Non-Equilibrium Phase Transitions (eBook)

Malte Henkel, born in 1960, received his Master's degree from the University of Bonn in 1984, and his PhD in 1987, when he also won the annual prize of the Minerva Foundation. From that year onward he has been a long-term visitor in many institutes, including the ITP at Santa Barbara, USA, the SPhT at Saclay, France, and the universities of Oxford, UK, Vienna, Austria, Padova, Italy, and Lisbon, Portugal. In 1995 he was appointed a professor at the University of Nancy I. His current research encompasses equilibrium and non-equilibrium phase transitions, using field-theoretical and numerical methods in general. In particular, his current focus is on dynamical scaling behaviour realised in ageing phenomena far from equilibrium. He has published well over a hundred articles and three monographs, one of which is Volume I of this set.

Theoretical and Mathematical Physics 1
TitlePage 2
Copyright Page 3
Preface 5
Contents 13
Chapter 1: Ageing Phenomena 20
1.1 Introduction 20
1.1.1 Ageing in Mechanically Deformed Polymers 21
1.1.2 Correlations and Responses 27
1.1.3 Ageing in Spin Glasses 29
1.1.4 Ageing in Simple Magnets 33
1.1.5 Mean-field Theory 37
1.1.6 Breaking of the Fluctuation-dissipation Theorem: Experiments 40
1.1.7 Breaking of the Fluctuation-dissipation Theorem: TwoSimple Solvable Models 46
1.1.8 Outline 51
1.2 Phase-ordering Kinetics 52
1.2.1 Linear Stability Analysis 54
1.2.2 Domain Walls 55
1.2.3 The Allen-Cahn Equation 55
1.2.4 Topological Defects 57
1.2.5 Porod’s Law 59
1.2.6 Bray-Rutenberg Theory for the Growth Law 62
1.2.7 Exact Result in Two Dimensions 66
1.2.8 Conserved Order-parameter: Phase-separation 68
1.2.9 Critical Dynamics 70
1.3 Phenomenology of Ageing 70
1.3.1 Scaling Forms 71
1.3.2 Passage into the Ageing Regime 73
1.3.3 Kurchan’s Lemma 76
1.3.4 The Yeung-Rao-Desai Inequalities 78
1.4 Scaling Behaviour of Integrated Responses 80
1.4.1 Thermoremanent Susceptibility 81
1.4.2 Zero-field Cooled Susceptibility 83
1.4.3 Intermediate Susceptibility 86
1.4.4 Alternating Susceptibility 86
1.5 Values of Non-equilibrium Exponents 86
1.5.1 Values of the Ageing Exponents a and b 86
1.5.2 Values of the Critical Autocorrelation Exponent 92
1.5.3 Values of the Autocorrelation Exponent Below Tc 99
1.5.4 Values of the Autoresponse Exponent 100
1.6 Global Persistence 101
Problems 108
Chapter 2: Exactly Solvable Models 114
2.1 One-dimensional Glauber-Ising Model 114
2.1.1 Two-time Correlation Function 116
2.1.2 Two-time Response Function 120
2.1.3 Low-temperature Initial States 123
2.1.4 Comparison With the 1D Ginzburg-Landau Equation 124
2.2 A Non-Glauberian Kinetic Ising Model 124
2.2.1 Definition 124
2.2.2 Calculation of the Dynamical Exponent 125
2.2.3 Global Response Functions 126
2.2.4 Global Correlation Functions 128
2.3 The Free Random Walk 130
2.4 The Spherical Model 130
2.4.1 Definition and Formalism 130
2.4.2 Solution of the Volterra Integral Equation 134
2.4.3 Dynamical Scaling Behaviour 135
2.5 The Long-range Spherical Model 137
2.5.1 Definition and Composite Observables 137
2.5.2 Long-range Initial Correlations 141
2.5.3 Magnetised Initial State 142
2.6 XY Model in Spin-wave Approximation 144
2.6.1 Outline of the Method and Applicability 144
2.6.2 Two-time Correlations 145
2.6.3 Two-time Responses 147
2.6.4 Numerical Tests and Extensions 148
2.6.5 Comparison With the Clock Model 150
2.6.6 Fluctuation-dissipation Relations in the XY Model 150
2.7 OJK Approximation 151
2.8 Further Solvable Models 154
Problems 154
Chapter 3: Simple Ageing: an Overview 159
3.1 Non-equilibrium Critical Dynamics 159
3.1.1 Purely Relaxational Dynamics (Model A) 159
3.1.2 Conserved Energy-density (Model C) 162
3.1.3 Effects of Initial Long-range Correlations 163
3.2 Ordered Initial States 163
3.2.1 Scaling Theory 164
3.2.2 Application to the Ising Model 166
3.2.3 Vector Order-parameter With n = 2 Components 167
3.2.4 Global Persistence 169
3.2.5 Semi-ordered Initial States 171
3.3 Conserved Order-parameter (Model B) 172
3.4 Fully Frustrated Systems 178
3.5 Disordered Systems I: Ferromagnets 182
