Statistical Physics of Liquids at Freezing and Beyond

Cover......Page 1
Title......Page 5
Copyright......Page 6
Decaition......Page 7
Contents......Page 9
Preface......Page 15
Acknowledgements......Page 18
1 Statistical physics of liquids......Page 21
1.1.1 Thermodynamic functions......Page 23
1.1.2 The classical N-particle system......Page 26
1.1.3 The BBGKY hierarchy equations......Page 27
1.1.4 The Boltzmann equation......Page 29
1.2 Equilibrium properties......Page 31
1.2.1 The Gibbs H-theorem......Page 32
1.2.2 The equilibrium ensembles......Page 34
1.2.3 The static structure factor......Page 39
1.2.4 Integral equations for g(r)......Page 44
1.3 Time correlation functions......Page 56
1.3.1 The density correlation function......Page 58
A1.3.2 Evaluation of the integrals......Page 61
1.3.3 The linear response function......Page 67
1.4.1 The Langevin equation......Page 70
1.4.2 The Stokes--Einstein relation......Page 73
A1.2 The force--force correlation......Page 75
A1.3.1 The noise correlation......Page 76
2.1 The density-functional approach......Page 78
2.1.1 A thermodynamic extremum principle......Page 80
2.1.2 An approximate free-energy functional......Page 84
2.1.3 The Ramakrishnan--Yussouff model......Page 88
2.2 Weighted density functionals......Page 92
2.2.1 The modified weighted-density approximation......Page 95
2.2.2 Gaussian density profiles......Page 96
2.2.3 The hard-sphere system......Page 98
2.3 Fundamental measure theory......Page 105
2.3.1 Density-independent weight functions......Page 106
2.3.2 The free-energy functional......Page 108
2.4 Applications to other systems......Page 110
2.4.1 Long-range interaction potentials......Page 111
2.4.2 The solid-liquid interface......Page 119
Appendix to Chapter 2......Page 125
A10.1 Matrix identity......Page 0
A2.2 The Ramakrishnan--Yussouff model......Page 126
A2.3 The weighted-density-functional approximation......Page 129
A2.4 The modified weighted-density-functional approximation......Page 133
A2.5 The Gaussian density profiles and phonon model......Page 135
3.1 Classical nucleation theory......Page 137
3.1.1 The free-energy barrier......Page 138
3.1.2 The nucleation rate......Page 141
3.1.3 Heterogeneous nucleation......Page 149
3.2 A simple nonclassical model......Page 151
3.2.1 The critical nucleus......Page 153
3.2.2 The free-energy barrier......Page 154
3.3.1 The square-gradient approximation......Page 157
3.3.2 The critical nucleus......Page 161
3.3.3 The weighted-density-functional approach......Page 165
3.4 Computer-simulation studies......Page 170
3.4.1 Comparisons with CNT predictions......Page 172
3.4.2 The structure of the nucleus......Page 175
Appendix to Chapter 3......Page 180
A3.1.2 The free-energy barrier......Page 181
A3.2 The excess free energy in the DFT model......Page 182
4.1 The liquid--glass transition......Page 184
4.1.1 Characteristic temperatures of the glassy state......Page 185
4.1.2 The free-volume model......Page 190
4.1.3 Self-diffusion and the Stokes--Einstein relation......Page 191
4.2 Glass formation vs. crystallization......Page 195
4.2.1 The minimum cooling rate......Page 196
4.2.2 The kinetic spinodal and the Kauzmann paradox......Page 197
4.3 The landscape paradigm......Page 201
4.3.1 The potential-energy landscape......Page 202
4.3.2 The free-energy landscape......Page 208
4.4 Dynamical heterogeneities......Page 212
4.4.1 Computer-simulation results......Page 213
4.4.2 Dynamic length scales......Page 218
5 Dynamics of collective modes......Page 224
5.1.1 The microscopic balance equations......Page 225
5.1.2 Euler equations of hydrodynamics......Page 227
5.1.3 Dissipative equations of hydrodynamics......Page 229
5.1.4 Tagged-particle dynamics......Page 231
5.1.5 Two-component systems......Page 232
5.2 Hydrodynamic correlation functions......Page 235
5.2.1 Self-diffusion......Page 237
5.2.2 Transport coefficients......Page 238
5.3.1 The generalized Langevin equation......Page 245
5.3.2 The liquid-state dynamics......Page 256
