Preface ......................................................... 5
1 Glassy dynamics of liquids-facets of the phenomenon .......... 1
1.1 Stretching of the dynamics .............................. 1
1.2 Power-law relaxation .................................... 8
1.3 Superposition principles ............................... 15
1.4 Two-step relaxation through a plateau .................. 21
1.5 The cage effect ........................................ 24
1.6 Crossover phenomena .................................... 30
1.7 Hard-sphere systems: the paradigms ..................... 37
1.8 Hard-sphere systems with short-range attraction ........ 44
2 Correlation functions ....................................... 51
2.1 The evolution of dynamical variables ................... 54
2.2 Correlation-function description of the dynamics ....... 61
2.3 Spectral representations ............................... 66
2.4 Memory-kernel descriptions of correlators .............. 72
2.4.1 Zwanzig-Mori equations .......................... 72
2.4.2 Models for correlation functions ................ 78
2.5 Linear-response theory ................................. 82
2.6 The arrested parts of correlation functions ............ 89
3 Elements of liquid dynamics ................................. 96
3.1 Preliminaries .......................................... 96
3.1.1 Homogeneous isotropic systems without
chirality ....................................... 96
3.1.2 Densities and density fluctuations ............. 100
3.2 Tagged-particle dynamics .............................. 107
3.2.1 Basic concepts and general equations ........... 107
3.2.2 Tagged-particle diffusion ...................... 112
3.2.3 The friction coefficient ....................... 120
3.2.4 The cage effect and glassy-dynamics
precursors of the velocity correlations ........ 125
3.3 Densities and currents in simple liquids .............. 132
3.3.1 Definitions and general equations .............. 132
3.3.2 Transverse-current diffusion ................... 142
3.3.3 The generalized-hydrodynamics description of
transverse-current correlations ................ 145
3.3.4 Visco-elastic features and glassy-dynamics
precursors of the transverse-current
correlators .................................... 149
3.3.5 Representations of the density correlators
in terms of relaxation kernels ................. 154
3.3.6 Sound waves and heat diffusion ................. 161
3.3.7 Visco-elastic features and glassy-dynamics
precursors of the density-fluctuation
correlators .................................... 169
4 Foundations of the mode-coupling theory for the
evolution of glassy dynamics in liquids .................... 177
4.1 Self-consistent-current-relaxation approaches ......... 178
4.1.1 The factorization ansatz ....................... 178
4.1.2 Self-consistency equations for density
correlators .................................... 185
4.2 A mode-coupling theory ................................ 191
4.2.1 Equations of motion and fixed-point
equations ...................................... 191
4.2.2 Mode-coupling-theory models .................... 196
4.2.3 The basic version of microscopic mode-
coupling theories .............................. 204
4.2.4 An elementary mode-coupling-theory model ....... 209
4.3 Glass-transition singularities ........................ 217
4.3.1 Regular and critical states .................... 217
4.3.2 Examples for bifurcation diagrams .............. 224
4.3.3 Classification of the critical states .......... 235
4.3.4 Correlation arrest near А2 singularities ....... 242
4.3.5 Density-fluctuation arrest in hard-sphere-
like systems ................................... 250
4.3.6 Arrest in systems with short-ranged-
attraction ..................................... 256
4.4 Dynamics near glass-transition singularities .......... 267
4.4.1 Relaxation through plateaus .................... 267
4.4.2 Below-plateau relaxation ....................... 281
4.4.3 Structure and structure relaxation ............. 289
4.4.4 Descriptions of some glassy-dynamics data ...... 292
5 Extensions of the mode-coupling theory for the evolution
of glassy dynamics of liquids .............................. 304
5.1 Extensions of the MCT for simple systems .............. 305
5.1.1 MCT equations for the glassy shear dynamics .... 305
5.1.2 Glassy-relaxation features of shear
correlations ................................... 308
5.1.3 MCT equations for the tagged-particle
dynamics ....................................... 320
5.1.4 Idealized transitions from diffusion to
localization ................................... 331
5.1.5 Glassy-dynamics features of tagged-particle
motions ........................................ 345
5.2 A mode-coupling theory for mixtures of spherical
particles ............................................. 356
5.2.1 The equations of motion ........................ 357
5.2.2 Density-fluctuation arrest ..................... 366
5.2.3 Hard-sphere mixtures ........................... 375
5.2.4 Sodium-disilicate melts ........................ 383
5.3 A mode-coupling theory for molecular liquids .......... 386
5.3.1 A theory for interaction-site-density
correlators .................................... 387
5.3.2 Systems of symmetric dumbbells ................. 395
5.3.3 Glassy Rouse dynamics .......................... 427
5.4 Some addenda .......................................... 432
6 Asymptotic relaxation laws ................................. 437
6.1 Dynamics of the first-scaling-law regime .............. 438
6.1.1 Reformulation of the MCT equations of motion ... 438
6.1.2 The critical dynamics .......................... 443
6.1.3 Asymptotic description of the А2-bifurcation
dynamics ....................................... 470
6.1.4 The scaling-limit description of the generic
liquid-glass-transition dynamics ............... 492
6.1.5 Extended scaling-limit description of
the generic А2-bifurcation dynamics ............. 502
6.2 Dynamics of the second-scaling-law regime ............. 513
6.2.1 Equations of motion for the second-scaling-
law regime ..................................... 514
6.2.2 The second-scaling-law description of
the liquid dynamics ............................ 519
6.2.3 Asymptotic corrections for the second scaling
limit .......................................... 533
6.3 Relaxation near higher-order singularities ............ 538
6.3.1 Correlation arrest near higher-order
singularities .................................. 539
6.3.2 Logarithmic relaxation ......................... 553
A Mathematical miscellanies .................................. 577
A.l Laplace transforms .................................... 577
A.2 Fourier transforms .................................... 580
A.3 Positive-definite and positive-analytic functions ..... 583
A.4 Harmonic-oscillator correlators ....................... 588
A.5 Matrix correlators .................................... 591
A.6 Product correlators ................................... 594
A.7 Power-law variations .................................. 596
A.8 Logarithmic variations ................................ 601
В Symmetries of fluctuation correlators ...................... 603
С Smoothened correlators ..................................... 608
D Theorems on MCT equations .................................. 611
D.l Convergence of the approximant sequences .............. 611
D.2 Completely monotonic approximants ..................... 613
D.3 The maximum-eigenvalue inequality ..................... 616
D.4 Further properties of stability matrices .............. 618
Bibliography .................................................. 621
Index ......................................................... 635
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