Slow relaxations and nonequilibrium dynamics in condensed matter : École d'Été de Physique des Houches, session LXXVII, 1-26 July 2002 ; NATO Advanced Study Institute, Euro Summer School, École Thématique du CNRS = Relaxations lentes et dynamiques ho

front-matter Title Conférenciers ÉCOLE DE PHYSIQUE DES HOUCHES Lecturers Participants Préface Preface CONTENTS 01 1 Introduction 1.1 Nonequilibrium steady states 1.2 Problems with usual thermodynamic concepts 2 Phase transitions far from equilibrium 2.1 Differences from equilibrium – constructing models with NESS 2.2 Generation of long-range interactions – nonlocal dynamics 2.3 Generation of long-range interactions – dynamical anisotropies 2.4 Driven lattice gases, surface growth 2.5 Flocking behavior 3 Where do the power-laws come from? 3.1 Self-organized criticality (SOC) 3.2 Absorbing state transitions and their connection to SOC 4 Distribution functions in nonequilibrium steady states 4.1 Power laws and universality of nonequilibrium distributions 4.2 Picture gallery of scaling functions 4.3 Upper critical dimension of the KPZ equation 5 Quantum phase transitions 5.1 Spin chains with .uxes 5.2 Effective interactions 6 Outlook References 02 1 Introduction 1.1 Colloidal glasses? 1.2 Model and real colloids: Interactions 1.2.1 Interactions between model colloids 1.2.2 More common colloidal systems 1.3 Gels or glasses: Various kinds of soft solids? 1.4 Wrapping up the introduction 2 Experimental facts 1: Soft solids that flow and age 2.1 Concentrating colloidal suspensions: From a “viscous liquid” to a “soft-solid” behaviour 2.2 Probing the system in its “soft-solid” phase 2.3 Mechanical aging 3 A class of simple models 3.1 A Maxwell model with one scalar internal variable 3.2 Relation to other models in the literature 3.2.1 Bond models for gels 3.2.2 Activated hopping 3.2.3 A 4-state model 3.2.4 A concluding remark 3.3 Predictions 3.3.1 Mechanical aging 3.3.2 Non-linear “steady-state” rheology 3.3.3 More complex topics: Oscillating rheology and transients 3.4 Intermediary conclusion 4 Experimental facts 2: Soft solids that flow in a strange way 4.1 Avalanches and “viscosity bifurcation” 4.2 Parallel with flow induced transitions: Heterogeneous “banded” flows 4.3 Description within the simple class of models 4.3.1 Adapting the model 4.3.2 Fixed stress $\Sigma$ 4.3.3 Fixed shear rate $\.\Gamma$ 4.3.4 Transients 4.4 Intermediary conclusion 5 Criticism of the model and perspectives 5.1 A classification of the phenomenologies 5.2 Successes and failures of these models 5.3 Better models: More variables? which collective physics? 5.4 Back to facts References 03 1 Soft condensed matter 2 Rheology 2.1 Stress tensor 2.2 Statistical mechanics of stress 2.3 Strain and strain rate 3 Linear rheology 3.1 Step-strain response 3.2 Oscillatory flow 3.3 Steady shear 3.4 Typical cases 3.4.1 “Normal” viscoelastic fluids and solids 3.4.2 Power law fluids 3.4.3 Soft glasses 3.5 Linear creep measurements 3.6 Simple forms for G(t) 4 Linear viscoelasticity of polymers 4.1 Entanglements 4.2 Entropic elasticity 4.3 Escape from the tube 5 Nonlinear rheology of polymers 5.1 Typical experiments 5.1.1 Nonlinear step-strain 5.1.2 Flow curve 5.2 Nonlinear relaxation of polymers 5.3 Constitutive equation 5.4 Why are polymers tractable? 6 Dumbing down 6.1 Dumbell model 6.2 Scalarisation 7 What rheologists want 8 Rheology of soft glasses 8.1 The effective temperature problem 8.2 Phenomenology 8.3 Bouchaud’s trap model 9 The SGR model 9.1 Features of the model 9.2 Constitutive equation for SGR 9.3 Tensorial SGR models 10 Rheological aging 10.1 Step stress and step strain 10.2 Oscillatory flows 10.3 AOFOT? 