5.4 Hydrodynamics of a solid......Page 266
Appendix to Chapter 5......Page 280
A5.1.1 The Euler equations......Page 283
A5.1.2 The entropy-production rate......Page 284
A5.2 The second fluctuation--dissipation relation......Page 289
6.1.1 Coupling of collective modes......Page 291
6.1.2 Nonlinear Langevin equations......Page 294
6.2.1 The one-component fluid......Page 307
6.2.2 The nonlinear diffusion equation......Page 313
6.2.3 A two-component fluid......Page 315
6.2.4 The solid state......Page 320
6.3.1 Smoluchowski dynamics......Page 323
6.3.2 Fokker--Planck dynamics......Page 328
Appendix to Chapter 6......Page 330
7 Renormalization of the dynamics......Page 338
7.1 The Martin--Siggia--Rose theory......Page 339
7.1.1 The MSR action functional......Page 340
7.2 The compressible liquid......Page 347
7.2.1 MSR theory for a compressible liquid......Page 348
7.2.2 Correlation and response functions......Page 350
7.3 Renormalization......Page 354
7.3.1 Fluctuation--dissipation relations......Page 355
7.3.2 Nonperturbative results......Page 359
7.3.3 One-loop renormalization......Page 363
Appendix to Chapter 7......Page 368
A7.1 The Jacobian of MSR fields......Page 370
A7.2 The MSR field theory......Page 371
A7.3 Invariance of the MSR action......Page 375
A7.4 The memory-function approach......Page 377
A7.4.1 The projection-operator method......Page 378
A7.4.2 The mode-coupling approximation......Page 381
8.1 Mode-coupling theory......Page 383
8.1.1 The schematic model......Page 385
8.1.2 Effects of structure on dynamics......Page 389
8.1.3 Tagged-particle dynamics......Page 396
8.1.4 Dynamical heterogeneities and MCT......Page 403
8.1.5 Linking DFT with MCT......Page 408
8.1.6 Dynamic density-functional theory......Page 411
8.2.1 Testing with schematic MCT......Page 420
8.2.2 Glass transition in colloids......Page 426
8.2.3 Molecular-dynamics simulations......Page 427
8.2.4 Discussion......Page 429
8.3 Ergodicity-restoring mechanisms......Page 431
8.3.1 Ergodic behavior in the NFH model......Page 432
8.3.2 The hydrodynamic limit......Page 434
8.3.3 Numerical solution of NFH equations......Page 437
8.4 Spin-glass models......Page 439
8.4.1 The p-spin interaction model......Page 440
8.4.2 MCT and mean-field theories......Page 446
Appendix to Chapter 8......Page 450
A8.2 Field-theoretic treatment of the DDFT model......Page 452
A8.3 The one-loop result for  Sigma v v (0, 0)......Page 462
9.1.1 A generalized fluctuation--dissipation relation......Page 463
9.1.2 Computer-simulation studies......Page 464
9.2 The effective temperature......Page 469
9.2.2 A simple thermometer......Page 471
9.3 A mean-field model......Page 476
9.3.1 The mode-coupling approximation......Page 479
9.3.2 The FDT regime......Page 480
9.3.3 The aging regime......Page 485
9.3.4 Quasi-ergodic behavior......Page 489
9.4.1 Thermalization......Page 491
9.4.2 Aging dynamics: experiments......Page 493
Appendix to Chapter 9......Page 499
A9.2 Evaluation of integrals......Page 501
A9.2.2 Integrals for the aging solution......Page 502
10.1 The entropy crisis......Page 506
10.1.1 The Adam--Gibbs theory......Page 507
10.1.2 Dynamics near TK......Page 508
10.2 First-order transitions......Page 510
10.2.1 Metastable aperiodic structures......Page 511
10.2.2 Random first-order transition theory......Page 514
10.3.1 Effective potential and overlap functions......Page 518
10.3.2 A model calculation......Page 521
10.4 Spontaneous breaking of ergodicity......Page 525
10.4.1 The replica method for self-generated disorder......Page 527
10.4.2 Free energy of the Replicated liquid......Page 529
10.4.3 An example: the &psi\' 4 model......Page 534
10.5 The amorphous solid......Page 539
10.5.1 The Mézard--Parisi model......Page 542
Appendix to Chapter 10......Page 551
A10.2 Matrix identity II......Page 552
A10.3 Computation of the vibrational contribution mathcal I v......Page 553
A10.4 Computation of Tr lnM*......Page 556
References......Page 560
Index......Page 578