非平衡相变 第2卷 英文PDF电子书下载

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1 Ageing Phenomena 1

1.1 Introduction 1

1.1.1 Ageing in Mechanically Deformed Polymers 2

1.1.2 Correlations and Responses 8

1.1.3 Ageing in Spin Glasses 10

1.1.4 Ageing in Simple Magnets 14

1.1.5 Mean-field Theory 18

1.1.6 Breaking of the Fluctuation-dissipation Theorem：Experiments 21

1.1.7 Breaking of the Fluctuation-dissipation Theorem：Two Simple Solvable Models 27

1.1.8 Outline 32

1.2 Phase-ordering Kinetics 33

1.2.1 Linear Stability Analysis 35

1.2.2 Domain Walls 36

1.2.3 The Allen-Cahn Equation 36

1.2.4 Topological Defects 38

1.2.5 Porod’s Law 40

1.2.6 Bray-Rutenberg Theory for the Growth Law 43

1.2.7 Exact Result in Two Dimensions 47

1.2.8 Conserved Order-parameter： Phase-separation 49

1.2.9 Critical Dynamics 51

1.3 Phenomenology of Ageing 51

1.3.1 Scaling Forms 52

1.3.2 Passage into the Ageing Regime 54

1.3.3 Kurchan’s Lemma 57

1.3.4 The Yeung-Rao-Desai Inequalities 59

1.4 Scaling Behaviour of Integrated Responses 61

1.4.1 Thermoremanent Susceptibility 62

1.4.2 Zero-eld Cooled Susceptibility 64

1.4.3 Intermediate Susceptibility 67

1.4.4 Alternating Susceptibility 67

1.5 Values of Non-equilibrium Exponents 67

1.5.1 Values of the Ageing Exponents a and b 67

1.5.2 Values of the Critical Autocorrelation Exponent 73

1.5.3 Values of the Autocorrelation Exponent Below Tc 80

1.5.4 Values of the Autoresponse Exponent 81

1.6 Global Persistence 82

Problems 89

2 Exactly Solvable Models 95

2.1 One-dimensional Glauber-Ising Model 95

2.1.1 Two-time Correlation Function 97

2.1.2 Two-time Response Function 101

2.1.3 Low-temperature Initial States 104

2.1.4 Comparison With the 1D Ginzburg-Landau Equation 105

2.2 A Non-Glauberian Kinetic Ising Model 105

2.2.1 Definition 105

2.2.2 Calculation of the Dynamical Exponent 106

2.2.3 Global Response Functions 107

2.2.4 Global Correlation Functions 109

2.3 The Free Random Walk 111

2.4 The Spherical Model 111

2.4.1 Definition and Formalism 111

2.4.2 Solution of the Volterra Integral Equation 115

2.4.3 Dynamical Scaling Behaviour 116

2.5 The Long-range Spherical Model 118

2.5.1 Definition and Composite Observables 118

2.5.2 Long-range Initial Correlations 122

2.5.3 Magnetised Initial State 123

2.6 XY Model in Spin-wave Approximation 125

2.6.1 Outline of the Method and Applicability 125

2.6.2 Two-time Correlations 126

2.6.3 Two-time Responses 128

2.6.4 Numerical Tests and Extensions 129

2.6.5 Comparison With the Clock Model 131

2.6.6 Fluctuation-dissipation Relations in the XY Model 131

2.7 OJK Approximation 132

2.8 Further Solvable Models 135

Problems 135

3 Simple Ageing： an Overview 141

3.1 Non-equilibrium Critical Dynamics 141

3.1.1 Purely Relaxational Dynamics （Model A） 141

3.1.2 Conserved Energy-density （Model C） 144

3.1.3 Effects of Initial Long-range Correlations 145

3.2 Ordered Initial States 145

3.2.1 Scaling Theory 146

3.2.2 Application to the Ising Model 148

3.2.3 Vector Order-parameter With n ≥ 2 Components 149

3.2.4 Global Persistence 151

3.2.5 Semi-ordered Initial States 153

3.3 Conserved Order-parameter （Model B） 154

3.4 Fully Frustrated Systems 160

3.5 Disordered Systems Ⅰ： Ferromagnets 164

3.5.1 Phenomenological Description 164

3.5.2 Exact Results 167

3.5.3 Simulational Studies 168

3.5.4 Superuniversality 173

3.6 Disordered Systems Ⅱ： Critical Glassy Systems 174

3.6.1 Critical Ising Spin Glasses 175

3.6.2 Gauge Glass 179

3.6.3 Interacting Flux Lines 180

3.7 Surface Effects 182

3.8 Ageing with Absorbing Steady-states Ⅰ 188

3.8.1 Contact Process 190

3.8.2 Experimental Results for Directed Percolation 196

3.8.3 Non-equilibrium Kinetic Ising Model 199

3.9 Ageing with Absorbing Steady-states Ⅱ 199

3.9.1 Bosonic Contact and Pair-contact Processes 199

3.9.2 Bosonic Particle-reaction Models with Levy Flights 205

3.10 Reversible Reaction-diffusion Systems 208

3.11 Growth Processes 215

Problems 219

4 Local Scale-invariance Ⅰ： z = 2 221

4.1 Introduction 221

4.2 The Schrodinger Group 224

4.2.1 Dynamical Conformal Invariance 224

4.2.2 Definition of the Schrodinger Group 224

4.2.3 Physical Examples of Schrodinger-invariance 229

4.2.4 Simple Consequences of Schrodinger-invariance 234

4.3 From Schrodinger-invariance to Ageing 237