3.5.1 Phenomenological Description 182
3.5.2 Exact Results 185
3.5.3 Simulational Studies 186
3.5.4 Superuniversality 191
3.6 Disordered Systems II: Critical Glassy Systems 192
3.6.1 Critical Ising Spin Glasses 193
3.6.2 Gauge Glass 197
3.6.3 Interacting Flux Lines 198
3.7 Surface Effects 200
3.8 Ageing with Absorbing Steady-states I 206
3.8.1 Contact Process 208
3.8.2 Experimental Results for Directed Percolation 214
3.8.3 Non-equilibrium Kinetic Ising Model 217
3.9 Ageing with Absorbing Steady-states II 217
3.9.1 Bosonic Contact and Pair-contact Processes 217
3.9.2 Bosonic Particle-reaction Models with L´evy Flights 223
3.10 Reversible Reaction-diffusion Systems 226
3.11 Growth Processes 233
Problems 237
Chapter 4: Local Scale-invariance I: z = 2 239
4.1 Introduction 239
4.2 The Schr¨odinger Group 242
4.2.1 Dynamical Conformal Invariance 242
4.2.2 Definition of the Schr¨odinger Group 242
4.2.3 Physical Examples of Schr¨odinger-invariance 247
4.2.4 Simple Consequences of Schr¨odinger-invariance 252
4.3 From Schr¨odinger-invariance to Ageing 255
4.3.1 Ageing-invariance 255
4.3.2 Example: Application to Mean-field Theory 256
4.4 Conformal Invariance and Ageing 257
4.4.1 Conformal Invariance of the Free Diffusion Equation 257
4.4.2 Parabolic Subalgebras 259
4.4.3 Non-relativistic Limits 261
4.4.4 Causality 263
4.4.5 Spinors and Supersymmetric Generalisations 264
4.5 Galilei-invariance 266
4.5.1 Galilei-invariance in Deterministic Systems 266
4.5.2 Galilei-invariance in Langevin Equations 270
4.5.3 Extensions 273
4.6 Calculation of Two-time Response and Correlation Functions 275
4.6.1 Ageing-invariant Response 275
4.6.2 Ageing-invariant Autocorrelators 276
4.6.3 Conformal Invariance 279
4.7 Tests of Ageing- and Conformal-invariance for z = 2 282
4.7.1 One-dimensional Glauber-Ising Model 283
4.7.2 XY Model in Spin-wave Approximation 285
4.7.3 Mean-field Theory and the Free Random Walk 286
4.7.4 Spherical Model 286
4.7.5 Ising Model in Two and Three Dimensions 288
4.7.6 XY Model in Two and Three Dimensions 294
4.7.7 Two-dimensional Ising and Potts Models 294
4.7.8 Bosonic Contact Processes 297
4.7.9 Bosonic Pair-contact Process 300
4.7.10 Reversible Reaction-diffusion Systems 301
4.7.11 Surface Growth: Edwards-Wilkinson Model 301
4.8 Nonrelativistic AdS/CFT Correspondence 302
4.8.1 Holographic Construction 302
4.8.2 Relationship with Cold Atoms 304
Problems 305
Chapter 5: Local Scale-invariance II: z =/ 2 309
5.1 Axioms of Local Scale-invariance 309
5.2 Construction of the Infinitesimal Generators 310
5.2.1 Generators Without Mass Terms 310
5.2.2 On Geometrical Interpretations of Local Scaling 313
5.2.3 Generators With Generalised Mass Terms 315
5.2.4 Some Basic Facts 316
5.3 Generalised Bargman Superselection Rule 317
5.4 Calculation of Two-time Responses 319
5.5 Calculation of Two-time Correlators 322
5.6 Tests of Local Scale-invariance With z 6= 2 324
5.6.1 Surface Growth: Mullins-Herring Model 325
5.6.2 Spherical Model With Long-range Interactions 326
5.6.3 Critical Conserved Spherical Model 327
5.6.4 Critical Ising Model 328
5.6.5 Critical XY Model 331
5.6.6 Phase-ordering Kinetics in the 2D Ising Model 331
5.6.7 Phase-ordering in the 2D Disordered Ising Model 334
5.6.8 Critical Ising Spin Glass I: Thermoremanent Susceptibilities 335
5.6.9 Critical Ising Spin Glass II: Alternating Susceptibilities 335
5.6.10 Critical Particle-reaction Models 338
5.6.11 Bosonic Particle-reaction Models 339
5.6.12 Surface Effects 340
5.7 Global Time-reparametrisation-invariance 341
5.8 Concluding Remarks 344
Problems 351
Chapter 6: Lifshitz Points: Strongly AnisotropicEquilibrium Critical Points 355
6.1 Phenomenology 355
6.2 Critical Exponents at Lifshitz Points 361
6.3 A Different Type of Local Scale-transformation 368
6.3.1 Infinitesimal Generators 368
6.3.2 Covariant Two-point Function 370
6.3.3 Solution in the Case N = 4 371
6.3.4 Solution in the Case N . 4 375
6.4 Application to Lifshitz Points 378
6.4.1 ANNNS Model 379
6.4.2 ANNNI Model 380
6.5 Conclusions 384
Problems 385
Appendices 387
Solutions 445
References 507
List of Figures 546
Index 550