10.4 Weak long term memory 10.5 The G$^{\prime\prime}$ problem 10.6 Aging scenarios 10.7 Nonlinear aging 10.8 Ongoing work on aging and rheology 11 Shear thickening and jamming 11.1 Nonmonotonic flow curves 11.2 Shear thickening mechanisms 11.3 Jamming SGR model 12 Rheological instability and oscillation 12.1 A simpler model for rheo-instability 13 Rheochaos 14 More fundamental approaches 15 Conclusion References 04 1 Introduction 1.1 Basic phenomenology 1.2 Theoretical issues 1.2.1 Static properties 1.2.2 Tapping and non thermal ensembles 2 The scalar model I: Discrete version 2.1 Definition and motivation 2.2 Stress distribution and the exponential tail 2.3 The “critical” case 3 The scalar model II: Continuous limit and perturbation theory 3.1 Continuous limit of the scalar model 3.2 Calculation of the averaged response and correlation functions 3.3 Further results: The un-averaged response function 3.4 The scalar model with bias: Edwards’ picture of arches 4 Static indeterminacy; elasticity and isostaticity 4.1 Elasticity and response functions 4.2 Indeterminacy at the grain level and isostaticity 4.3 Numerical simulations and Edwards’ assumption 5 A stress-only approach to granular media 5.1 A vectorial q-model 5.2 A constitutive relation between stress components 5.3 Some simple situations 5.4 Symmetries and constitutive relations 5.5 Boundary conditions and “fragility” 6 Experimental and numerical determination of the stress response function 7 Force chains scattering I: Weak disorder limit 7.1 A stochastic wave equation 7.2 Calculation of the averaged response function 7.3 Generalized wave equations 8 Force chains scattering II: Strong disorder limit 8.1 Introduction and numerical results 8.2 A Boltzmann description of force chain splitting 8.3 The role of chain merging 9 Statics of granular materials: Concluding remarks and open questions 10 Glassy dynamics in granular media: A brief survey 10.1 Slow compaction 10.2 Self-inhibitory dynamics and dynamical heterogeneities 10.2.1 Non exponential relaxation 10.2.2 Dynamical heterogeneities 10.2.3 Another point of view: Edwards postulate 10.3 Granular dynamics and the trap model References 05 1 Introduction 2 Supercooled liquids and the glass transition: Important facts and concepts 3 The mode-coupling theory of the glass transition 3.1 The Mori-Zwanzig formalism 3.2 Application of the Mori-Zwanzig formalism to glass-forming systems 4 Computer simulations of glass-forming systems 5 The relaxation dynamics of glass-forming liquids as investigated by computer simulations 5.1 Static and dynamic properties of a simple liquid with Newtonian dynamics 5.2 The relaxation dynamics of a simple liquid with stochastic dynamics 5.3 Static and dynamic properties of a network forming liquid 6 Summary and perspectives References 06 1 Introduction 1.1 General considerations 1.2 Glassiness, metastability and hysteresis 1.3 The organization of these lectures 2 The random energy model 2.1 The definition of the model 2.2 Equilibrium properties of the model 2.3 The properties of the low temperature phase 2.4 A careful analysis of the high temperature phase 2.5 The replica method 2.6 Dynamical properties of the model 3 Models with partially correlated energy 3.1 The definition of the models 3.2 The replica solution 3.3 The physical interpretation 3.4 The two susceptibilities 3.5 The cavity method 4 Complexity 4.1 The basic definitions 4.2 Computing the complexity 4.3 Complexity and replicas 4.4 A summary of the results 4.5 Some consideration on the free energy landscape and on the dynamics 4.6 Small fluctuations 5 Structural relations 5.1 Stochastic stability 5.2 A simple consequence of stochastic stability 5.3 Fluctuation dissipation relations 6 A short introduction to glasses 