4.3.1 Ageing-invariance 237

4.3.2 Example： Application to Mean-field Theory 238

4.4 Conformal Invariance and Ageing 239

4.4.1 Conformal Invariance of the Free Diffusion Equation 239

4.4.2 Parabolic Subalgebras 241

4.4.3 Non-relativistic Limits 243

4.4.4 Causality 245

4.4.5 Spinors and Supersymmetric Generalisations 246

4.5 Galilei-invariance 248

4.5.1 Galilei-invariance in Deterministic Systems 248

4.5.2 Galilei-invariance in Langevin Equations 252

4.5.3 Extensions 255

4.6 Calculation of Two-time Response and Correlation Functions 257

4.6.1 Ageing-invariant Response 257

4.6.2 Ageing-invariant Autocorrelators 258

4.6.3 Conformal Invariance 261

4.7 Tests of Ageing- and Conformal-invariance for z = 2 264

4.7.1 One-dimensional Glauber-Ising Model 265

4.7.2 XY Model in Spin-wave Approximation 267

4.7.3 Mean-field Theory and the Free Random Walk 268

4.7.4 Spherical Model 268

4.7.5 Ising Model in Two and Three Dimensions 270

4.7.6 XY Model in Two and Three Dimensions 276

4.7.7 Two-dimensional Ising and Potts Models 276

4.7.8 Bosonic Contact Processes 279

4.7.9 Bosonic Pair-contact Process 282

4.7.10 Reversible Reaction-diffusion Systems 283

4.7.11 Surface Growth： Edwards-Wilkinson Model 283

4.8 Nonrelativistic AdS/CFT Correspondence 284

4.8.1 Holographic Construction 284

4.8.2 Relationship with Cold Atoms 286

Problems 287

5 Local Scale-invariance Ⅱ： z ≠ 2 291

5.1 Axioms of Local Scale-invariance 291

5.2 Construction of the Infinitesimal Generators 292

5.2.1 Generators Without Mass Terms 292

5.2.2 On Geometrical Interpretations of Local Scaling 295

5.2.3 Generators With Generalised Mass Terms 297

5.2.4 Some Basic Facts 298

5.3 Generalised Bargman Superselection Rule 299

5.4 Calculation of Two-time Responses 301

5.5 Calculation of Two-time Correlators 304

5.6 Tests of Local Scale-invariance With z ≠ 2 306

5.6.1 Surface Growth： Mullins-Herring Model 307

5.6.2 Spherical Model With Long-range Interactions 308

5.6.3 Critical Conserved Spherical Model 309

5.6.4 Critical Ising Model 310

5.6.5 Critical XY Model 313

5.6.6 Phase-ordering Kinetics in the 2D Ising Model 313

5.6.7 Phase-ordering in the 2D Disordered Ising Model 316

5.6.8 Critical Ising Spin Glass Ⅰ： Thermoremanent Susceptibilities 317

5.6.9 Critical Ising Spin Glass Ⅱ： Alternating Susceptibilities 317

5.6.10 Critical Particle-reaction Models 320

5.6.11 Bosonic Particle-reaction Models 321

5.6.12 Surface Effects 322

5.7 Global Time-reparametrisation-invariance 323

5.8 Concluding Remarks 326

Problems 333

6 Lifshitz Points： Strongly Anisotropic Equilibrium Critical Points 337

6.1 Phenomenology 337

6.2 Critical Exponents at Lifshitz Points 343

6.3 A Different Type of Local Scale-transformation 350

6.3.1 Infinitesimal Generators 350

6.3.2 Covariant Two-point Function 352

6.3.3 Solution in the Case N = 4 353

6.3.4 Solution in the Case N ？ 4 357

6.4 Application to Lifshitz Points 360

6.4.1 ANNNS Model 361

6.4.2 ANNNI Model 362

6.5 Conclusions 366

Problems 367

Appendices 369

A Equilibrium Models 369

A.1 Potts Model 369

A.2 Clock Model 370

A.3 Turban Model 371

A.4 Baxter-Wu Model 371

A.5 Blume-Capel Model 372

A.6 XY Model 372

A.7 O（n） Model 373

A.8 Double Exchange Model 374

A.9 Hilhorst-van Leeuven Model 375

A.10 Frustrated Spin Models 375

A.11 Weakly Random Spin Systems 377

A.12 Logarithmic Sub-scaling Exponents 378

A.13 Ising Spin Glasses 381

A.14 Gauge Glass 383

D Langevin Equations and Path Integrals 384

I Cluster Algorithms： Competing Interactions 386

J Fractional Derivatives 388

J.1 Singular Fractional Derivatives 390

J.2 Fractional Laplacians 392

K Conformally Invariant Interacting Fields 393

K.1 Conformal Invariance and Coupling Constants 394

K.2 Conformally Conserved Currents 395

L Lie Groups and Lie Algebras： a Reminder 397

L.1 Finite groups 397

L.2 Continuous Groups and Lie Groups 399

L.3 From Lie Groups to Lie Algebras and Back 401

L.4 Matrix Representations and the Cartan-Weyl Basis 408

L.5 Function-space Representations 412

L.6 Central Extensions 415

M On the Central Limit Theorem 420

Q Lexique/Lexikon 424

Solutions 427

Frequently Used Symbols 487

Abbreviations 488

References 489

List of Tables 527

List of Figures 532

Index 533

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