Non-Equilibrium Phase Transitions

by Henkel, Malte; Hinrichsen, Haye; Lubeck, Sven

• ISBN13: 9781402087646
• ISBN10: 1402087640
• Format: Hardcover
• Copyright: 2009-05-01
• Publisher: Springer Verlag

Summary

This book is Volume 1 of a two-volume set describing two main classes of non-equilibrium phase-transitions. It covers the statics and dynamics of transitions into an absorbing state. Volume 2 will cover dynamical scaling in far-from equilibrium relaxation behaviour and ageing.

Table of Contents

Introduction	p. 1
Problems	p. 6
Survey of Equilibrium Critical Phenomena	p. 7
Phase Transitions in Equilibrium Systems	p. 8
Notations	p. 8
Phase Transitions in Ferromagnetic Systems	p. 8
Power-law Scaling	p. 11
Scale-Invariance and Universality	p. 13
Scale-Invariance	p. 13
Scaling Functions and Data Collapses	p. 14
Universality Classes	p. 17
Experimental Evidence of Universality	p. 19
Mean-Field and Renormalisation Group Methods	p. 21
Mean-Field Theory of Ferromagnetic Systems	p. 21
Universal Amplitude Ratios	p. 22
Remarks on Renormalisation-Group Theory	p. 23
Scaling Laws Induced by Renormalisation-Group Theory	p. 27
Field-Theory and ¿-Expansion	p. 30
Surface Critical Phenomena	p. 31
Finite-Size Scaling	p. 34
Fluctuation-Dissipation Theorem	p. 38
From Scale-Invariance to Conformal Invariance	p. 44
Problems	p. 54
Directed Percolation	p. 59
Directed Percolation at First Glance	p. 59
Directed Percolation as a Stochastic Process	p. 64
Basic Scaling Behaviour	p. 64
Universality and the DP Conjecture	p. 66
Simple Mean-Field Approximation	p. 66
Phenomenological Langevin Equation	p. 68
Update Schemes and Evolution Equations	p. 69
Lattice Models of Directed Percolation	p. 73
Domany-Kinzel Automaton	p. 73
Contact Process	p. 77
Pair-Contact Process	p. 79
Threshold Transfer Process	p. 81
Ziff-Gulari-Barshad Model	p. 82
Further Non-equilibrium Phenomena Related to DP	p. 84
Experiments Related to Directed Percolation	p. 87
Experiments Resembling DP Dynamics	p. 87
Growth Processes Related to DP	p. 90
Intermittent Turbulence	p. 94
Discussion	p. 96
Problems	p. 98
Scaling Properties of Absorbing Phase Transitions	p. 101
Scaling in the Steady-State	p. 101
Order Parameters	p. 103
Rapidity-Reversal Symmetry of Directed Percolation	p. 104
The Correlation Lengths ¿⊥ and ¿	p. 105
Scale-Invariance	p. 107
Two-Point Correlation Function in the Steady-State	p. 108
Empty-Interval Probabilities in the Steady-State	p. 109
The External Field h	p. 112
Fluctuations of the Order-Parameter in the Steady-State	p. 116
Finite-Size Scaling in the Steady-State	p. 118
Dynamical Scaling Behaviour	p. 121
Homogeneously Active Initial State	p. 121
Pair-Connectedness Function, I	p. 122
Spreading Profile at Criticality	p. 123
Clusters Generated from a Single Seed	p. 124
Properties of Clusters in the Absorbing Phase	p. 126
Pair-Connectedness Function, II	p. 127
Response Function	p. 129
Early-Time Behaviour and Critical Initial Slip	p. 131
Fractal Initial Conditions	p. 133
Influence of an External Field	p. 135
Finite-Size Scaling	p. 136
Universality of Finite-Size Amplitudes	p. 138
Methods of Analysis	p. 142
Exact Diagonalisation	p. 142
Yang-Lee and Fisher Zeros	p. 145
Series Expansion	p. 147
Field-Theoretical Methods	p. 149
Methods for Exact Solution	p. 157
Monte Carlo Simulations	p. 159
Universal Moment Ratios	p. 166
Density-Matrix Renormalisation-Group Methods	p. 167
Other Critical Properties	p. 180
Surface Critical Behaviour	p. 180
Persistence Exponents	p. 187
Problems	p. 193
Universality Classes Different from Directed Percolation	p. 197
Parity-Conserving Universality Class	p. 198
Voter Universality Class	p. 199
The Classical Voter Model	p. 199
Voter-Type Phase Transitions	p. 201
Compact Directed Percolation	p. 203
Tricritical Directed Percolation	p. 205
Mean-Field Approximation of TDP	p. 206
Numerical Simulations of TDP	p. 208
Dynamical Percolation	p. 210
Long-Range Interactions	p. 213
DP with Spatial Lévy Flights	p. 214
DP with Temporal Long-Range Interactions	p. 218
Other Models with Long-Range Interactions by Levy Flights	p. 220
Simulating Models with Long-Range Interactions	p. 220
Manna Universality Class	p. 221
Manna Model	p. 222
Conserved Threshold Transfer Process (CTTP)	p. 223
Conserved Lattice Gas (CLG) and Other Reaction-Diffusion Processes	p. 225
Scaling Properties	p. 226
Relationship Between Absorbing Phase Transitions and Self-Organised Criticality (SOC)	p. 228
Absorbing Phase Transitions and SOC: Mean-Field Approximation	p. 229
Relating Critical Exponents of SOC and Absorbing Phase Transitions	p. 231
Pair-Contact Process with Diffusion	p. 232
First-Order Phase Transitions	p. 234
Stabilisation by Elimination of Minority Islands	p. 236
First-Order Transitions in One Spatial Dimension	p. 237
Impossibility of Discontinuous Phase Transitions in Fluctuating One-Dimensional Systems	p. 238
Phase Coexistence and Hysteresis Cycles	p. 241
Crossover Phenomena	p. 244
Crossover from DP to TDP	p. 245
Crossover from DP to CDP	p. 246
Crossover to Mean-Field Scaling Behaviour	p. 248
Quenched Disorder	p. 252
Temporally Quenched Disorder	p. 252
Spatially Quenched Disorder	p. 253
Attempts of Classification	p. 254
Some Open Questions	p. 258
Problems	p. 258
Appendices	p. 261
Equilibrium Models	p. 261
Potts Model	p. 261
Clock Model	p. 262
Turban Model	p. 263
Baxter-Wu Model	p. 263
Blume-Capel Model	p. 264
XY Model	p. 264
O(n) Model	p. 265
Double Exchange Model	p. 266
Frustrated Spin Models	p. 266
Hilhorst-van Leeuven Model	p. 267
Scaling Laws for Absorbing Phase Transitions	p. 268
Diagonalisation of Time-Evolution Operators	p. 270
Langevin Equations and Path Integrals	p. 272
Mean-Field Approximations	p. 274
Simple Mean-Field/Site Approximation	p. 275
Pair-Approximation	p. 276
The 'Hop-Away' Mean-Field Approximation	p. 277
Finite-Size Scaling Techniques	p. 279
Sequences of Finite-Size Estimates	p. 279
Sequence Extrapolation	p. 283
Numerical Methods	p. 286
Simulational Techniques	p. 286
Computation of Response Functions	p. 290
Fractal Dimensions	p. 298
Solutions	p. 301
Frequently Used Symbols	p. 343
Abbreviations	p. 344
References	p. 345
List of tables	p. 371
List of figures	p. 373
Index	p. 377
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