7 The replica approach to structural glasses: General formalism 7.1 The partition function 7.2 Molecular bound states 7.3 The small cage expansion 7.3.1 Zeroth order term 7.3.2 First order term 7.3.3 Higher orders 7.3.4 Harmonic resummation 7.3.5 Without replicas 8 The replica approach to structural glasses: Some results 8.1 Three approximation schemes 8.2 Critical temperature and effective temperature 8.3 Cage size 8.4 Free energy, specific heat and configurational entropy 8.5 The missing dynamical transition 8.6 Lenhard-Jones binary mixtures 9 Discussion and perspectives Appendices References 07 1 Introduction 2 Some physical systems out of equilibrium 2.1 Domain growth 2.2 Glasses 2.3 Spin-glasses 2.4 Quantum fluctuations 2.5 Rheology and granular matter 2.6 Elastic manifolds in random potentials 2.7 Aging 2.8 Summary 3 Theoretical approach 4 Systems in contact with environments 4.1 Modeling the coupled system 4.1.1 Statics 4.1.2 Dynamics 5 Observables and averages 5.1 Classical systems 5.2 Quantum problems 5.3 Average over disorder 6 Time dependent probability distributions 6.1 The Fokker–Planck and Kramers equations 6.2 Approach to equilibrium 6.3 Equilibrium dynamics 7 The fluctuation – dissipation theorem (FDT) 7.1 Static FDT 7.2 Dynamic FDT 7.3 Quantum FDT 7.4 Examples 7.4.1 Harmonic oscillator and diffusion 7.4.2 A driven system 7.4.3 No Einstein relation 7.4.4 A complex bath 8 Dynamic generating functionals 8.1 Classical models 8.2 Supersymmetry (SUSY) 8.3 Connection with the replica formalism 8.4 Quantum models 8.5 Average over disorder 9 Dynamic equations 9.1 A useful derivation for fully-connected models 9.1.1 Classical systems 9.1.2 Quantum models 9.2 Beyond fully-connected models 9.2.1 Classical models 9.2.2 Quantum models 9.3 Field equations 9.4 The thermodynamic limit and time-scales 9.5 Single spin equation 10 Diagrammatic techniques 10.1 Perturbative solution 10.2 The mode coupling approximation (MCA) 10.3 MCA and disordered models 10.4 MCA for super-cooled liquids and glasses 11 Glassy dynamics: Generic results 11.1 The weak-ergodicity breaking scenario 11.2 The weak long-term memory scenario 11.3 Slow time-reparametrization invariant dynamics 11.4 Correlation scales 11.4.1 Properties 11.4.2 Definition of a characteristic time 11.5 Modifications of FDT 11.5.1 Time domain 11.5.2 Frequency domain 11.5.3 Time-reparametrization invariant formulation 11.5.4 FDT part 11.5.5 Diffusion 12 Solution to mean-field models 12.1 Numerical solution 12.2 Solution at high temperatures 12.3 Solution at low-T 12.3.1 The Lagrange multiplier 12.3.2 The stationary regime 12.3.3 The aging regime 12.3.4 The Edwards-Anderson parameter 12.3.5 Fluctuation – dissipation relation 12.3.6 Discontinuous classical transition 12.3.7 The classical threshold level 12.3.8 Two $p$ models 12.3.9 ${\sc sk}$ model and similar 12.3.10 Mode dependence 12.3.11 Quantum fluctuations 12.3.12 Driven dynamics 13 Modifications of FDT in physical systems 13.1 Domain growth 13.2 Structural glasses 13.3 Spin-glasses 13.4 Rheology 13.5 Vibrated models and granular matter 13.6 Driven vortex systems 13.7 Quantum fluctuations 13.8 Systems of finite size: Preasymptotic behavior 13.9 Critical dynamics 13.10 Connection with equilibrium 14 Effective temperatures 14.1 Thermodynamical tests 14.1.1 How to measure a temperature 14.1.2 Zeroth law 14.1.3 Auxiliary thermal baths 14.2 Temperature fixing by SUSY breaking 14.3 Fictive temperatures 14.4 Nonequilibrium thermodynamics 14.5 Statistical mechanics 15 Metastable states 15.1 Equilibrium 15.2 Static TAP approach 15.3 The TAP equations 15.4 Stability of, and barriers between, the TAP solutions 15.5 Index dependent complexity 15.6 Weighted sums over TAP solutions 15.7 Accessing metastable states with replicas 15.8 Dynamics and quantum systems 16 Conclusions 17 Perspectives Appendix A Generalized Langevin equations B The Kubo formula C The response in a Langevin process D Grassmann variables and supersymmetry E Integrals in the aging regime References 08 1 Introduction 2 Equilibrium states 2.1 Infinite system ground states 2.2 Positive temperature states 2.3 Number of equilibrium states? 