1 Fractals |
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1 | (54) |
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1.1 The Concepts of Scale Invariance and Self-Similarity |
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1 | (3) |
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1.2 Measure Versus Dimensionality |
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4 | (8) |
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1.3 Self-Similarity (Scale Invariance) as the Origin of the Fractal Dimension |
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12 | (2) |
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14 | (2) |
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16 | (3) |
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1.6 The Geometrical Support of Multifractals |
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19 | (4) |
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1.7 Multifractals, Examples |
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23 | (18) |
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23 | (2) |
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1.7.2 The General Case of the Cantor Set |
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25 | (1) |
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1.7.3 Dimensions of the Subsets |
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26 | (3) |
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1.7.4 Lengths of the Subsets |
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29 | (5) |
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1.7.5 Measures of the Subsets |
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34 | (5) |
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1.7.6 Analogy with Statistical Physics |
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39 | (1) |
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1.7.7 Subsets η Versus Subsets α |
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40 | (1) |
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41 | (1) |
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1.8 The General Formalism of Multifractals |
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41 | (7) |
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1.9 Moments of the Measure Distribution |
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48 | (4) |
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52 | (3) |
2 Ensemble Theory in Statistical Physics: Free Energy Potential |
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55 | (94) |
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55 | (2) |
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57 | (6) |
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2.3 Microcanonical Ensemble |
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63 | (6) |
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2.4 MCE: Fluctuations as Nonequilibrium Probability Distributions |
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69 | (11) |
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2.5 Free Energy Potential of the MCE |
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80 | (8) |
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2.6 MCE: Free Energy Minimization Principle (Entropy Maximization Principle) |
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88 | (2) |
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90 | (5) |
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2.8 Nonequilibrium Fluctuations of the Canonical Ensemble |
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95 | (4) |
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2.9 Properties of the Probability Distribution of Energy Fluctuations |
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99 | (7) |
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2.10 Method of Steepest Descent |
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106 | (10) |
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2.11 Entropy of the CE. The Equivalence of the MCE and CE |
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116 | (2) |
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2.12 Free Energy Potential of the CE |
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118 | (6) |
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2.13 Free Energy Minimization Principle |
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124 | (2) |
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126 | (13) |
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2.15 Fluctuations as the Investigator's Tool |
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139 | (3) |
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2.16 The Action of the Free Energy |
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142 | (4) |
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146 | (3) |
3 The Ising Model |
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149 | (76) |
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3.1 Definition of the Model |
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149 | (3) |
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3.2 Microstates, MCE, CE, Order Parameter |
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152 | (3) |
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3.3 Two-Level System Without Pair Spins Interactions |
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155 | (4) |
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3.4 A One-Dimensional Nonideal System with Short-Range Pair Spin Interactions: The Exact Solution |
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159 | (6) |
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3.5 Nonideal System with Pair Spin Interactions: The Mean-Field Approach |
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165 | (5) |
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170 | (30) |
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3.6.1 The Equation of State |
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170 | (2) |
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3.6.2 The Minimization of Free Energy |
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172 | (4) |
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3.6.3 Stable, Metastable, Unstable States, and Maxwell's Rule |
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176 | (4) |
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180 | (2) |
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182 | (4) |
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3.6.6 Equilibrium Free Energy |
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186 | (2) |
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3.6.7 Classification of Phase Transitions |
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188 | (1) |
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3.6.8 Critical and Spinodal Slowing Down |
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189 | (6) |
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3.6.9 Heterogeneous System |
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195 | (5) |
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200 | (7) |
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207 | (10) |
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3.9 Antiferromagnet on a Triangular Lattice. Frustration |
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217 | (2) |
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3.10 Mixed Ferromagnet-Antiferromagnet |
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219 | (2) |
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221 | (4) |
4 The Theory of Percolation |
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225 | (34) |
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4.1 The Model of Percolation |
