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9781860945045

Complexity And Criticality

by ;
  • ISBN13:

    9781860945045

  • ISBN10:

    186094504X

  • Format: Hardcover
  • Copyright: 2005-12-30
  • Publisher: World Scientific Pub Co Inc
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Supplemental Materials

What is included with this book?

Summary

- The book is self-contained and accessible to readers not familiar with the concepts within cooperative phenomena- Can form the basis for advanced undergraduate as well as postgraduate courses in cooperative phenomena- An introductory reference for researchers in various fields- The text is accompanied by exercises with solutions and various visual interactive simulations for all the models considered to allow the reader to experience for themselves the behavior of the models

Table of Contents

Preface vii
Percolation
1(114)
Introduction
1(4)
Definition of site percolation
3(1)
Quantities of interest
3(2)
Percolation in d = 1
5(10)
Cluster number density
6(2)
Average cluster size
8(5)
Transition to percolation
13(1)
Correlation function
13(1)
Critical occupation probability
14(1)
Percolation on the Bethe Lattice
15(15)
Definition of the Bethe lattice
15(1)
Critical occupation probability
16(1)
Average cluster size
16(3)
Transition to percolation
19(3)
Cluster number density
22(7)
Correlation function
29(1)
Percolation in d = 2
30(9)
Transition to percolation
31(2)
Average cluster size
33(2)
Cluster number density -- exact
35(1)
Cluster number density -- numerical
35(4)
Cluster Number Density -- Scaling Ansatz
39(12)
Scaling function and data collapse
41(1)
Scaling function and data collapse in d = 1
42(2)
Scaling function and data collapse on the Bethe lattice
44(5)
Scaling function and data collapse in d = 2
49(2)
Scaling Relations
51(4)
Geometric Properties of Clusters
55(14)
Self-similarity and fractal dimension
55(3)
Mass of a large but finite cluster at p = pc
58(4)
Correlation length
62(1)
Mass of the percolating cluster for p > pc
63(6)
Finite-Size Scaling
69(9)
Order parameter
70(3)
Average cluster size and higher moments
73(2)
Cluster number density
75(3)
Non-Universal Critical Occupation Probabilities
78(3)
Universal Critical Exponents
81(1)
Real-Space Renormalisation
82(20)
Self-similarity and the correlation length
82(1)
Self-similarity and fixed points
83(2)
Coarse graining and rescaling
85(2)
Real-space renormalisation group procedure
87(4)
Renormalisation in d = 1
91(4)
Renormalisation in d = 2 on a triangular lattice
95(3)
Renormalisation in d = 2 on a square lattice
98(2)
Approximation via the truncation of parameter space
100(2)
Summary
102(13)
Exercises
104(11)
Ising Model
115(134)
Introduction
115(9)
Definition of the Ising model
116(3)
Review of equilibrium statistical mechanics
119(4)
Thermodynamic limit
123(1)
System of Non-Interacting Spins
124(7)
Partition function and free energy
125(2)
Magnetisation and susceptibility
127(2)
Energy and specific heat
129(2)
Quantities of Interest
131(9)
Magnetisation
131(2)
Response functions
133(1)
Correlation length and spin-spin correlation function
134(1)
Critical temperature and external field
135(3)
Symmetry breaking
138(2)
Ising Model in d = 1
140(16)
Partition function
141(2)
Free energy
143(2)
Magnetisation and susceptibility
145(4)
Energy and specific heat
149(2)
Correlation function
151(3)
Critical temperature
154(2)
Mean-Field Theory of the Ising Model
156(13)
Partition function and free energy
157(1)
Magnetisation and susceptibility
158(8)
Energy and specific heat
166(3)
Landau Theory of the Ising Model
169(6)
Free energy
170(2)
Magnetisation and susceptibility
172(3)
Specific heat
175(1)
Landau Theory of Continuous Phase Transitions
175(4)
Ising Model in d = 2
179(9)
Partition function
179(1)
Magnetisation and susceptibility
180(4)
Energy and specific heat
184(2)
Critical temperature
186(2)
Widom Scaling Ansatz
188(5)
Scaling ansatz for the free energy
190(1)
Scaling ansatz for the specific heat
191(1)
Scaling ansatz for the magnetisation
191(1)
Scaling ansatz for the susceptibility
192(1)
Scaling ansatz for the spin-spin correlation function
192(1)
Scaling Relations
193(2)
Widom Scaling Form and Critical Exponents in d = 1
195(3)
Non-Universal Critical Temperatures
198(1)
Universal Critical Exponents
199(1)
Ginzburg Criterion
200(2)
Real-Space Renormalisation
202(20)
Kadanoff's block spin transformation
202(4)
Kadanoff's block spin and the free energy
206(3)
Kadanoff's block spin and the correlation function
209(2)
Renormalisation in d = 1
211(4)
Renormalisation in d = 2 on a square lattice
215(7)
Wilson's Renormalisation Group Theory
222(15)
Coupling space and renormalisation group flow
222(5)
Self-similarity and fixed points
227(2)
Basin of attraction of fixed points
229(1)
RG flow in coupling and configurational space
230(1)
Universality and RG flow near fixed point
231(4)
Widom scaling form
235(2)
Summary
237(12)
Exercises
241(8)
Self-Organised Criticality
249(92)
Introduction
249(6)
Sandpile metaphor
250(5)
BTW Model in d = 1
255(9)
Algorithm of the BTW model in d = 1
256(1)
Transient and recurrent configurations
257(2)
Avalanche time series
259(1)
Avalanche-size probability
260(4)
Mean-Field Theory of the BTW Model
264(3)
Random neighbour BTW model
264(1)
Algorithm of the random neighbour BTW model
264(1)
Steady state and the average avalanche size
265(2)
Branching Process
267(6)
Branching ratio
267(1)
Avalanche-size probability -- exact
268(2)
Avalanche-size probability -- scaling form
270(3)
Avalanche-Size Probability -- Scaling Ansatz
273(2)
Scaling Relations
275(1)
Moment Analysis of Avalanche-Size Probability
276(2)
BTW Model in d = 2
278(7)
Algorithm of the BTW model in d = 2
278(1)
Steady state and the average avalanche size
279(1)
Avalanche time series
280(1)
Avalanche-size probability
281(4)
Ricepile Experiment and the Oslo Model
285(17)
Ricepile experiment
285(2)
Ricepile avalanche time series
287(2)
Ricepile avalanche-size probability density
289(1)
Ricepile modelling
290(1)
Algorithm of the Oslo model
291(1)
Transient and recurrent configurations
292(1)
Avalanche time series
293(1)
Avalanche-size probability
294(8)
Earthquakes and the OFC Model
302(23)
Earthquake mechanism
302(1)
Earthquake time series
303(1)
Earthquake-size frequency
304(1)
Earthquake modelling
305(4)
Algorithm of the OFC model
309(1)
Steady state and the average avalanche size
310(4)
Avalanche time series
314(4)
Avalanche-size probability
318(7)
Rainfall
325(6)
Rainfall mechanism
325(1)
Rainfall time series
326(3)
Rainfall-size number density
329(2)
Summary
331(10)
Exercises
334(7)
Appendix A Taylor Expansion 341(2)
Appendix B Hyperbolic Functions 343(2)
Appendix C Homogeneous and Scaling Functions 345(6)
Appendix D Fractals 351(4)
Appendix E Data Binning 355(4)
Appendix F Boltzmann Distribution 359(2)
Appendix G Free Energy 361(2)
Appendix H Metropolis Algorithm 363(2)
Bibliography 365(6)
List of Symbols 371(10)
Index 381

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