical temperature
2.9 Widom Scaling Ansatz
2.9.1 Scaling ansatz for the free energy
2.9.2 Scaling ansatz for the specific heat
2.9.3 Scaling ansatz for the magnetisation
2.9.4 Scaling ansatz for the susceptibility
2.9.5 Scaling ansatz for the spin-spin correlation function
2.10 Scaling Relations
2.11Widom Scaling Form and Critical Exponents in d = 1
2.12 Non-Universal Critical Temperatures
2.13 Universal Critical Exponents
2.14 Ginzburg Criterion
2.15 Real-Space Renormalisation
2.15.1 Kadanoffs block spin transformation
2.15.2 Kadanoffs block spin and the free energy
2.15.3 Kadanoffs block spin and the correlation function
2.15.4 Renormalisation in d = 1
2.15.5 Renormalisation in d = 2 on a square lattice
2.16 Wilsons Renormalisation Group Theory
2.16.1 Coupling space and renormalisation group flow
2.16.2 Self-similarity and fixed points
2.16.3 Basin of attraction of fixed points
2.16.4 RG flow in coupling and configurational space
2.16.5 Universality and RG flow near fixed point
2.16.6 Widom scaling form
2.17 Summary
Exercises

3. Self-Organised Criticality
3.1 Introduction
3.1.1 Sandpile metaphor
3.2 BTW Model in d = 1
3.2.1 Algorithm of the BTW model in d = 1
3.2.2 Transient and recurrent configurations
3.2.3 Avalanche time series
3.2.4 Avalanche-size probability
3.3 Mean-Field Theory of the BTW Model
3.3.1 Random neighbour BTW model
3.3.2 Algorithm of the random neighbour BTW model
3.3.3 Steady state and the average avalanche size
3.4 Branching Process
3.4.1 Branching ratio
3.4.2 Avalanche-size probability - exact
3.4.3 Avalanche-size probability - scaling form
3.5 Avalanche-Size Probability- Scaling Ansatz
3.6 Scaling Relations
3.7 Moment Analysis of Avalanche-Size Probability
3.8 BTW Model in d = 2
3.8.1 Algorithm of the BTW model in d = 2
3.8.2 Steady state and the average avalanche size
3.8.3 Avalanche time series
3.8.4 Avalanche-size probability
3.9 Ricepile Experiment and the Oslo Model
3.9.1 Ricepile experiment
3.9.2 Ricepile avalanche time series
3.9.3 Ricepile avalanche-size probability density
3.9.4 Ricepile modelling
3.9.5 Algorithm of the Oslo model
3.9.6 Transient and recurrent configurations
3.9.7 Avalanche time series
3.9.8 Avalanche-size probability
3.10 Earthquakes and the OFC Model
3.10.1 Earthquake mechanism
3.10.2 Earthquake time series
3.10.3 Earthquake-size frequency
3.10.4 Earthquake modelling
3.10.5 Algorithm of the OFC model
3.10.6 Steady state and the average avalanche size
3.10.7 Avalanche time series
3.10.8 Avalanche-size probability
3.11 Rainfall
3.11.1 Rainfall mechanism
3.11.2 Rainfall time series
3.11.3 Rainfall-size number density
3.12 Summary
Exercises

Appendix A Taylor Expansion
Appendix B Hyperbolic Functions
Appendix C Homogeneous and Scaling Functions
Appendix D Fractals
Appendix E Data Binning
Appendix F Boltzmann Distribution
Appendix G Free Energy
Appendix H Metropolis Algorithm
Bibliography
List of Symbols
Copyright © 2020 by topbester.com.
All Rights Reserved.
沪ICP备14027842号-1