Mode-coupling theory is an approach to the study of complex behavior in the supercooled liquids which developed from the idea of a nonlinear feedback mechanism. From the coupling of slowly decaying correlation functions the theory predicts the existence of a characteristic temperature T c above the experimental glass transition temperature T g for the liquid. This article discusses the various methods used to obtain the model equations and illustrates the effects of structure on dynamics and scaling behavior over different time scales using a wave-vector-dependent model. It compares the theoretical predictions, experimental observations, and computer simulation results, and also considers phenomenological extensions of mode-coupling theory. Numerical solutions of the model equations to study the dynamics from a nonperturbative approach are also reviewed. The review looks briefly at recent observations from landscape studies of model systems of structural glasses and their relation to the mode-coupling temperature T c. The equations for the mean-field dynamics driven by the p-spin interaction Hamiltonian are similar to those of mode-coupling theory for structural glasses. Related developments in the nonequilibrium dynamics and generalization of the fluctuation-dissipation relation for the structural glasses are briefly touched upon. The review ends with a summary of the open questions and possible future direction of the field.

No abstract

We show how kinetic theory, the statistics of classical particles obeying Newtonian dynamics, can be formulated as a field theory. The field theory can be organized to produce a self-consistent perturbation theory expansion in an effective interaction potential. The need for a self-consistent approach is suggested by our interest in investigating ergodic-nonergodic transitions in dense fluids.The formal structure we develop has been implemented in detail for the simpler case of Smoluchowski dynamics. One aspect of the approach is the identification of a core problem spanned by the variables ρ the number density and B a response density. In this paper we set up the perturbation theory expansion with explicit development at zeroth and first order. We also determine all of the cumulants in the noninteracting limit among the core variables ρ and B.0

A1.1 The Gibbs inequality A1.2 The force-force correlation A1.3 Brownian motion A1.3.1 The noise correlation A1.3.2 Evaluation of the integrals 2 The freezing transition 2.1 The density-functional approach 2.1.1 A thermodynamic extremum principle vii viii Contents 2.1.2 An approximate free-energy functional 2.1.3 The Ramakrishnan-Yussouff model 2.2 Weighted density functionals 2.2.1 The modified weighted-density approximation 2.2.2 Gaussian density profiles 2.2.3 The hard-sphere system 2.3 Fundamental measure theory 2.3.1 Density-independent weight functions 2.3.2 The free-energy functional 2.4 Applications to other systems 2.4.1 Long-range interaction potentials 91 2.4.2 The solid-liquid interface Appendix to Chapter 2 105 A2.1 Correlation functions for the inhomogeneous solid A2.2 The Ramakrishnan-Yussouff model A2.3 The weighted-density-functional approximation A2.4 The modified weighted-density-functional approximation 113 A2.5 The Gaussian density profiles and phonon model 3 Crystal nucleation Appendix to Chapter 3 A3.1 The schematic model for nucleation A3.1.1 Critical nucleus A3.1.2 The free-energy barrier 161 A3.2 The excess free energy in the DFT model 4 The supercooled liquid 164 4.1 The liquid-glass transition 4.1.1 Characteristic temperatures of the glassy state 165 Contents ix 4.1.2 The free-volume model 170 4.1.3 Self-diffusion and the Stokes-Einstein relation 171 4.2 Glass formation vs. crystallization 175 4.2.1 The minimum cooling rate 4.2.2 The kinetic spinodal and the Kauzmann paradox 177 4.3 The landscape paradigm 181 4.3.1 The potential-energy landscape 182 4.3.2 The free-energy landscape 188 4.4 Dynamical heterogeneities 192 4.4.1 Computer-simulation results 193 4.4.2 Dynamic length scales 198 5 Dynamics of collective modes 204 5.1 Conservation laws and dissipation 205 5.1.1 The microscopic balance equations 205 5.1.2 Euler equations of hydrodynamics 207 5.1.3 Dissipative equations of hydrodynamics 209 5.1.4 Tagged-particle dynamics 5.1.