Non-Equilibrium Phase Transitions: Volume 2: Ageing and Dynamical Scaling Far from Equilibrium (Theoretical and Mathematical Physics)

C......Page 1
Theoretical and Mathematical Physics......Page 3
Non-Equilibrium Phase Transitions......Page 5
Preface......Page 9
Contents......Page 17
1.1 Introduction......Page 25
1.1.1 Ageing in Mechanically Deformed Polymers......Page 26
1.1.2 Correlations and Responses......Page 32
1.1.3 Ageing in Spin Glasses......Page 34
1.1.4 Ageing in Simple Magnets......Page 38
1.1.5 Mean-field Theory......Page 42
1.1.6 Breaking of the Fluctuation-dissipation Theorem: Experiments......Page 45
1.1.7 Breaking of the Fluctuation-dissipation Theorem: TwoSimple Solvable Models......Page 51
1.1.8 Outline......Page 56
1.2 Phase-ordering Kinetics......Page 57
1.2.1 Linear Stability Analysis......Page 59
1.2.3 The Allen-Cahn Equation......Page 60
1.2.4 Topological Defects......Page 62
1.2.5 Porod’s Law......Page 64
1.2.6 Bray-Rutenberg Theory for the Growth Law......Page 67
1.2.7 Exact Result in Two Dimensions......Page 71
1.2.8 Conserved Order-parameter: Phase-separation......Page 73
1.3 Phenomenology of Ageing......Page 75
1.3.1 Scaling Forms......Page 76
1.3.2 Passage into the Ageing Regime......Page 78
1.3.3 Kurchan’s Lemma......Page 81
1.3.4 The Yeung-Rao-Desai Inequalities......Page 83
1.4 Scaling Behaviour of Integrated Responses......Page 85
1.4.1 Thermoremanent Susceptibility......Page 86
1.4.2 Zero-field Cooled Susceptibility......Page 88
1.5.1 Values of the Ageing Exponents a and b......Page 91
1.5.2 Values of the Critical Autocorrelation Exponent......Page 97
1.5.3 Values of the Autocorrelation Exponent Below Tc......Page 104
1.5.4 Values of the Autoresponse Exponent......Page 105
1.6 Global Persistence......Page 106
Problems......Page 113
2.1 One-dimensional Glauber-Ising Model......Page 119
2.1.1 Two-time Correlation Function......Page 121
2.1.2 Two-time Response Function......Page 125
2.1.3 Low-temperature Initial States......Page 128
2.2.1 Definition......Page 129
2.2.2 Calculation of the Dynamical Exponent......Page 130
2.2.3 Global Response Functions......Page 131
2.2.4 Global Correlation Functions......Page 133
2.4.1 Definition and Formalism......Page 135
2.4.2 Solution of the Volterra Integral Equation......Page 139
2.4.3 Dynamical Scaling Behaviour......Page 140
2.5.1 Definition and Composite Observables......Page 142
2.5.2 Long-range Initial Correlations......Page 146
2.5.3 Magnetised Initial State......Page 147
2.6.1 Outline of the Method and Applicability......Page 149
2.6.2 Two-time Correlations......Page 150
2.6.3 Two-time Responses......Page 152
2.6.4 Numerical Tests and Extensions......Page 153
2.6.6 Fluctuation-dissipation Relations in the XY Model......Page 155
2.7 OJK Approximation......Page 156
Problems......Page 159
3.1.1 Purely Relaxational Dynamics (Model A)......Page 165
3.1.2 Conserved Energy-density (Model C)......Page 168
3.2 Ordered Initial States......Page 169
3.2.1 Scaling Theory......Page 170
3.2.2 Application to the Ising Model......Page 172
3.2.3 Vector Order-parameter With n ≥ 2 Components......Page 173
3.2.4 Global Persistence......Page 175
3.2.5 Semi-ordered Initial States......Page 177
3.3 Conserved Order-parameter (Model B)......Page 178
3.4 Fully Frustrated Systems......Page 184
3.5.1 Phenomenological Description......Page 188
3.5.2 Exact Results......Page 191
3.5.3 Simulational Studies......Page 192
3.5.4 Superuniversality......Page 197
3.6 Disordered Systems II: Critical Glassy Systems......Page 198
3.6.1 Critical Ising Spin Glasses......Page 199
3.6.2 Gauge Glass......Page 203