3 Dynamics and barriers 3.1 Domain coarsening 3.2 Dynamics and local equilibrium 4 Spin glasses 4.1 Scaling scenario 4.2 Observation of many states? 5 Non-random systems: True glasses References 09 1 Introduction 2 Aging, memory and rejuvenation 2.1 Aging range: A simple quench experiment 2.2 Memory and rejuvenation 2.2.1 Historical background 2.2.2 Memory experiment on PMMA 2.2.3 Advanced memory experiments 2.2.4 Deleting memory 2.2.5 Double memory 2.2.6 Direct comparison with the memory effects in spin-glasses 2.3 Discussion on the memory effect 3 Effective temperature of an aging material 3.1 The X-ray scattering experiment 3.2 Supercooled liquid experiment 3.3 Gel electric properties 3.3.1 The experimental apparatus 3.3.2 FDR measurements 3.3.3 Statistical analysis of the noise 3.4 Polycarbonate dielectric properties 3.4.1 The experimental apparatus 3.4.2 FDR measurements 3.4.3 Statistical analysis of the noise in Polycarbonate 3.5 Rheological measurements 3.5.1 Experimental apparatus 3.5.2 FDR on the rheology of Laponite 3.6 Discussion and conclusions on the effective temperature 4 General conclusions References 10 1 Introduction 2 Theoretical background 3 Experimental 3.1 Measurement of magnetic fluctuations 3.2 Principle of measurement: An absolute thermometer 3.3 Experimental results 4 Discussion 5 Conclusion References 12 1 Introduction 2 Overview of protein architectures and discussion of physical background of their natural selection 2.1 Protein structures 2.2 Physical selection of protein structures 3 Thermodynamic aspects of protein folding 3.1 Reversible denaturation of protein structures 3.2 What do denatured proteins look like? 3.3 Why denaturation of a globular protein is the first-order phase transition 3.4 “Gap” in energy spectrum: The main characteristic that distinguishes protein chains from random polymers 4 Kinetic aspects of protein folding 4.1 Protein folding in vivo 4.2 Protein folding in vitro (in the test-tube) 4.2.1 The Levinthal paradox 4.2.2 Folding pathways and folding intermediates 4.2.3 “Two-state” protein folding 4.2.4 Folding nucleus 4.3 Theory of protein folding rates and solution of the Levinthal paradox References 13 1 Introduction 2 1D multijunction SQUID with random location of the junctions 3 Single-junction SQUID 4 1D sandpile model 5 The simplified model of 1D multijunction SQUID 6 Computer simulation results 7 Conclusions References 14 1 Introduction 2 Experimental facts 2.1 Rejuvenation 2.2 Overaging and underaging 2.3 Memory effect of the first kind, or “Kovacs effect” 2.4 Memory effect of the second kind 2.5 Need for a generic and robust phenomenology 3 Two mean-field theoretical approaches 3.1 Trap and multi-trap models 3.2 Infinite-range models 4 Spatial approaches 4.1 Domain growth 4.2 A minimal phenomenology 4.3 Back to experiments 4.4 Droplets and chaos in spin glasses 4.5 Surfing on a critical line 5 Two experiments 5.1 Anderson insulator 5.2 Colloidal suspension 6 Conclusion References 15 1 Introduction and overview 2 The Fredrickson-Andersen model 3 The Kob-Andersen model 4 The $ABC$ model 5 Conclusions and outlook References 16 1 Introduction 2 Universal behavior of the force distribution? 3 Colloidal glasses and gels 4 Jamming transitions 5 Conclusion References back-matter