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226 | (3) |
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4.2 One-Dimensional Percolation |
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229 | (4) |
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233 | (4) |
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237 | (10) |
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247 | (6) |
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4.6 The Moments of the Cluster-Size Distribution |
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253 | (3) |
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256 | (3) |
5 Damage Phenomena |
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259 | (30) |
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5.1 The Parameter of Damage |
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259 | (2) |
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5.2 The Fiber-Bundle Model with Quenched Disorder |
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261 | (2) |
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5.3 The Ensemble of Constant Strain |
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263 | (4) |
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267 | (3) |
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5.5 The Ensemble of Constant Stress |
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270 | (7) |
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5.6 Spinodal Slowing Down |
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277 | (4) |
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5.7 FBM with Annealed Disorder |
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281 | (2) |
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283 | (6) |
6 Correlations, Susceptibility, and the Fluctuation-Dissipation Theorem |
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289 | (76) |
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6.1 Correlations: The One-Dimensional Ising Model with Short-Range Interactions |
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290 | (6) |
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6.2 Correlations: The Mean-Field Approach for the Ising Model in Higher Dimensions |
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296 | (15) |
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6.3 Magnetic Systems: The Fluctuation-Dissipation Theorem |
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311 | (7) |
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6.4 Magnetic Systems: The Ginzburg Criterion |
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318 | (5) |
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6.5 Magnetic Systems: Heat Capacity as Susceptibility |
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323 | (5) |
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6.6 Percolation: The Correlation Length |
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328 | (5) |
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6.7 Percolation: Fluctuation-Dissipation Theorem |
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333 | (3) |
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6.8 Percolation: The Hyperscaling Relation and the Scaling of the Order Parameter |
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336 | (5) |
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6.9 Why Percolation Differs from Magnetic Systems |
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341 | (2) |
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6.10 Percolation: The Ensemble of Clusters |
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343 | (6) |
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6.11 The FBM: The Fluctuation-Dissipation Theorem |
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349 | (3) |
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352 | (3) |
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6.13 The FBM: The epsilon-Ensemble |
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355 | (1) |
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6.14 The FBM: The σ-Ensemble |
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356 | (7) |
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363 | (2) |
7 The Renormalization Group |
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365 | (56) |
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366 | (2) |
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7.2 RG Approach of a Single Survivor: One-Dimensional Magnetic Systems |
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368 | (14) |
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7.3 RG Approach of a Single Survivor: Two-Dimensional Magnetic Systems |
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382 | (4) |
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7.4 RG Approach of Representation: Two-Dimensional Magnetic Systems in the Absence of Magnetic Field |
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386 | (12) |
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7.5 RG Approach of Representation: Two-Dimensional Magnetic Systems in the Presence of Magnetic Field |
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398 | (8) |
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406 | (8) |
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414 | (2) |
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7.8 Why does the RG Transformation Return only Approximate Results? |
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416 | (2) |
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418 | (3) |
8 Scaling: The Finite-Size Effect and Crossover Effects |
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421 | (74) |
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8.1 Percolation: Why Is the Cluster-Size Distribution Hypothesis Wrong? |
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421 | (7) |
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8.2 Percolation: The Finite-Size Effect |
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428 | (15) |
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8.3 Magnetic Systems: The Scaling of Landau Theory |
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443 | (10) |
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8.4 Magnetic Systems: Scaling Hypotheses |
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453 | (6) |
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8.5 Magnetic Systems: Superseding Correction |
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459 | (5) |
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8.6 Crossover Effect of Magnetic Field |
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464 | (4) |
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8.7 Magnetic Systems: Crossover Phenomena |
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468 | (1) |
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8.8 Magnetic Systems: The Finite-Size Effect |
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469 | (3) |
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8.9 The Illusory Asymmetry of the Temperature |
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472 | (3) |
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8.10 The Formalism of General Homogeneous Functions |
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475 | (3) |
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8.11 The Renormalization Group as the Source of Scaling |
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478 | (12) |
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8.12 Magnetic Systems: Spinodal Scaling |
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490 | (2) |
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492 | (3) |
Index |
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495 | |