3.6.3 Interacting Flux Lines......Page 204
3.7 Surface Effects......Page 206
3.8 Ageing with Absorbing Steady-states I......Page 212
3.8.1 Contact Process......Page 214
3.8.2 Experimental Results for Directed Percolation......Page 220
3.9.1 Bosonic Contact and Pair-contact Processes......Page 223
3.9.2 Bosonic Particle-reaction Models with L磂vy Flights......Page 229
3.10 Reversible Reaction-diffusion Systems......Page 232
3.11 Growth Processes......Page 239
Problems......Page 243
4.1 Introduction......Page 245
4.2.2 Definition of the Schr╫dinger Group......Page 248
4.2.3 Physical Examples of Schr╫dinger-invariance......Page 253
4.2.4 Simple Consequences of Schr╫dinger-invariance......Page 258
4.3.1 Ageing-invariance......Page 261
4.3.2 Example: Application to Mean-field Theory......Page 262
4.4.1 Conformal Invariance of the Free Diffusion Equation......Page 263
4.4.2 Parabolic Subalgebras......Page 265
4.4.3 Non-relativistic Limits......Page 267
4.4.4 Causality......Page 269
4.4.5 Spinors and Supersymmetric Generalisations......Page 270
4.5.1 Galilei-invariance in Deterministic Systems......Page 272
4.5.2 Galilei-invariance in Langevin Equations......Page 276
4.5.3 Extensions......Page 279
4.6.1 Ageing-invariant Response......Page 281
4.6.2 Ageing-invariant Autocorrelators......Page 282
4.6.3 Conformal Invariance......Page 285
4.7 Tests of Ageing- and Conformal-invariance for z = 2......Page 288
4.7.1 One-dimensional Glauber-Ising Model......Page 289
4.7.2 XY Model in Spin-wave Approximation......Page 291
4.7.4 Spherical Model......Page 292
4.7.5 Ising Model in Two and Three Dimensions......Page 294
4.7.7 Two-dimensional Ising and Potts Models......Page 300
4.7.8 Bosonic Contact Processes......Page 303
4.7.9 Bosonic Pair-contact Process......Page 306
4.7.11 Surface Growth: Edwards-Wilkinson Model......Page 307
4.8.1 Holographic Construction......Page 308
4.8.2 Relationship with Cold Atoms......Page 310
Problems......Page 311
5.1 Axioms of Local Scale-invariance......Page 315
5.2.1 Generators Without Mass Terms......Page 316
5.2.2 On Geometrical Interpretations of Local Scaling......Page 319
5.2.3 Generators With Generalised Mass Terms......Page 321
5.2.4 Some Basic Facts......Page 322
5.3 Generalised Bargman Superselection Rule......Page 323
5.4 Calculation of Two-time Responses......Page 325
5.5 Calculation of Two-time Correlators......Page 328
5.6 Tests of Local Scale-invariance With z 6= 2......Page 330
5.6.1 Surface Growth: Mullins-Herring Model......Page 331
5.6.2 Spherical Model With Long-range Interactions......Page 332
5.6.3 Critical Conserved Spherical Model......Page 333
5.6.4 Critical Ising Model......Page 334
5.6.6 Phase-ordering Kinetics in the 2D Ising Model......Page 337
5.6.7 Phase-ordering in the 2D Disordered Ising Model......Page 340
5.6.9 Critical Ising Spin Glass II: Alternating Susceptibilities......Page 341
5.6.10 Critical Particle-reaction Models......Page 344
5.6.11 Bosonic Particle-reaction Models......Page 345
5.6.12 Surface Effects......Page 346
5.7 Global Time-reparametrisation-invariance......Page 347
5.8 Concluding Remarks......Page 350
Problems......Page 357
6.1 Phenomenology......Page 361
6.2 Critical Exponents at Lifshitz Points......Page 367
6.3.1 Infinitesimal Generators......Page 374
6.3.2 Covariant Two-point Function......Page 376
6.3.3 Solution in the Case N = 4......Page 377
6.3.4 Solution in the Case N ? 4......Page 381
6.4 Application to Lifshitz Points......Page 384
6.4.1 ANNNS Model......Page 385
6.4.2 ANNNI Model......Page 386
6.5 Conclusions......Page 390
Problems......Page 391
Appendices......Page 393